Calculus of variations is utilized to minimize the elastic energy arising from the curvature squared while maximizing the van der Waals energy. Firstly, the shape of folded graphene sheets is investigated, and an arbitrary constant arising by integrating the Euler–Lagrange equation is determined. In this study, the structure is assumed to have a translational symmetry along the fold, so that the problem may be reduced to a two dimensional problem with reflective symmetry across the fold. Secondly, both variational calculus technique and least squared minimization procedure are employed to determine the joining structure involved a C60 fullerene and a carbon nanotube, namely a nanobud. We find that these two methods are in reasonable overall agreement. However, there is no experimental or simulation data to determine which procedure gives the more realistic results.

We discuss various optimisation-based approaches to machine learning. Tasks include regression, clustering, and classification. We discuss frequently used terms like 'unsupervised learning,' 'penalty methods,' and 'dual problem.' We motivate our discussion with simple examples and visualisations.

We investigate the construction of multidimensional prolate spheroidal wave functions using techniques from Clifford analysis. The prolates are eigenfunctions of a time-frequency limiting operator, but we show that they are also eigenfunctions of a differential operator. In an effort to compute solutions of this operator, we prove a Bonnet formula for a class of Clifford-Gegenbauer polynomials.

The models of collective decision-making considered in this presentation are nonlinear interconnected systems with saturating interactions, similar to Hopfield newtorks. These systems encode the possible outcomes of a decision process into different steady states of the dynamics. When the model is cooperative, i.e., when the underlying adjacency graph is Metzler, then the system is characterized by the presence of two main attractors, one positive and the other negative, representing two choices of agreement among the agents, associated to the Perron-Frobenius eigenvector of the system. Such equilibria are achieved when there is a sufficiently high 'social commitment' among the agent (here interpreted as a bifurcation parameter). When instead cooperation and antagonism coexist, the resulting signed graph is in general not structurally balanced, meaning that Perron-Frobenius theorem does not apply directly. It is shown that the decision-making process is affected by the distance to structural balance, in the sense that the higher the frustration of the graph, the higher the commitment strength at which the system bifurcates. In both cases, it is possible to give conditions on the commitment strength beyond which other equilibria start to appear. These extra bifurcations are related to the algebraic connectivity of the graph.

Sea ice acts as a refrigerator for the world. Its bright surface reflects solar heat, and the salt it expels during the freezing process drives cold water towards the equator. As a result, sea ice plays a crucial role in our climate system. Antarctic sea-ice extent has shown a large degree of regional variability, in stark contrast with the steady decreasing trend found in the Arctic. This variability is within the ranges of natural fluctuations, and may be ascribed to the high incidence of weather extremes, like intense cyclones, that give rise to large waves, significant wind drag, and ice deformation. The role exerted by waves on sea ice is still particular enigmatic and it has attracted a lot of attention over the past years. Starting from theoretical knowledge, new understanding based on experimental models and computational fluid dynamics is presented. But exploration of waves-in-ice cannot be exhausted without being on the field. And this is why I found myself in the middle of the Southern Ocean during a category five polar cyclone to measure waves…

Motivating in constructing conformal field theories Jones recently discovered a very general process that produces actions of the Thompson groups $F$,$T$ and $V$ such as unitary representations or actions on $C^{\ast}$-algebras. I will give a general panorama of this construction along with many examples and present various applications regarding analytical properties of groups and, if time permits, in lattice theory (e.g. quantum field theory).

Let $t$ be the the multiplicative inverse of the golden mean. In 1995 Sean Cleary introduced the irrational-slope Thompson's group $F_t$, which is the group of piecewise-linear maps of the interval $[0,1]$ with breaks in $Z[t]$ and slopes powers of $t$. In this talk we describe this group using tree-pair diagrams, and then demonstrate a finite presentation, a normal form, and prove that its commutator subgroup is simple. This group is the first example of a group of piecewise-linear maps of the interval whose abelianisation has torsion, and it is an open problem whether this group is a subgroup of Thompson's group $F$.

A Jonsson-Tarski algebra is a set X endowed with an isomorphism $X\to XxX$. As observed by Freyd, the category of Jonsson-Tarski algebras is a Grothendieck topos - a highly structured mathematical object which is at once a generalised topological space, and a generalised universe of sets.
In particular, one can do algebra, topology and functional analysis inside the Jonsson-Tarski topos, and on doing so, the following objects simply pop out: Cantor space; Thompson's group V; the Leavitt algebra L2; the Cuntz semigroup S2; and the reduced $C^{\ast}-algebra of S2. The first objective of this talk is to explain how this happens.
The second objective is to describe other "self-similar toposes" associated to, for example, self-similar group actions, directed graphs and higher-rank graphs; and again, each such topos contains within it a familiar menagerie of algebraic-analytic objects. If time permits, I will also explain a further intriguing example which gives rise to Thompson's group F and, I suspect, the Farey AF algebra.
No expertise in topos theory is required; such background as is necessary will be developed in the talk.

It is commonly expected that $e$, $\log 2$, $\sqrt{2}$, among other « classical » numbers, behave, in many respects, like almost all real numbers. For instance, their decimal expansion should contain every finite block of digits from $\{0, \ldots , 9\}$. We are very far away from establishing such a strong assertion. However, there has been some small recent progress in that direction. Let $\xi$ be an irrational real number. Its irrationality exponent, denoted by $\mu (\xi)$, is the supremum of the real numbers $\mu$ for which there are infinitely many integer pairs $(p, q)$ such that $|\xi - \frac{p}{q}| < q^{-\mu}$. It measures the quality of approximation to $\xi$ by rationals. We always have $\mu (\xi) \ge 2$, with equality for almost all real numbers and for irrational algebraic numbers (by Roth's theorem). We prove that, if the irrationality exponent of $\xi$ is equal to $2$ or slightly greater than $2$, then the decimal expansion of $\xi$ cannot be `too simple', in a suitable sense. Our result applies, among other classical numbers, to badly approximable numbers, non-zero rational powers of ${{\rm e}}$, and $\log (1 + \frac{1}{a})$, provided that the integer $a$ is sufficiently large. It establishes an unexpected connection between the irrationality exponent of a real number and its decimal expansion.

I introduce and demonstrate the Coq assisted theorem prover.

The old joke is that a topologist can’t distinguish between a coffee cup and a doughnut. A recent variant of Homology, called Persistent Homology, can be used in data analysis to understand the shape of data. I will give an introduction to persistent Homology and describe two example applications of this tool.

Imagine a world, where physical and chemical laboratories are unnecessary, because all experiments can be simulated accurately on a computer. In principle this is possible, solving the quantum mechanical Schrödinger equation. Unfortunately, this is far from trivial and practically impossible for large and complex materials and reactions. In 1998, Walter Kohn and John A Pople won the Nobel Prize in Chemistry for developing the density-functional theory (DFT). DFT allows to find solutions for the Schrödinger equation much more efficiently than ab-initio and similar approaches, thus enabling the computation of materials properties in an unprecedented way. In this seminar, I will introduce quantum mechanical principles and the basic idea of the DFT. Then, I will present an example of the computational elucidation of a reaction mechanism in materials science.

This project aims to investigate algebraic objects known as 0-dimensional groups, which are a mathematical tool for analysing the symmetry of infinite networks. Group theory has been used to classify possible types of symmetry in various contexts for nearly two centuries now, and 0-dimensional groups are the current frontier of knowledge. The expected outcome of the project is that the understanding of the abstract groups will be substantially advanced, and that this understanding will shed light on structures possessing 0-dimensional symmetry. In addition to being cultural achievements in their own right, advances in group theory such as this also often have significant translational benefits. This will provide benefits such as the creation of tools relevant to information science and researchers trained in the use of these tools.

The project aims to develop novel techniques to investigate Geometric analysis on infinite dimensional bundles, as well as Geometric analysis of pathological spaces with Cantor set as fibre, that arise in models for the fractional quantum Hall effect and topological matter, areas recognised with the 1998 and 2016 Nobel Prizes. Building on the applicant's expertise in the area, the project will involve postgraduate and postdoctoral training in order to enhance Australia's position at the forefront of international research in Geometric Analysis. Ultimately, the project will enhance Australia's leading position in the area of Index Theory by developing novel techniques to solve challenging conjectures, and mentoring HDR students and ECRs.

This project aims to solve hard, outstanding problems which have impeded our ability to progress in the area of quantum or noncommutative calculus. Calculus has provided an invaluable tool to science, enabling scientific and technological revolutions throughout the past two centuries. The project will initiate a program of collaboration among top mathematical researchers from around the world and bring together two separate mathematical areas into a powerful new set of tools. The outcomes from the project will impact research at the forefront of mathematical physics and other sciences and enhance Australia's reputation and standing.

Mahler's method in number theory is an area wherein one answers questions surrounding the transcendence and algebraic independence of both power series $F(z)$, which satisfy the functional equation $$a_0(z)F(z)+a_1(z)F(z^k)+\cdots+a_d(z)F(z^{k^d})=0$$ for some integers $k\geqslant 2$ and $d\geqslant 1$ and polynomials $a_0(z),\ldots,a_d(z)$, and their special values $F(\alpha)$, typically at algebraic numbers $\alpha$. The most important examples of Mahler functions arise from important sequences in theoretical computer science and dynamical systems, and many are related to digital properties of sets of numbers. For example, the generating function $T(z)$ of the Thue-Morse sequence, which is known to be the fixed point of a uniform morphism in computer science or equivalently a constant-length substitution system in dynamics, is a Mahler function. In 1930, Mahler proved that the numbers $T(\alpha)$ are transcendental for all non-zero algebraic numbers $\alpha$ in the complex open unit disc. With digital computers and computation so prevalent in our society, such results seem almost second nature these days and thinking about them is very natural. But what is one really trying to communicate by proving that functions or numbers such as those considered in Mahler's method?

In this talk, highlighting work from the very beginning of Mahler's career, we speculate---and provide some variations---on what Mahler was really trying to understand. This talk will combine modern and historical methods and will be accessible to students.

In this talk, we will present a brief overview of mathematical diffraction of structures with no translational symmetry but are not ruled out to exhibit long-range order. We introduce aperiodic tilings as toy models for such structures and discuss the relevant measure-theoretic formulation of the diffraction analysis. In particular, we focus on the component of the diffraction that suggests stochasticity but can be non-trivial for deterministic systems, and how its absence can be confirmed using some techniques involving Lyapunov exponents and Mahler measures. This is joint work with Michael Baake, Michael Coons, Franz Gaehler and Uwe Grimm.

The problem of packing space with regular tetrahedra has a 2000 year history. This talk surveys the history of work on the problem. It includes work by mathematicians, computer scientists, physicists, chemists, and materials scientists. Much progress has been made on it in recent years, yet there remain many unsolved problems.

In this talk, I will show how to build $C^*$-algebras using a family of local homeomorphisms. Then we will compute the KMS states of the resulted algebras using Laca-Neshveyev machinery. Then I will apply this result to $C^*$-algebras of $K$-graphs and obtain interesting $C^*$-algebraic information about $k$-graph algebras. This talk is based on a joint work with Astrid an Huef and Iain Raeburn.

The KMS condition for equilibrium states of C*-dynamical systems has been around since the 1960’s. With the introduction of systems arising from number theory and from semigroup dynamics following pioneering work of Bost and Connes, their study has accelerated significantly in the last 25 years. I will give a brief introduction to C*-dynamical systems and their KMS states and discuss two constructions that exhibit fascinating connections with key open questions in mathematics such as Hilbert’s 12th problem on explicit class field theory and Furstenberg’s x2 x3 conjecture.

Using a variant of the Laca-Raeburn program for calculating KMS states, Laca, Raeburn, Ramagge and Whittaker showed that, at any inverse temperature above a critical value, the KMS states arising from self-similar actions of groups (or groupoids) $G$ are parameterised by traces on C*(G). The parameterisation takes the form of a self-mapping \chi of the trace space of C*(G) that is built from the structure of the stabilisers of the self-similar action. I will outline how this works, and then sketch how to see that \chi has a unique fixed-point, which picks out the ``preferred" trace of C*(G) corresponding to the only KMS state that persists at the critical inverse temperature. The first part of this will be an exposition of results of Laca-Raeburn-Ramagge-Whittaker. The second part is joint work with Joan Claramunt.

Zombies are a popular figure in pop culture/entertainment and they are usually portrayed as being brought about through an outbreak or epidemic. Consequently, we model a zombie attack, using biological assumptions based on popular zombie movies. We introduce a basic model for zombie infection, determine equilibria and their stability, and illustrate the outcome with numerical solutions. We then refine the model to introduce a latent period of zombification, whereby humans are infected, but not infectious, before becoming undead. We then modify the model to include the effects of possible quarantine or a cure. Finally, we examine the impact of regular, impulsive reductions in the number of zombies and derive conditions under which eradication can occur. We show that only quick, aggressive attacks can stave off the doomsday scenario: the collapse of society as zombies overtake us all.

During my study leave in 2018 I have applied nonlinear stability analysis techniques to the Douglas-Rachford Algorithm, with the aim of shedding light on the interesting non-convex case, where convergence is often observed but seldom proven. The Douglas-Rachford Algorithm can solve optimisation and feasibility problems, provably converges weakly to solutions in the convex case, and constitutes a practical heuristic in non-convex cases. Lyapunov functions are stability certificates for difference inclusions in nonlinear stability analysis. Some other recent nonlinear stability results are showcased as well.

Almost by definition, the main tool and goal of Geometric Group Theory is to find and exploit correspondences between geometric and algebraic features of groups. Following this philosophy, I will focus on the question: what does it mean for a sub(space/group) to "sit nicely" inside a bigger (space/group)? Focusing on groups, for a subgroup H of a group G, possible answers for the above question are when the subgroup H is: quasi-isometrically embedded, undistorted, normal/malnormal, finitely generated, geometrically separated...
Many of the above are equivalent when H is a quasiconvex subgroup of a hyperbolic group G, providing very successful correspondences between geometric and algebraic properties of subgroups.
The goal of this talk is to review quasiconvexity in hyperbolic spaces and try to generalize several of those features in a broader setting, namely the class of hierarchically hyperbolic groups (HHG). This is a joint work with Hung C. Tran and Jacob Russell.

The QUT porous media modelling group has developed a number of fruitful collaborations with industry partners over the last 20 years where large-scale computations were utilised to investigate and optimise operations. In this lecture I will reflect on the rich experience of working with industry — from commercial research to work integrated learning for final year students. A pleasing outcome is the impact our research has had on industry practices. A selection of our past modelling projects will be reviewed, including:

• Forecasting the value of a southern pine plantation.
• Drying of lignocellulosic materials.
• Groundwater modelling.

I will also provide a brief survey of the computational solution strategies employed for the models.

Brief Biography: Ian Turner is a professor of computational mathematics in the School of Mathematical Sciences at the Queensland University of Technology. His main research interests are in the fields of computational mathematics and numerical analysis, where he has over thirty years experience in solving systems of coupled, nonlinear partial differential equations that govern flow in porous media. He has published over 250 research articles in a wide cross section of journals spanning science and engineering, and his multidisciplinary research demonstrates a strong interaction with industry. He is also a former Head of School of Mathematical Sciences at QUT. Recently, Professor Turner was named in the 2015, 2016 Thomson Reuters and 2017 Clarivate Analytics Web of Science list of Highly Cited Researchers.

Hausdorff dimension has become a standard tool to measure the "size" of fractals in real space. However, it can be defined on any metric space and therefore can be used to measure the "size" of subgroups of, say, pro-p groups (with respect to a chosen metric). This line of investigation was started 20 years ago by Barnea and Shalev, who showed that p-adic analytic groups do not have any "fractal" subgroups, and asked whether this characterises them among finitely generated pro-p groups. I will explain what all of this means and report on joint work with Oihana Garaialde and Benjamin Klopsch in which, while trying to solve this problem, we ended up showing an analogue of a theorem of Schreier in the context of pro-p groups of positive rank gradient: any finitely generated infinite normal subgroup of a pro-p group of positive rank gradient is of finite index. I will also explain what "positive rank gradient" means, and why pro-p groups with such a property are "free-like".

In the first part of this seminar, I will present some geometric cocycles associated to trees and ways to compute their norms. Similar construction exists for Euclidean buildings but no satisfactory estimates of the norm is currently known. In the second part, I will discuss some ongoing research with Thibaut Pillon on actions the infinite cyclic group by piecewise translations on locally compact group. Piecewise translation actions have been well studied for finitely generated groups, e.g. by Whyte, and provide positive answers to the von-Neumann-Day problem or the Burnside problem. The generalization to LC-groups was introduced by Schneider. The topic seems to have interesting implications for tdlc-groups

The divergence of a pair of geodesics in a metric space measures how fast they spread apart. For example, in Euclidean space all pairs of geodesics diverge linearly, while in hyperbolic space all pairs of geodesics diverge exponentially. In the 1980s Gromov proved that in symmetric spaces of non-compact type, the only possible divergence rates are linear or exponential, and he asked whether the same dichotomy holds in CAT(0) spaces. Soon afterwards, Gersten used these ideas to define a quasi-isometry invariant, also called divergence, which measures the "worst" rate of divergence. Gersten and others have since found many examples of finitely generated groups with quadratic divergence. We study divergence in right-angled Coxeter groups with triangle-free defining graphs. Using the structure of certain flats in the associated Davis complex, which is a CAT(0) square complex, we characterise such groups with linear and quadratic divergence, and construct examples of right-angled Coxeter groups with divergence polynomial of arbitrary degree. This is joint work with Pallavi Dani (Louisiana State University).

Given a group one of the most natural things one can study about it is its subgroup lattice, and the maximal subgroups take a prominent role. If the group is infinite, one can ask whether all maximal subgroups have finite index or whether there are some (and how many) of infinite index. After telling some historical developments on this question, I will motivate the study of maximal subgroups of groups of intermediate growth and report on joint work with Dominik Francoeur where we give a complete description of all maximal subgroups of some "siblings" of Grigorchuk's group.

Let X be an algebraic curve over an algebraically closed field of characteristic two. We will prove that for any such curve X, there exists a tamely ramified morphism from X to the projective line. The assertion is closely related to Belyi's theorem. In this talk, we first recall Belyi's theorem and its positive characteristic analogue. Next we introduce a key notion called ``pseudo-tame", which plays an important role in our proof and we prove that there exists a pseudo-tamely ramified morphism from X to the projective line by showing that an obstruction class vanishes. If time permits, we give a way to construct a tamely ramified morphism from X to the projective line by using a pseudo-tamely ramified morphism. This is a joint work with Seidai Yasuda of Osaka university.

We consider a class of linear operator equations that includes systems of ordinary differential equations, difference equations and fractional-order ordinary differential equations. This class also includes Fredholm integral equations, operator exponentials and powers, as well as eigenvalue problems. We generalise the idea of a fundamental matrix and provide an explicit method for obtaining an exact series solution to these types of operator equations, together with sufficient conditions for convergence and error bounds. Illustrative examples are also given.

I will describe the relationship between self-similar groups, permutational bimodules and virtual group endomorphisms. Based on chapter 2 of Nekrashevych’s book.

In this talk, I will discuss a simple modification of the forward-backward splitting method for finding a zero in the sum of two monotone operators. This modified method converges under the same assumptions as Tseng's forward-backward-forward method, namely, it does not require cocoercivity of the single-valued operator but rather only Lipschitz continuity. Each of iteration of the method only requires one forward evaluation rather than two as is the case in Tseng's method. Variants incorporating a linesearch, an inertial term, or a structured three operator inclusion will also be discussed. Based on joint work with Yura Malitsky (University of Göttingen).

In this talk, we will introduce a class of tree automorphism groups known as bounded automata. From this definition, we will see that many of the interesting examples of self-similar groups in the literature are members of this class.
A problem in group theory is classifying groups based on the difficulty of solving their co-word problems, that is, classifying them by the computational difficulty to decide if a word is not equivalent to the identity. Some well-known results in this study are that a group has a co-word problem given by a regular language if and only if it is finite, a deterministic context-free language if and only if it is virtually free, and a deterministic one-counter machine if and only if it is virtually cyclic. Each of these language classes corresponds to a natural and well-studied model of computation.
We will show that the class of bounded automata groups has a co-word problem given by an ET0L language – a class of formal language which has recently gained popularity in areas of group theory. This strengthens a recent result of Holt and Röver (who showed this result for a less restrictive class of language) and extends a result of Ciobanu-Elder-Ferov (who proved this result for the first Grigorchuk group).

Every compactly generated t.d.l.c. group acts vertex transitively on a locally finite graph with compact open vertex stabilisers. Such a graph is called a rough Cayley graph and, up to quasi-isometry, is an invariant for the group. This allows us to define Gromov hyperbolic t.d.l.c. groups and their Gromov boundary in a way analogous to the finitely generated case. The space of directions of a t.d.l.c. group is a metric space 'at infinity' obtained by analysing the action of the group on the set of compact open subgroups. It is particularly useful for detecting flat subgroups, think subgroups that look like $\mathbb{Z}^n$.
In my talk, I will introduce these two concepts of boundary and give some new results which relate them. Time permitting, I may also give details about the proofs.

The existence of so-called dark energy and matter in the universe implies that the conventional accounting of mass and energy is incorrect. Here, we use the framework of special relativity and validation through Lorentz invariance to develop an alternative accounting of mass and energy. We assume the usual Einstein relations of special relativity, but we make the distinction between the particle energy e = mc^2 and the actual work done by the particle E*, and we adopt the perspective that it is not just the momentum vector p = mu that contributes to the work done E*, but rather the intrinsic particle energy e itself plays an important role through the combined potentials (p, e/c) as a well-defined four vector within special relativity. The resulting formulation provides a natural extension of Newton's second law, emerges as a fully consistent development of special relativity that is properly invariant under the Lorentz group, and yields an extension of Einstein's famous equation for the work done involving new terms. The new work done expressions can involve the log function, and possibly generate extremely large energies that might well represent the first formal indication of the origin of dark energy. Two alternative expressions are both well defined as a limiting case for energy-mass waves travelling at the speed of light, and are in complete accord with well-established theory for photons and light for which energy is known to vary linearly with momentum. The present formulation suggests that large energies might be generated even for slowly moving systems, and that dark energy might arise in consequence of conventional mechanical theory neglecting the work done in the direction of time.

Given a profinite group $G$, we can consider the semigroup $\mathrm{End}(G)$ of continuous homomorphisms from $G$ to itself. In general $\lambda \in\mathrm{End}(G)$ can be injective but not surjective, or vice versa: consider for instance the case when $G$ is the group $F_p[[t]$ of formal power series over a finite field, $n$ is an integer, and $\lambda_n$ is the continuous endomorphism that sends $t^k$ to $t^{k+n}$ if $k+n \ge 0$ and $0$ otherwise. However, when $G$ has only finitely many open subgroups of each index (for instance, if $G$ is finitely generated), the structure of endomorphisms is much more restricted: given $\lambda \in\mathrm{End}(G)$, then $G$ can be written as a semidirect product $N \rtimes H$ of closed subgroups, where $\lambda$ acts as an automorphism on $H$ and a contracting endomorphism on $N$. When $\lambda$ is open and injective, the structure of $N$ can be restricted further using results of Glöckner and Willis (including the very recent progress that George told us about a few weeks ago). This puts some restrictions on the profinite groups that can appear as a '$V_+$' group for an automorphism of a t.d.l.c. group.

We introduce the notion of self-similarity for groups acting on regular rooted trees as well as their description using automata and wreath iteration. Following the definition of Grigorchuk's group we show that it is an infinite, finitely generated 2-group. The proof illustrates the use of self-similarity.

Lax-Philips scattering theory is a method to solve for scattering as an expansion over the singularities of the analytic extension of the scattering problem to complex frequencies. I will show how a complete theory can be developed in the case of simple scattering problems. I will illustrate how this theory can be used to find a numerical solution and I will illustrate the method by applying it to the vibration of ice shelves.

(joint work with R. Grigorchuk ad D. Horadam) The tree representation theorem represents a certain group associated with the scale of an automorphism of a t.d.l.c. group as acting by symmetries of a regular (unrooted) tree. It shows that groups acting on regular trees are a fundamental part of the theory of t.d.l.c. groups.
There is also an extensive theory of self-similar and self-replicating groups of symmetries of rooted trees which has developed from the discovery (or creation) of examples such as the Grigorchuk groups.
It will be seen in this talk that these two branches of research are studying essentially the same groups.

See here for an abstract.

In this presentation, I describe and reflect upon teaching the mathematics sequence within the Bachelor of Science (Extended), the BSc (Ext), program at the University of Melbourne. In the introduction to the presentation, I give a brief overview of the BSc (Ext) which was established in 2015 at the University of Melbourne as a pathway program to enable Indigenous students to successfully transition into their studies in science and related areas. In the main part of the presentation, I present the key aspects of the high-expectations teaching approach that I use. This consists, firstly, of constructing a classroom context that affirms the mathematical capacities of Indigenous students, based on insights from Australian First Nations educator, Dr Chris Sarra and Professor Russell Bishop, a First Nations scholar from New Zealand. I present feedback from students and other materials to illustrate the importance of setting this learning framework. In the final section, I present student attempts and feedback from a 'Learner Generated Example' question within a specific mathematics assignment. This illustrates the advanced learning that is possible once a high-expectations context has been established.

(joint work with Federico Berlai) A natural way to study infinite groups is via looking at their finite quotients. A subset S of a group G is then said to be (finitely) separable in G if we can recognise it in some finite quotient of G, meaning that for every g outside of S there is a finite quotient of G such that the image of g under the canonical projection does not belong to the image of S. We can then describe classes of groups by specifying which types of subsets do we require to be separable: residually finite groups have separable singletons, conjugacy separable groups have separable conjugacy classes of elements, cyclic subgroup separable groups have separable cyclic subgroups and so on... We could also restrict our attention only to some class of quotients, such as finite p-groups, solvable, alternating... Properties of this type are called separability properties. In case when the class of admissible quotients has reasonable closure properties we can use topological methods.
We prove that the property of being cyclic subgroup separable, that is having all cyclic subgroups closed in the profinite topology, is preserved under forming graph products.
Furthermore, we develop the tools to study the analogous question in the pro-p case. For a wide class of groups we show that the relevant cyclic subgroups - which are called p-isolated - are closed in the pro-p topology of the graph product. In particular, we show that every p-isolated cyclic subgroup of a right-angled Artin group is closed in the pro-p topology and, consequently, we show that maximal cyclic subgroups of a right-angled Artin group are p-separable for every p.

Let $M(n)$ be the number of distinct entries in the multiplication table for integers smaller than $n$. More precisely, $M(n) := |\{ij \mid\ 0<= i,j <n\}|$. The order of magnitude of $M(n)$ was established in a series of papers by various authors, starting with Erdös (1950) and ending with Ford (2008), but an asymptotic formula for $M(n)$ is still unknown. After describing some of the history of $M(n)$ I will consider two algorithms for computing $M(n)$ exactly for moderate values of $n$, and several Monte Carlo algorithms for estimating $M(n)$ accurately for large $n$. This leads to consideration of algorithms, due to Bach (1985-88) and Kalai (2003), for generating random factored integers - integers $r$ that are uniformly distributed in a given interval, together with the complete prime factorisation of $r$. The talk will describe ongoing work with Carl Pomerance (Dartmouth, New Hampshire) and Jonathan Webster (Butler, Indiana).

Bio: Richard Brent is a graduate of Monash and Stanford Universities. His research interests include analysis of algorithms, computational complexity, parallel algorithms, structured linear systems, and computational number theory. He has worked at IBM Research (Yorktown Heights), Stanford, Harvard, Oxford, ANU and the University of Newcastle (NSW). In 1978 he was appointed Foundation Professor of Computer Science at ANU, and in 1983 he joined the Centre for Mathematical Analysis (also at ANU). In 1998 he moved to Oxford, returning to ANU in 2005 as an ARC Federation Fellow. He was awarded the Australian Mathematical Society Medal (1984), the Hannan Medal of the Australian Academy of Science (2005), and the Moyal Medal (2014). Brent is a Fellow of the Australian Academy of Science, the Australian Mathematical Society, the IEEE, ACM, IMA, SIAM, etc. He has supervised twenty PhD students and is the author of two books and about 270 papers. In 2011 he retired from ANU and moved to Newcastle to join CARMA, at the invitation of the late Jon Borwein.

The use of various methods to obtain close to optimal quantization leads to interesting questions about the behavior of random processes, Diophantine approximation, ergodic maps, shrinking targets, and other related constructions. The goal in all of these approaches to quantization is the speed of decrease of the error, coupled with the simplicity and concreteness of the process employed.

I will discuss the various completed, ongoing, and planned mathematics visualisation projects within CARMA's SeeLab visualisation laboratory.

Bio: Michael Assis was awarded a PhD in Statistical Mechanics at Stony Brook University in 2014, and then took a postdoctoral fellowship at the University of Melbourne. In 2017 he held a computational mathematics postdoctoral position within CARMA, and earlier this year he worked to develop CARMA's Seelab mathematics visualisation laboratory together with David Allingham.

There is an intriguing analogy between number fields and function fields. If we view classical Number Theory as the study of the ring of integers and its extensions, then function field arithmetic is the study of the ring of polynomials over a finite field and its extensions. According to this analogy, most constructions and phenomena in classical Number Theory, ranging from the elementary theorems of Euler, Fermat and Wilson, to the Riemann Hypothesis, Elliptic curves, class field theory and modular forms all have their function field analogues. I will give a panoramic tour of some of these constructions and highlight their similarities and differences to their classical counterparts.

This lecture should be accessible to advanced undergraduate students.

The Discrete Element Method (DEM) is a very powerful numerical method for the simulation of unbonded and bonded granular materials, such as soil and rock. One of the unique features of this approach is that it explicitly considers the individual grains or particles and all their interactions. The DEM is an extension of the Molecular Dynamics (MD) approach. The motion of the particles is governed by Newton's second law and the rigid body dynamic equations are generally solved by applying an explicit time-stepping algorithm. Spherical particles are usually used, as this results in most efficient contact detection. Nevertheless, with the increase of computing power non-spherical particles are becoming more popular. In addition, great effort is made for coupling the method with other continuum methods to model multiphase materials. The talk discusses recent developments of the DEM in Geomechanics based on the open-source framework YADE and some of its ongoing challenges.

Bio: Klaus has more than 10 years' experience in the development of cutting-edge numerical tools for geotechnical engineering and rock mechanics applications. He obtained his PhD in civil engineering from Graz University of Technology (Austria). After moving to Australia, he expanded his initial research experience on continuum-based numerical modelling with the Boundary Element Method (BEM) and Finite Element Method (FEM) by taking on the Discrete Element Method (DEM), a discontinuum-based method. He is an active developer of the open-source DEM framework YADE (https://yade-dem.org), an efficient numerical tool for the dynamic simulation of geomaterials. Lately he has been concentrating on the development of a highly innovative framework for the modelling of deformable discrete elements.

Experimental discovery has long played an important role in research mathematics, even before the advent of modern computational tools. Many methods of antiquity are familiar to all of us, including the drawing of pictures to gain geometric insights and exhaustively solving similar problems in order to identify patterns. I will share a variety modern computational tools and techniques which I have used for my research at CARMA. The contexts of the discoveries will be varied -- including number theory, non-Euclidean geometry, complex analysis, and optimization -- and so the emphasis will be on the strategies employed rather than specific outcomes.

Bio: Scott Lindstrom received his master's degree from Portland State University. In September, 2015, he came to CARMA at University of Newcastle as a PhD student of Jonathan Borwein. Following Professor Borwein's untimely passing, he has continued as a student of Brailey Sims, Heinz Bauschke, and Bishnu Lamichhane. In October he will begin a postdoctoral fellowship at Hong Kong Polytechnic University. His principal research area is experimental mathematics with particular emphasis in optimization and nonlinear convex analysis. He is a member of the AustMS special interest group Mathematics of Computation and Optimization (MoCaO) and organizes the Borwein Meetings for RHD students and postdocs at CARMA.

An enduring topic of research interest relates to the heritability of mental traits, such as intelligence. Some of the work on this topic has focussed on genetic contributions to the speed of cognitive processing, by examination of response times in psychometric tests. An important limitation of previous work is the underlying assumption that variability in response times solely reflects variability in the speed of cognitive processing. This assumption has been problematic in other domains, due to the confounding effects of caution and motor execution speed on observed response times. We extend a cognitive model of decision-making to account for the relatedness structure in a twin study paradigm. This approach has the potential to separately quantify different contributions to the heritability of response time: contributions from cognitive processing speed, caution, and motor execution speed. In some ways, this is a typical usage of an evidence accumulation model, and it throws up all the typical problems that we struggle with in data visualisation. Those problems will become evident during the talk, as we discuss data from the Human Connectome Project. We find that caution is both highly heritable and highly influenced by the environment, while cognitive processing speed is moderately heritable with little environmental influence, and motor execution speed appears to have no strong influence from either. Our study suggests that the assumption made in previous studies of the heritability being within mental processing speed is incorrect, with response caution actually being the most heritable part of the decision process.

Complex virtual environments are used for entertainment in the form of games and are also fundamental in training and simulation environments. Apart from the visual representation of reality, these environments, and the interactions occurring between users within them, are a source of a wide variety of data. These data cover interactions such as spatio-temporal positional tracking within 3D virtual environments, to the measurement of physiological responses of users to in game events. Of particular interest are measures of visual complexity, and how these measures might be useful in determining minimum realism for affective virtual environments. This talk will consider these different data types and sources and highlight some active research areas in the analysis and visualisation of this data.

About the speaker: Dr Karen Blackmore is a Senior Lecturer in Computing at the School of Electrical Engineering and Computing, The University of Newcastle, Australia. She received her BIT (Spatial Science) With Distinction and PhD (2008) from Charles Sturt University, Australia. Dr Blackmore is a spatial scientist with research expertise in the modelling and simulation of complex social and environmental systems. Her research interests cover the use of agent-based models for simulation of socio-spatial interactions, and the use of simulation and games for serious purposes. Her research is cross-disciplinary and empirical in nature, and extends to exploration of the ways that humans engage and interact with models and simulations. Before joining the University of Newcastle, Dr Blackmore was a Research Fellow in the Department of Environment and Geography at Macquarie University, Australia and a Lecturer in the School of Information Technology, Computing and Mathematics at Charles Sturt University.

In teaching mathematics, we are interested in improving students' understanding of core concepts. Students enter our classrooms as relative novices in their understanding of mathematics and one of our goals is to help them build expert understanding of mathematics. This presents us with two related problems: (1) creating effective teaching strategies designed to evolve novice thinking to expert thinking, and (2) designing and validating measures capable of assessing whether different teaching interventions improve students' conceptual understanding of mathematics. Many usual approaches to these problems make use of scoring rubrics for student work. I will discuss an experiment that highlights some of the difficulties of using scoring rubrics for this work, and then I will present an alternative approach to these problems that makes use of the law of comparative judgment, which is based on the principle of that humans are better at comparing two things against one another than they are at comparing one thing against a set of criteria (Thurstone's Law of Comparative Judgment, 1927). As part of this presentation, I will demonstrate ComPAIR, a new online tool for supporting student learning with peer feedback. ComPAIR was co-developed with a group of colleagues from the Faculty of Science, the Faculty of Arts, and the Centre for Teaching and Learning Technology at the University of British Columbia.

Constructive methods for the controller design for dynamical systems subject to bounded state constraints have only been investigated by a limited number of researchers. The construction of robust control laws is significantly more difficult compared to unconstrained problems due to the necessity of discontinuous feedback laws. A rigorous understanding of the problem is however important in obstacle or collision avoidance for mobile robots, for example. In this talk we present preliminary results on the controller design for obstacle avoidance of linear systems based on the notation of hybrid systems. In particular, we derive a discontinuous feedback law, globally stabilizing the origin while avoiding a neighborhood around an obstacle. In this context, additionally an explicit bound on the maximal size of the obstacle is provided.

The lattice Boltzmann method is used to carry out a direct numerical simulation of laminar and turbulent flows in a smooth and rough wall channel or pipe at critical and subcritical Reynolds number. The velocity field is solved using the Lattice Boltzmann Method (LBM) as an alternative numerical approach to computational Fluid dynamics. The method is successfully used to simulate more complex fluid dynamics such as thermal transportation, jet flows, electrokinetic flows and so on. The basic idea of LBM is to construct a simplified kinetic model that incorporates the essential physics of microscopic average properties, which obey the desired Navier-Stokes equations. The computation and visualization will be discussed in this seminar.

About the speaker: Dr Nisat Nowroz Anika completed her Bachelor and MSc in Applied Mathematics from Khulna University in Bangladesh at the year of 2011 and 2013 respectively. She is currently undertaking a Ph.D. in Mechanical Engineering at the University of Newcastle under the supervision of Professor Lyazid Djenidi. The major focus of her research is mixing at low Reynolds number by generating turbulence.

The late Professor Jonathan Borwein was fascinated by the constant $\pi$. Some of his talks on this topic can be found on the CARMA website.
This homage to Jon is based on my talk at the Jonathan Borwein Commemorative Conference. I will describe some algorithms for the high-precision computation of $\pi$ and the elementary functions, with particular reference to the book Pi and the AGM by Jon and his brother Peter Borwein.
Here "AGM" is the arithmetic-geometric mean of Gauss and Legendre. Because the AGM has second-order convergence, it can be combined with FFT-based fast multiplication algorithms to give fast algorithms for the \hbox{$n$-bit} computation of $\pi$.
I will survey a few of the results and algorithms that were of interest to Jon. In several cases they were either discovered or improved by him. If time permits, I will also mention some new results that would have been of interest to Jon.

The finite element method has become the most powerful approach in solving partial differential equations arising in modern engineering and physical applications. We present computation and visualisation of the solutions of some applied partial differential equations using the finite element method for most of our examples. Our examples come from solid and fluid mechanics, image processing and heat conduction in sliding meshes.

About the speaker: Dr Lamichhane was awarded the MSc in Industrial Mathematics from the University of Kaiserslautern in 2001, and the PhD in Mathematics from the University of Stuttgart in 2006. He took a postdoctoral fellow at the Australian National University in 2008 and is now a senior lecturer at the University of Newcastle. Dr Lamichhane’s main interests are numeral analysis, differential equations and applied mathematics and his recent research focus is on the approximation of solutions of partial differential equations using the finite element method.

Multi-objective optimization provides decision-makers with a complete view of the trade-offs between their objective functions that are attainable by feasible solutions. Since many problems can be formulated as integer programs, the development of efficient and reliable multi-objective integer programming solvers may have significant benefits for problem solving in industry and government. However, the conjunction of multiple objectives and integrality yields problems that can be challenging to solve. So, this talk provides an overview of a few new exact as well as heuristic algorithms for this class of optimization problems. In particular, the talk focuses on computing the nondominated frontier and also the problem of optimization over the frontier. It is worth mentioning that all of the algorithms and their corresponding open-source software packages are developed in Multi-Objective Optimization Laboratory at the University of South Florida.

The rapid increase in available information has led to many attempts to automatically locate patterns in large, abstract, multi-attributed information spaces. These techniques are often called data mining and have met with varying degrees of success. An alternative approach to automatic pattern detection is to keep the user in the exploration loop by developing displays that enhance their natural sensory abilities to detect patterns. This approach, whether visual, auditory, or touch based, can assist a domain expert to search their data for useful relationships. However, designing models of the abstract data and defining appropriate sensory mappings are critical tasks in building such a system. Intuitive multi-sensory displays (visual, auditory, touch) of abstract data are difficult to design and the process needs to carefully consider human perceptual and cognitive abilities. This talk will introduce a taxonomy that helps designers consider the range of sensory mappings, along with appropriate guidelines, when building such multisensory displays. To illustrate this process a case study in the domain of stock market data is also presented.

About the speaker: Keith completed his Bachelor's degree in Mathematics at Newcastle University in 1988 and his Masters in Computing in 1993. Between 1989-1999, Keith worked on applied computer research for BHP Research. His PhD examined the design of multi-sensory displays for stock market data and was completed at Sydney University in 2003. His work has received international recognition, being selected among the best visualisations and consequently exhibited at a number of international locations and reviewed in the prestigious journal Science. In 2007 he completed a post-doctoral year in Boston working at the New England Complex Systems Institute visualising health related data. He has expertise in the fields of Human Interface Design, Computer Games, Virtual Reality, Immersive Analytics, and the theory of Perception and Cognition related to the design of multi-sensory user interfaces. Keith currently works in the school of School of Electrical Engineering and Computing at the University of Newcastle, Australia where he teaches Computer Games and Programming. While his background is in Computer Science, he has also exhibited his paintings in 11 exhibits and provided lyrics for 5 CDS and a musical. You can find more about his art and science at www.knesbitt.com.

Expanding the 1993 paper by Hohn and Skoruppa, and a brief exploration of Mahler measure optimal conditions.

About the speaker: Elijah Moore is a summer research student under the supervision of Wadim Zudilin.

The human brain is still one of the most powerful and at the same time most energy efficient computers. Artificial neural networks (ANN) are inspired by their biological counterparts and the workings of biological nervous systems. ANNs were among the most popular machine learning algorithms in the 1980-90s. However, after 2000 other algorithms came to be regarded as more accurate and practical. In 2012 ANNs came back with a big bang: a new form of biologically-inspired ANNs, deep convolutional neural networks, showed surprisingly good performances on image classification and object detection tasks, far superior to all other methods available. Since then deep networks have breaken records in many application domains, from object detection for autonomous vehicles to playing the game of Go and skin health diagnostics. Deep networks are currently revolutionising machine learning in academia and industry. They can be regarded as the most disruptive technology in any industry that involves machine learning, artificial intelligence, pattern recognition, data mining or control. This seminar aims at providing an overview of ANNs - old and new - with a special view towards how visualisations could help to explain how they work.

About the speaker: Stephan Chalup (Ph.D., Dipl.-Math.) is an associate professor at the University of Newcastle in Australia, where he is leading the Interdisciplinary Machine Learning Research Group and the Newcastle Robotics Lab. He studied mathematics with neuroscience at the University of Heidelberg and completed his Ph.D. in Computing Science at the Machine Learning Research Centre at Queensland University of Technology (QUT) in 2002. Stephan has published 100 research articles and is on the editorial boards of several journals. He is member of the University of Newcastle's Priority Research Centre CARMA.

Groups of rooted tree automorphisms, and (weakly) branch groups in particular, have received considerable attention in the last few decades, due to the examples with unexpected properties that they provide, and their connections to dynamics and automata theory. These groups also showcase interesting phenomena in profinite group theory. I will discuss some of these and other profinite completions that one can use to study these groups, and how to find them. All these concepts will be defined in the talk.

This presentation will discuss the megatrends, both technological and societal, that are impacting the modern supply chain. In particular, the balance between people and machines will be explored in the context of future of work within supply chains. What are the appropriate roles for robotics within the supply chain of the future? What is the future for people in the supply chain? Examples of existing and emerging technologies will be presented to show that the future supply chain is close at hand.

The Chebyshev conjecture is a 59-year-old open problem in the fields of analysis, optimisation, and approximation theory, positing that Chebyshev subsets of a Hilbert space must be convex. Inspired by the work of Asplund, Ficken and Klee, we investigate an equivalent formulation of this conjecture involving Chebyshev subsets of the unit sphere. We show that such sets have superior structure and use the Radon-Nikodym Property to extract some local structural results about such sets.

We present $h$ and $p$-versions of the time domain boundary element method for boundary and screen problems for the wave equation in $\mathbb{R}^3$. First, graded meshes are shown to recover optimal approximation rates for solution in the presence of edge and corner singularities on screens. Then an a posteriori error estimate is presented for general discretizations, and it gives rise to adaptive mesh refinement procedures. We also discuss preliminary results for $p$ and $hp$-versions of the time domain boundary element method. Numerical experiments illustrate the theory. Joint with H. Gimperlein and D. Stark, Heriot-Watt University, Edinburgh.

One of the most contentious areas in Indigenising Curriculum is the Maths and Sciences. This presentation considers how Maths and Statistics can provide a solid and meaningful response to the Indigenising imperative that will fulfil the two criteria of socially just education:

1. The inclusion and success of Aboriginal and Torres Strait Islander students and;
2. The education of all students on Aboriginal and Torres Strait Islander epistemologies and ontologies

Suggestions on both content areas and student recruitment, retention and success will be discussed. Examples will be based on the presenters experiences as cultural facilitators in education from Foundations to the tertiary sector.

Associate Professor Kathy Butler and Ms Tammy Small are employed in the Office of the Pro Vice-Chancellor Indigenous Education and Research at the University of Newcastle. With Professor Steve Larkin, Tammy and Kathy are currently examining ways for the University to provide cultural competency training as a whole-of-university initiative.

This presentation is to assist academics consider how to adapt programs and course content and delivery to incorporate, be mindful of and better appeal to people with Indigenous backgrounds and interests.

We consider variations on the commutative diagram consisting of the Fourier transform, the Sampling Theorem and the Paley-Wiener Theorem. We start from a generalization of the Paley-Wiener theorem and consider entire functions with specific growth properties along half-lines. Our main result shows that the growth exponents are directly related to the shape of the corresponding indicator diagram, e.g., its side lengths. Since many results from sampling theory are derived with the help from a more general function theoretic point of view (the most prominent example for this is the Paley-Wiener Theorem itself), we motivate that a closer examination and understanding of the Bernstein spaces and the corresponding commutative diagrams can—via a limiting process to the straight line interval [−A,A]—yield new insights into the Lp(R)-sampling theory. This is joint work with Gunter Semmler, Technische Universität Bergakademie Freiberg, Germany.

Schoenberg’s polynomial cardinal B-splines of order $n$ provide a family of compactly supported $C^{n-2}$-functions. We present several generalizations of these B-splines, discuss their properties, and relate them to fractional difference and differentiation operators. Potential applications are mentioned.

Given a sequence of integers, one would like to understand the pattern which generates the sequence, as well as its asymptotics. If the sequence is viewed as the coefficients of the series expansion of a function, called its generating function, many questions regarding the sequence can be answered more easily. If the generating function satisfies a linear ODE or a nonlinear algebraic DE, the differential equation can be found if enough terms in the sequence are given. In this talk I'll discuss my implementation in C of such a search, applications, and a systematic search of the entire Online Encyclopedia of Integer Sequences (OEIS) for generating functions.

I will present a brief survey of some recent results that deal with the characterization of hyperbolic dynamics in terms of the existence of appropriate Lyapunov functions. The main novelty of these results lies in the fact that they consider noninvertible and infinite-dimensional dynamics. This is a joint work with L. Barreira, C. Preda and C. Valls.

In this talk I will briefly introduce the mixed finite element method and show their applications. I consider Poisson, elasticity, Stokes and biharmonic equations for the applications of the mixed finite element method. The mixed finite element method also arises naturally in Stokes flow, multi-physics problems as well as when we consider non-conforming discretisation techniques. I will also present my recent works on the mixed finite element method for biharmonic and Reissner-Mindlin plate equations.

Colour images are represented by functions of 2 variables that output 3 variables, and analysing them requires tools that can handle these dimensions. One method is to use Clifford Algebras and their recently discovered Fourier Transform. We prove the Clifford Fourier Transform has a Hardy Space, and that it's Paley-Wiener Space and Bernstein Spaces are identical. Another method is to find 2 dimensional wavelets that are non-separable. We achieve this through the use of the Douglas-Rachford Projection Algorithm, and hope to achieve it through the use of Proximal Alternating Linear Methods. This talk briefly overviews these methods and the path to completion.

I will discuss how to relate regular origami tilings to vertex models in statistical mechanics. The Miura-ori origami pattern has found many uses in engineering as an auxetic metamaterial. I analyze the effect of crease assignment defects on the long-range order properties of the Miura-ori and 4 other foldable lattices. These defects are known to affect the material's compressibility properties, so my exact results help to understand how easy it is to tune an origami metamaterial to have desired compressibility properties by introducing a set density of defects. I have found that certain origami patterns are more easily tunable than others, and conversely, the long-range ordering of some are more stable with respect to defect formation. I have also found analytical expressions for the locations of phase transition points with respect to crease assignment ordering as well as layer ordering.

The aim of this workshop is to bring together the world's foremost experts on the theory of semigroups and their relationships to other fields of mathematics such as operator algebras and totally disconnected locally compact groups. This workshop will allow the international leaders in the field to come to Australia to teach young Australian ECRs, and to forge new collaborations with Australian mathematicians.

Details are available on the conference website.

Operator algebras associated to semigroups can be traced back to a famous theorem of Coburn from the 1960s. The theory has recently been reinvigorated through Xin Li's construction of semigroup C*-algebras. Li's construction has introduced new and interesting classes of C*-algebras, which have deep connections to number theory and dynamical systems. One connection that will be thoroughly explored through this meeting is that to the representation theory of totally disconnected locally compact groups.

This presentation will outline my research into fitness for purpose of tertiary algebra textbooks used in Iraq in the teaching of undergraduate algebra courses with regard to the training of pre-service teachers. The project draws on work done in textbook analysis, and work done into the teaching and learning of abstract algebra and the nature of proof.

It is well recognised that for many students learning abstract algebra and the nature of proof is difficult (Selden, 2010). Courses in abstract algebra are central to many tertiary pre-service mathematics teacher programs, including in Iraq. Capaldi (2012) suggests that abstract algebra textbooks can lay the foundation for a course and greatly influence student understanding of the material. However, it has been found that there can be large differences in textbooks used, at the school level at least, in different cultures. (Alajmi A. H., 2012, Fan and Zhu, 2007, Pepin and Haggarty, 2001). For instance, Mayer and Sims, (1995) Japanese texts feature many more worked out examples than texts used in the United States for mathematics.

I will be examining the textbooks in light of theories by Harel and Sowder, and Stacey and Vincent, regarding types and proof and modes of reasoning (Stacey and Vincent, 2009) and Capaldi (2012) regarding reader's relationships with books.

The textbooks will also be examined to try to infer the underlying assumptions about pedagogies and knowledge made by the author(s). Baxter-Magolda's theory, linking forms of assessment to underlying theories of knowledge (Baxter-Magolda, 1992) will be helpful in this pursuit.

The theory of minimal surfaces (a.k.a. soap films) goes back to Euler’s discovery in 1741 that the catenoid is area-minimising. It is still a remarkably vibrant area of research. I will describe recent joint work with Franc Forstneric of the University of Ljubljana, Slovenia. We assemble all minimal surfaces with a given shape into a space. It is an infinite-dimensional space. What does it look like? We have been able to determine its "rough shape". I will explain what we mean by "rough shape" and describe the ingredients from complex analysis, differential topology, and homotopy theory that go into our result.

Lyapunov's second or direct method provides an easy-to-check sufficient condition for stability properties of equilibria. The converse question - given a stability property, does there exist an appropriate Lyapunov function? - has been fundamental in differentiating and classifying different stability properties, particularly with regards to "uniform" stability.

In this talk, I will review the usual textbook definitions for Lyapunov functions for time-varying systems and describe where they are deficient. Some interesting new sufficient (and probably necessary) conditions pop up along the way.

The research interest in pattern avoiding permutations is inspired by Donald Knuth’s work in stack-sorting. According to Knuth, a permutation can be sorted by passing through a single infinite stack if and only if it avoids a sub-permutation pattern 231. Murphy extended Knuth’s research by using two infinite stacks in series and found out that the basis for generated permutations is infinite but Elder proved that the basis is finite when one of the stack is limited to depth two and the permutations are algebraic. My research is to investigate the permutations generated by a stack of depth 3 and an infinite stack in series. It is to determine the basis and nature of the permutations in term of formal language.

We determine the Borel complexity of the topological isomorphism problem for profinite, t.d.l.c., and Roelcke precompact non-Archimedean groups, by showing it is equivalent to graph isomorphism.

For oligomorphic groups we merely establish this as an upper bound.

Joint work with Kechris and Tent.

Control Lyapunov functions (CLFs) for the control of dynamical systems have faded from the spotlight over the last years even though their full potential has not been explored yet. To reactivate research on CLFs we review existing results on Lyapunov functions and (nonsmooth) CLFs in the context of stability and stabilization of nonlinear dynamical systems. Moreover, we highlight open problems and results on CLFs for destabilization. The talk concludes with ideas on Complete CLFs, which combine the concepts of stability and instability. The results presented in the talk are illustrated and motivated on the examples of a nonholonomic integrator and Artstein's circles.

Part of my 2016 SSP included completion of a semi-historical review on the mathematics of W.N. Bailey, a familiar name in some combinatorics circles in relation with the "Bailey lemma" and "Bailey pairs." My personal encounters with the mathematician from the first half of the 20th century were somewhat different and more related to applications of special functions to number theory—the subject Bailey had never dealt with himself. One motivation for my writing was the place where I spent my SSP—details to be revealed in the talk. There will be some formulas displayed, sometimes scary, but they will serve as a background to historical achievements. Broad audience is welcome.

In this talk we discuss a new approach for the Hamilton cycle problem (HCP). The HCP is one of the classical problems in combinatorial mathematics. It can be stated as given a graph G, find a cycle that passes through every single vertex exactly once, or determine that such a cycle does not exist. In 1994, Filar and Krass developed a new model for HCP by embedding this problem into a Markov decision process. This approach was the motivation of a new line research which was extended by several other people afterwards. In this approach, a new polytope corresponding to a given graph G was constructed and searching for Hamiltonian cycles in a given Hamiltonian graph G was converted to searching for particular extreme points (called Hamiltonian extreme points) among extreme points of that polytope. In this research, we are going to design a Markov chain with certain properties to sample Hamiltonian extreme points of that polytope. More precisely, we would like to study a specific class of input graphs, the so-called random graphs. Some preliminary theoretical results are presented in this talk.

In this talk we consider a class of monotone operators which are appropriate for symbolic representation and manipulation within a computer algebra system. Various structural properties of the class (e.g., closure under taking inverses, resolvents) are investigated as well as the role played by maximal monotonicity within the class. In particular, we show that there is a natural correspondence between our class of monotone operators and the subdifferentials of convex functions belonging to a class of convex functions deemed suitable for symbolic computation of Fenchel conjugates which were previously studied by Bauschke & von Mohrenschildt and by Borwein & Hamilton. A number of illustrative computational examples utilising the introduced class of operators will be provided including computation of proximity operators, recovery of a convex penalty function associated with the hard thresholding operator, and computation of superexpectations, superdistributions and superquantiles with specialization to risk measures.

I am going to look at three unsolved graph theory problems for which the same family of graphs presents a barrier to either solving or making substantial progress on the problems. The graphs in this family are called honeycomb toroidal graphs. The three problems are not closely related.

Problem solving, communication and information literacy are just a few graduate attributes that employers value, yet it commonly appears that students upon graduating show only limited improvement in these areas. For instance, 3rd year students can still be thrown by relatively simple unfamiliar problems, even after working actively on numerous related exercises and problems throughout their degree. I will discuss some of the things I have implemented in my teaching to specifically target the development of student graduate attributes. My experience is with teaching mathematics, physics and engineering students, however much of my discussion will be non-discipline-specific.

In a way, mathematics can be seen as a language game, where we use symbols, together with some rewriting rules, to represent objects we are interested in and then ask what can be said about the sequences of symbols (languages) that capture certain phenomena. For example, given a group G with generators a and b, can we recognise (using a computer) the sequences of generators that correspond to non-trivial elements of G? If yes, how strong computer do we need, i.e. how complicated is the language we are studying?

There is a natural duality between various types of computational models and classes of languages that can be recognised by them. Until recently most problems/languages in group theory were classified within the Chomsky hierarchy, but there are more computational models to consider. In the talk I will briefly introduce L-systems, a family of classes of languages originally developed to model growth of algae, and show that the co-word problem in Grigorchuk's group, a group of particularly nice transformations of infinite binary tree, can be seen as a language corresponding to a fairly simple L-system.

Totally disconnected, locally compact (t.d.l.c.) groups are a large class of topological groups that arise from a few different sources, for instance as automorphism groups of a range of algebraic and combinatorial structures, or from the study of isomorphisms between finite index subgroups of a given group. A general theory has begun to emerge in recent years, based on the interaction between small-scale and large-scale structure in t.d.l.c. groups. I will give a survey of some ways in which these groups arise and some of the tools that have been developed for understanding them.

In recent joint work on equilibrium states on semigroup C*-algebras with Afsar, Brownlowe, and Larsen, we discovered that the structure of equilibrium states admits an elegant description in terms of substructures of the original semigroup. More precisely, we consider two almost contrary subsemigroups and related features to obtain a unifying picture for a number of predating case studies. Somewhat surprisingly, all the examples from the case studies satisfy a list of four abstract properties (and are then called admissible). The nature and presence of these properties is yet to be fully understood. In this talk, I will focus on a class of examples arising as Zappa-Szép products of right LCM semigroups which showcases some interesting features. No prerequisites in operator algebras are required to follow this talk.

This talk gives an outline of (mostly unfinished) work done collaboratively while on sabbatical in semester 2 last year. Join me as we travel through the USA, Germany, Belgium and Austria. Your guide will share off-the-beaten-track highlights such as quaternionic splines, prolate shift systems, higher-dimensional Hardy, Paley-Wiener and Bernstein spaces, the Clifford Fourier transform, multidimensional prolates, and a Jon Borwein-inspired optimization-based approach to the construction of multidimensional wavelets. Breakfast not included.

In this talk, we discuss a new approach to demand forecasting in supply chains. Demand forecasting is an inevitable task in supply chain management. Due to the endogenous and exogenous factors impact a supply chain, the regime of the supply chain may vary significantly. Such changes in the regime can bring a high volatility to demand time series and consequently, a single statistical model may not suffice to forecast the demand with a desirable level of precision. We develop a nexus between stochastic processes and statistical models to forecast the demand in supply chains with regime switching. The preliminary results on real world time series data sets are promising.

In this talk, I will describe the conditional value at risk (CVaR) measure used in modelling risk aversion in decision making problems.

CVaR is a highly consistent risk measure for modelling risk aversion.

I will then present two applications of CVaR. The first application considers all problems that are representable by decision trees. In this application, I show that these problems under the CVaR criterion can be solved efficiently by solving a linear program. In the second application, I consider a basic problem in the area of production planning with random yield. For this problem, I present a risk aversion model. The model is nonconvex. I present an efficient locally optimal solution method and then provide a sufficient optimality condition.

There is to date no overarching classification theorem for C*-algebras, which means the theory of C*-algebras is an example-driven field of mathematics. Perhaps the most important class of examples are group C*-algebras, which are as old as the field itself. An analogous construction of C*-algebras associated to semigroups has been an active area of research among operator algebraists since Coburn’s Theorem regarding the universality of the C*-algebra generated a single isometry appeared in the 1960s. In July this year, Newcastle will host the AMSI/AustMS sponsored event "Interactions between operator algebras and semigroups". In this talk I will give a gentle introduction to the theory of semigroup C*-algebras and perhaps it will convince some of you to come along and take part in the meeting.

Gamification refers to the use of elements of games in non-game contexts and has been applied in workplace, marketing, health programs and other areas, with mounting evidence of increased interest, involvement, satisfaction and performance of the participants. More recently gamification has been emerging as a teaching method that has a great potential to improve students’ motivation and engagement. Gamification in education should not be confused with playing educational games, as it only uses concepts such as points, leader boards, etc, rather than computer games themselves. In this talk we describe the gamification of a theoretical computer science course we performed in 2014/2015/2016 as well as our experience with two other STEM courses.

The talk will include general information on the current state of plasma fusion as an energy source and some more detailed aspects of this research area.

In 1975, culminating more than 40 years of published work by Paul Erdos on the problem, he and John Selfridge proved that the product of consecutive integers cannot be a nonzero perfect power. Their proof was a remarkable combination of elementary and graph theoretic arguments. Subsequently, Erdos conjectured that this result can be generalized to a product of consecutive terms in an arithmetic progression, under certain basic assumptions. In this talk, we discuss joint work with Samir Siksek in the direction of proving Erdos' conjecture. Our approach is via techniques based upon the modularity of Galois representations, bounds for the number of supersingular primes for elliptic curves, and analytic estimates for Dirichlet character sums.

In this talk, we discuss our new approach to design reverse logistics models for dairy industries, in particular whey products. Whey is a by-product of cheese making with many applications spanning from dairy and meat to pharmaceuticals. We develop a hierarchical location-routing model for a whey recovery network design. In this class of models, the location and routing decisions are made simultaneously. As the problem is NP-hard, it may not be possible to solve even the small-size instances efficiently. We suggest different approaches such as adding valid inequalities and improving lower and upper bounds to solve the problem in a reasonable amount of time.

Mathscraft is a workshop for junior high school students that aims to give them the experience of doing maths the way research mathematicians do. It is coordinated and sponsored by the ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), and sessions are conducted by Anthony Harradine, (Prince Alfred College, Adelaide).

In a Mathscraft session there are up to 10 groups, each comprising three students (years 7-10), one teacher and one mathematician. The teams are given mathematical problems and are guided through a problem-solving process. The problems and the process are designed to mimic the mathematics that is done by research mathematicians - exploring, noticing patterns, making conjectures, proving them, figuring out why, and thinking of ways to extend the problem.

In this talk I'll describe the design of the problems and process (with examples), and explain the motivations behind them. I'll also talk about a Professional Development workshop that we ran for teachers in November last year, which had the aim of training them to run Mathscraft sessions in their own local areas. This workshop was sponsored by ACEMS and MATRIX.

Jonathan Michael Borwein (20 May 1951 - 2 Aug 2016) had many talents, among which were his abilities to make discoveries in mathematics, to seek tenaciously for proofs of these, and to do both of those things in collegial concert with other workers. In this colloquium I shall give three examples of situation in which I had the pleasure of seeing those talents in action. They concern multiple zeta values, walks on lattices, and modular forms. In each case I shall give a notable identity, comment on its proof, and indicate further work that was provoked by the discovery. The identities in question are chosen to be comprehensible to anyone with an undergraduate education in mathematics and also to people, like myself, who lack that particular qualification.

A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. These conditions are usually expressed by on the basis of the weights of some evaluating function. Based on these conditions there are several types of graph labelings such as graceful, magic, antimagic, sum and irregular labeling. In this research, we looking at the H-supermagic labeling of firecracker, banana tree, flower and grid graphs; the exclusive sum labelling of trees; and the edge irregularity strength of the grid graphs.

Incremental stability describes the asymptotic behavior between any two trajectories of dynamical systems. Such properties are of interest, for example, in the study of observers or synchronization of chaos. In this paper, we develop the notions of incremental stability and incremental input-to-state stability (ISS) for discrete-time systems. We derive Lyapunov function characterizations for these properties as well as a useful summation-to-summation formulation of the incremental stability property.

I will discuss how to solve free group equations using a practical computer program. Ciobanu, Diekert and Elder recently gave a theoretical algorithm which runs in nondeterministic space $n\log n$, but implementing their method as an actual computer program presents many challenges, which I will describe.

Some Engel words and also commutators of commutators can be expressed as products of powers. I discuss recent work of Colin Ramsay in this area, using PEACE (Proof Extraction After Coset Enumeration), and in particular provide expressions for commutators of commutators as short products of cubes.

The Australian Council on Healthcare Standards collates data on measures of performance in a clinical setting in six-month periods. How can these data best be utilised to inform decision-making and systems improvement? What are the perils associated with collecting data in six-month periods, and how may these be addressed? Are there better ways to analyse, report and guide policy?

The Council for Aid to Education is one of many organisations internationally attempting to assess tertiary institutional performance. Value-add modelling is a technique intended to inform system performance. How valid and reliable are these techniques? Can they be improved?

Educational techniques and outreach activities are employed across the education system and the wider community for the purposes of increasing access, equity and understanding.

When new concepts are formed, a well-designed instrument to assess and provide evidence of their performance is required. Does immersion in professional experience activity enable pre-service teachers to achieve teaching standards? Do engagement activities for schools in remote and rural areas increase students’ aspirations and engagement with tertiary institutions?

Forensic anthropologists deal with the collection of bones and profiling individuals based on the remains found. How can statistics inform such decision-making?

Such questions and existing and potential answers will be discussed in the context of research collaborations with Taipei Medical University (Taiwan), Health Services Research Group, Australian Council on Healthcare Standards, Hunter Medical Research Institute, School of Education, Wollotuka Institute, School of Environmental Sciences and a Forensic Anthropologist.

A challenge with our large-enrolment courses is to manage assessment resources: questions, quizzes, assignments and exams. We want traditional in-class assessment to be easier, quicker and more reliable to produce, in particular where multiple versions of each assessment is required. Our approach is to

• Manage a library of past assessments
• Generate assessments from the library, with or without randomness
• Algorithmically generate new assessments

We have implemented this within standard software: LaTeX, Ruby, git, and our favourite mathematics software.

We will briefly show off our achievements in 2016, including new features of the software and how we've used them in our teaching. We then invite discussion on we we can do to help our colleagues use these tools.

For a two-coloring of the vertex set of a simple graph $G$ consider the following color-change rule: a red vertex is converted to blue if it is the only red neighbor of some blue vertex. A vertex set $S$ is called zero-forcing if, starting with the vertices in $S$ blue and the vertices in the complement $V \setminus S$ red, all the vertices can be converted to blue by repeatedly applying the color-change rule. The minimum cardinality of a zero-forcing set for the graph $G$ is called the zero-forcing number of $G$, denoted by $Z(G)$.

There is a conjecture connecting zero forcing number, minimum degree $d$ and girth $g$ as follows: "If G is a graph with girth $g \geq 3$ and minimum degree $d \geq 2$, then $Z(G) \geq d+ (d-2)(g-3)$".

I shall discuss a recent paper where the conjecture is proved to be true for all graphs with girth \leq 10.

Targeted Audience: All early career staff and PhD students; other staff welcome

Abstract: Many of us have been involved in discussions revolving around the problem of choosing suitable thesis topics and projects for post-graduate students, honours students and vacation research students. The panel is going to present some ideas that we hope people in the audience will find useful as they get ready for or continue with their careers.

About the Speakers: Professor Brian Alspach has supervised thirteen PhDs, twenty-five MScs, nine post-doctoral fellows and a dozen undergraduate scholars over his fifty-year career. Professor Eric Beh has 20 years' international experience in the analysis of categorical data with a focus on data visualisation. He has and has, or currently is, supervised about a 10 PhD students. Dr Mike Meylan has twenty years research experience in applied mathematics both leading projects and working with others. He has supervised 5 PhD students and three post-doctoral fellows.

Today's discrete mathematics seminar is dedicated to Mirka Miller. I am going to present the beautiful Hoffman-Singleton (1964) paper which established the possible values for valencies for Moore graphs of diameter 2, gave us the Hoffman-Singleton graph of order 50, and gave us one of the intriguing still unsettled problems in combinatorics. The proof is completely linear algebra and is a proof that any serious student in discrete mathematics should see sometime. This is the general area in which Mirka made many contributions.

In this talk I will present a class of C*-algebras known as "generalised Bunce-Deddens algebras" which were constructed by Kribs and Solel in 2007 from directed graphs and sequences of natural numbers. I will present answers to questions asked by Kribs and Solel about the simplicity and the classification of these C*-algebras. These results are from my PhD thesis supervised by Dave Robertson and Aidan Sims.

This afternoon (31 October) we shall complete the discussion about vertex-minimal graphs with dihedral automorphism groups. I have attached an outline of what was covered in the first two weeks.

Learning to rank is a machine learning technique broadly used in many areas such as document retrieval, collaborative filtering or question answering. We present experimental results which suggest that the performance of the current state-of-the-art learning to rank algorithm LambdaMART, when used for document retrieval for search engines, can be improved if standard regression trees are replaced by oblivious trees. This paper provides a comparison of both variants and our results demonstrate that the use of oblivious trees can improve the performance by more than 2:2%. Additional experimental analysis of the inuence of a number of features and of a size of the training set is also provided and confirms the desirability of properties of oblivious decision trees.

About the Speaker: Dr Michal Ferov is a Postdoctoral Research Fellow in the School of Mathematical and Physical Sciences,Faculty of Science and Information Technology.

I am studying the complexity of solving equations over different algebraic objects, like free groups, virtually free groups, and hyperbolic groups. We have an NSPACE(n log n) algorithm to find solutions in free groups, which I will try to briefly explain. Applications include pattern recognition and machine learning, and first order theories in logic.

Maintenance plays a crucial role in the management of rail infrastructure systems as it ensures that infrastructure assets (e.g., tracks, signals, and rail crossings) are in a condition that allows safe, reliable, and efficient transport. An important and challenging problem facing planners is the scheduling of maintenance activities which must consider the movement and availability of the maintenance resources (e.g., equipment and crews). The problem can be viewed as an inventory routing problem (IRP) in which vehicles deliver product to customers so as to ensure that the customers have sufficient inventory to meet future demand. In the case of rail maintenance, the customers are the infrastructure assets, the vehicles correspond to the resources used to perform the maintenance, and the product that is in demand, the inventory of which is replenished by the vehicle, is the asset condition. To the best of our knowledge, such a viewpoint of rail maintenance has not been previously considered.

In this thesis we will study the IRP in the rail maintenance scheduling context. There are several important differences between the classical IRP and our version of the problem. Firstly, we need to differentiate between stationary and moving maintenance. Stationary maintenance can be thought of having demand for product at a specific location, or point, while moving maintenance is more like the demand for product being distributed along a line between two points. Secondly, when performing maintenance, trains may be subject to speed restrictions, be delayed, or be rerouted, all of which affect the infrastructure assets and their condition differently. Finally, the long-term maintenance schedules that are of interest are developed annually. IRPs with such a long planning horizon are intractable to direct solution approaches and therefore require the development of customised solution methodologies.

This week we shall continue by introducing the cast of characters to be used for producing minimal-order graphs with dihedral automorphism group.

Many governments and international finance organisations use a carbon price in cost-benefit analyses, emissions trading schemes, quantification of energy subsidies, and modelling the impact of climate change on financial assets. The most commonly used value in this context is the social cost of carbon (SCC). Users of the social cost of carbon include the US, UK, German, and other governments, as well as organisations such as the World Bank, the International Monetary Fund, and Citigroup. Consequently, the social cost of carbon is a key factor driving worldwide investment decisions worth many trillions of dollars.

The social cost of carbon is derived using integrated assessment models that combine simplified models of the climate and the economy. One of three dominant models used in the calculation of the social cost of carbon is the Dynamic Integrated model of Climate and the Economy, or DICE. DICE contains approximately 70 parameters as well as several exogenous driving signals such as population growth and a measure of technological progress. Given the quantity of finance tied up in a figure derived from this simple highly parameterized model, understanding uncertainty in the model and capturing its effects on the social cost of carbon is of paramount importance. Indeed, in late January this year the US National Academies of Sciences, Engineering, and Medicine released a report calling for discussion on the various types of uncertainty in the overall SCC estimation approach and addressing how different models used in SCC estimation capture uncertainty.

This talk, which focuses on the DICE model, essentially consists of two parts. In Part One, I will describe the social cost of carbon and the DICE model at a high-level, and will present some interesting preliminary results relating to uncertainty and the impact of realistic constraints on emissions mitigation efforts. Part one will be accessible to a broad audience and will not require any specific technical background knowledge. In Part Two, I will provide a more detailed description of the DICE model, describe precisely how the social cost of carbon is calculated, and indicate ongoing developments aimed at improving estimates of the social cost of carbon.

Konig (1936) asked whether every finite group G is realized as the automorphism group of a graph. Frucht answered the question in the affirmative and his answer involved graphs whose orders were substantially bigger than the orders of the groups leading to the question of finding the smallest graph with a fixed automorphism group. We shall discuss some of the early work on this problem and some recent results for the family of dihedral groups.

For over 25 years, Wolfram Research has been serving Educators and Researchers. In the past 5 years, we have introduced many award winning technology innovations like Wolfram|Alpha Pro, Wolfram SystemModeler, Wolfram Programming Lab, and Natural Language computation. Join Craig Bauling as he guides us through the capabilities of Mathematica. Craig will demonstrate the key features that are directly applicable for use in teaching and research. Topics of this technical talk include

• Natural Language Input
• Market Leading Statistical Analysis Functionality
• 2-D and 3-D information visualization and 3D Printing
• Creating interactive models that encourage student participation and learning
• Practical applications in Engineering, Chemistry, Physics, Finance, Biology, Economics and Mathematics
• On-demand Chemical, Biological, Economic, Finance and Social data
• Mathematica as a modern programming language

Prior knowledge of Mathematica is not required - new users are encouraged. Current users will benefit from seeing the many improvements and new features of Mathematica 11.

We discuss ongoing work in convex and non-convex optimization. In the convex setting, we use symbolic computation to study problems which require minimizing a function subject to constraints. In the non-convex setting, we use a variety of computational means to study the behavior of iterated Douglas-Rachford method to solve feasibility problems, finding an element in the intersection of several sets.

The formation of high-mass stars (> 8 times more massive than our sun) poses an enormous challenge in modern astrophysics. Theoretically, it is difficult to understand whether the final mass of a high-mass star is accreted locally or from afar. Observationally, it is difficult to observe the early cold stages because they have relatively short lifetimes and also occur in very opaque molecular clouds. These early stages, however, can be probed by emission from molecular lines emitting at centimetre, millimetre, and sub-millimetre wavelengths. Our recent work clearly demonstrates that dense molecular clumps embedded in the filamentary "Infrared Dark Clouds" spawn high-mass stars, and these the clumps evolve as star-formation activity progresses within them. We have now identified hundreds of clumps in the earliest "pre-stellar" stage. Our MALT90 and RAMPS surveys reveal that these clumps are collapsing, confirming a prediction from "competitive accretion" models. New observations with the ALMA telescope demonstrate that turbulence--and not gravity--dominates the structure of "the Brick", the Milky Way's most massive "pre-stellar" clump.

Come join us for a discussion and public forum on 'Creativity & Mathematics' at Newcastle Museum on Monday, 1st August. We've lined up world leading experts from a diverse set of disciplines to shed some light on the connection between creativity and mathematics.

It's free, but please register for catering purposes. It begins at 6:30 pm with finger food and a chat before the forum itself gets under way at 7 pm.

The panel discussion and forum will have lots of audience involvement. The panel members are from a diverse group of disciplines each concerned in some way with the relationship between creativity and mathematics. Prof. John Wilson (The University of Oxford), a leading expert on group theory, is intrigued by the similarities between mathematicians finding new ideas and composers creating new music. Prof. George Willis (University of Newcastle) will talk about the creativity of mathematics itself. Prof. Michael Ostwald will spin gold around mathematical constraints and architectural forms. A/Prof. Phillip McIntyre is an international expert on creativity and author of The Creative System in Action. He has been described as having a mind completely unpolluted by mathematics!

Come along and enjoy an evening of mental stimulation and unexpected insights. You never know: participants might walk away with a completely different view of mathematics and its place in the world.

The standard height function $H(\mathbf p/q) = q$ of simultaneous approximation can be calculated by taking the LCM (least common multiple) of the denominators of the coordinates of the rational points: $H(p_1/q_1,\ldots,p_d/q_d) = \mathrm{lcm}(q_1,\ldots,q_m)$. If the LCM operator is replaced by another operator such as the maximum, minimum, or product, then a different height function and thus a different theory of simultaneous approximation will result. In this talk I will discuss some basic results regarding approximation by these nonstandard height functions, as well as mentioning their connection with intrinsic approximation on Segre manifolds using standard height functions. This work is joint with Lior Fishman.

Dr Simmons is a visitor of Dr Mumtaz Hussain.

Let $\Sigma_d^{++}(\R)$ be the set of positive definite matrices with determinant 1 in dimension $d\ge 2$. Identifying two $SL_d(\Z)$-congruent elements in $\Sigma_d^{++}(\R)$ gives rise to the space of reduced quadratic forms of determinant one, which in turn can be identified with the locally symmetric space $X_d:=SL_d(\Z)\backslash SL_d(\R)\slash SO_d(\R)$. Equip the latter space with its natural probability measure coming from the Haar measure on $SL_d(\R)$. In 1998, Kleinbock and Margulis established very sharp estimates for the probability that an element of $X_d$ takes a value less than a given real number $\delta>0$ over the non-zero lattice points $\Z^d\backslash\{ \bm{0} \}$.

This talk will be concerned with extensions of such estimates to a large class of probability measures arising either from the spectral or the Cholesky decomposition of an element of $\Sigma_d^{++}(\R)$. The sharpness of the bounds thus obtained are also established for a subclass of these measures.

This theory has been developed with a view towards application to Information Theory. Time permitting, we will briefly introduce this topic and show how the estimates previously obtained play a crucial role in the analysis of the perfomance of communication networks.

This is work joint with Evgeniy Zorin (University of York). Dr Adiceam is a visitor of Dr Mumtaz Hussain.

The finite element method is a very popular technique to approximate solutions of partial differential equations. The mixed finite element method is a type of finite element method in which extra variables are introduced in the formulation. This introduction is useful for some problems where more than one unknowns are desirable. In this research, we will apply the mixed finite element method for some applications, such as Poisson equation, elasticity equation, and sixth-order problem. Furthermore, we also utilise the mixed finite element method to solve linear wave equation which arises from real world problem.

The density of 1's in the Kolakoski sequence is conjectured to be 1/2. Proving this is an open problem in number theory. I shall cast the density question as a problem in combinatorics, and give some visualisations which may suggest ways to gain further insight into the conjecture.

I continue the discussion of the Erdos-Szekeres conjecture about points in convex position with an outline of the recent proof of an asymptotic version of the conjecture.

In 1935 Erdos and Szekeres proved that there exists a function f such that among f(n) points in the plane in general position there are always n that form the vertices of a convex n-gon. More precisiely, they could prove a lower and an upper bound for f(n) and conjectured that the lower bound is sharp. After 70 years with very limited progress, there have been a couple of small improvements of the upper bound in recent years, and finally last month Andrew Suk announced a huge step forward: a proof of an asymptotic version of the conjecture.
I plan two talks on this topic: (1) a brief introduction to Ramsey theory, and (2) an outline of Suk's proof.

Zero forcing number, Z(G), of a graph G is the minimum cardinality of a set S of black vertices (whereas vertices in V (G)\S are colored white) such that V (G) is turned black after finitely many applications of "the color-change rule": a white vertex is converted black if it is the only white neighbor of a black vertex.

Zero forcing number was introduced by the "AIM Minimum Rank – Special Graphs Work Group". In this talk, I present an overview of the results obtained from their paper.

In 2000, after investigating the published literature(for which I had reason then), I realised that there was clearly confusion surrounding the question of how WW2 Japanese army and navy codes had been broken by the Allies.

Fourteen years later, my academic colleague Peter Donovan and I understood why that was so: the archival documents needed to perform this task, plus the mathematical understanding needed to interpret correctly these documents, had only exposed themselves through our combined researches over this long period. The result, apart from a number of research publications in journals, is our book, "Code Breaking in the Pacific", published by Springer International in 2014.

Both the Imperial Japanese Army (IJA) and the Imperial Japanese Navy (IJN) used an encryption system involving a code book and then a second stage encipherment, a system which we call an additive cipher system, for their major codes – not a machine cipher such as the Enigma machines used widely by German forces in ww2 or the Typex/Sigaba/ECM machines used by the Allies. Thus, the type of attack needed to crack such a system is very different to those described in books about Bletchley Park and its successes against Enigma ciphers.

However, there is a singular difference: while the IJN’s main coding system, known to us as JN-25, was broken from its inception and throughout the Pacific War, yielding for example the intelligence information that enabled the battles of the Coral Sea and Midway to occur, or the shooting down of Admiral Yamamoto to be planned, the many IJA coding systems in use were, with one exception, never broken!

I will describe the general structure of additive systems, the rational way developed to attack them and its usual failure in practice, and the "miracle" that enabled JN-25 to be broken - probably the best-kept secret of the entire Pacific War: multiples of three! Good maths, but not highly technical!

Lehmer's famous question concerns the existence of monic integer coefficient polynomials with Mahler measure smaller than a certain constant. Despite significant partial progress, the problem has not been fully resolved since its formulation in 1933. A powerful result independently proven by Lawton and Boyd in the 1980s establishes a connection between the classical Mahler measure of single variable polynomials and the generalized Mahler measure of multivariate polynomials. This led to speculation that it may be possible to answer Lehmer's question in the affirmative with a multivariate polynomial although the general consensus among researchers today is that no such polynomial exists. We show that each possible candidate among two variable polynomials corresponding to curves of genus 1 can be bi-rationally mapped onto a polynomial with Mahler measure greater than Lehmer's constant. Such bi-rational maps are expected to preserve the Mahler measure for large values of a certain parameter.

Milutin is a completing Honours Student of Wadim Zudilin.

A metric generator is a set W of vertices of a graph G such that for every pair of vertices u,v of G, there exists a vertex w in W with the condition that the length of a shortest path from u to w is different from the length of a shortest path from v to w. In this case the vertex w is said to resolve or distinguish the vertices u and v. The minimum cardinality of a metric generator for G is called the metric dimension. The metric dimension problem is to find a minimum metric generator in a graph G. In this talk I am discussing about the metric dimension and partition dimension of Cayley (di)graphs.

This week I shall finish my discussion of sequenceable and R-sequenceable groups.

I am now refereeing a manuscript on the above and I’ll tell you about its contents.

Start by placing piles of indistinguishable chips on the vertices of a graph. A vertex can fire if it's supercritical; i.e., if its chip count exceeds its valency. When this happens, it sends one chip to each neighbour and annihilates one chip. Initialize a game by firing all possible vertices until no supercriticals remain. Then drop chips one-by-one on randomly selected vertices, at each step firing any supercritical ones. Perhaps surprisingly, this seemingly haphazard process admits analysis. And besides having diverse applications (e.g., in modelling avalanches, earthquakes, traffic jams, and brain activity), chip-firing reaches into numerous mathematical crevices. The latter include, alphabetically, algebraic combinatorics, discrepancy theory, enumeration, graph theory, stochastic processes, and the list could go on (to zonotopes). I'll share some joint work—with Dave Perkins—that touches on a few items from this list. The talk'll be accessible to non-specialists. Promise!

B. Gordon (1961) defined sequenceable groups and G. Ringel (1974) defined R-sequenceable groups. Friedlander, Gordon and Miller conjectured that finite abelian groups are either sequenceable or R-sequenceable. The preceding definitions are special cases of what T. Kalinowski and I are calling an orthogonalizeable group, namely, a group for which every Cayley digraph on the group admits either an orthogonal directed path or an orthogonal directed cycle. I shall go over the history and current status of this topic along with a discussion about the completion of a proof of the FGM conjecture.

Mapping class groups are groups which arise naturally from homeomorphisms of surfaces. They are ubiquitous: from hyperbolic geometry, to combinatorial group theory, to algebraic geometry, to low dimensional topology, to dynamics. Even to this colloquium!

In this talk, I will give a survey of some of the highlights from this beautiful world, focusing on how mapping class groups interact with covering spaces of surfaces. In particular, we will see how a particular order 2 element (the hyperelliptic involution) and its centraliser (the hyperelliptic mapping class group) play an important role, both within the world of mapping class groups and in other areas of mathematics. If time permits, I will briefly touch on some recent joint work with Rebecca Winarski that generalises the hyperelliptic story.

No experience with mapping class groups will be assumed, and this talk will be aimed at a general mathematics audience.

The discrepancy of a graph measures how evenly its edges are distributed. I will talk about a lower bound which was proved by Bollobas and Scott in 2006, and extends older results by Erdos, Goldberg, Pach and Spencer. The proof provides a nice illustration of the probabilistic method in combinatorics. If time allows I will outline how this stuff can be used to prove something about convex hulls of bilinear functions.

In this talk, I will outline my interest in, and results towards, the Erdős Discrepancy Problem (EDP). I came about this problem as a PhD student sometime around 2007. At the time, many of the best number theorists in the world thought that this problem would outlast the Riemann hypothesis. I had run into some interesting examples of some structured sequences with very small growth, and in some of my early talks, I outlined a way one might be able to attack the EDP. As it turns out, the solution reflected quite a bit of what I had guessed. And I say 'guessed' because I was so young and naïve that my guess was nowhere near informed enough to actually have the experience behind it to call it a conjecture. In this talk, I will go into what I was thinking and provide proof sketches of what turn out to be the extremal examples of EDP.

How confident are you in your choice? Such a simple but important question for people to answer. Yet, capturing how people answer this question has proven challenging for mathematical models of cognition. Part of the challenge has been that these models assume confidence is a static variable based on the same information used to make a decision. In the first part of my talk, I will review my dynamic theory of confidence, two-stage dynamic signal detection theory (2DSD). 2DSD is based on the premise that the same evidence accumulation process that underlies choice is used to make confidence judgments, but that post-decisional processing of information contributes to confidence judgments. Thus, 2DSD correctly predicts that the resolution of confidence judgments, or their ability to discriminate between correct and incorrect choices, increases over time. However, I have also found that the dynamics of confidence is driven by other factors including the very act of making a choice. In the second of the part of the talk, I will show how 2DSD and other models derived from classical stochastic theories are unable to parsimoniously account for this stable interference effect of choice. In contrast, quantum random walk modes of evidence accumulation account for this property by treating judgments and decisions as a measurement process by which a definite state is created from an indefinite state. In summary, I hope to show how better understanding the dynamic nature of confidence can provide new methods for improving the accuracy of people’s confidence, but also reveal new properties of the deliberation process including perhaps the quantum nature of evidence accumulation.

I'll continue to discuss Frankl's union-closed sets conjecture. In particular I'll present two possible approaches (local configurations and averaging) and indicate obstacles to proving the general case using these methods.

We report upon insights gained into the BMath through: "Conversations: BMath Experiences". This is a project that was initiated by the BMath Convener in collaboration with NUMERIC. We invited first year BMath students to semi-structured conversations around their experiences in their degree. We will be sharing general insights that we have gained into the BMath through the project.

Speakers: Mike Meylan, Andrew Kepert, Liz Stojanovski and Judy-anne Osborn.

I’m going to give a summary of research projects I have been involved in over my study leave; they represent a shared theme: retailing. The projects which I’m going to talk about are:

• Value of Mixed Bundling with Random Store Traffic (with E. Li and S. Lu, University of Sydney, Australia)
• Oligopolistic Contracting: Channel Coordination under Competition (with G. Gallego, Columbia University, US)
• Optimal Pricing and Demand Learning with Strategic Customers (with N. Ghorbaniyan and M. Modarres, Sharif University of Technology, Iran)
• Markdown Budgets in Fashion Retailing (with A. Sen, Bilkent University, Turkey)
• Food and Beverage Supply Chain Optimization (with R. Berretta, A. Eshragh, and R. Middleton, University of Newcastle, Australia)

Start labelling the vertices of the square grid with 0's and 1's with the condition that any pair of neighbouring vertices cannot both be labelled 1. If one considers the 1's to be the centres of small squares (rotated 45 degrees) then one has a picture of square-particles that cannot overlap.

This problem of "hard-squares" appears in different areas of mathematics - for example it has appeared separately as a lattice gas in statistical mechanics, as independent sets in combinatorics and as the golden-mean shift in symbolic dynamics. A core question in this model is to quantify the number of legal configurations - the entropy. In this talk I will discuss the what is known about the entropy and describe our recent work finding rigorous and precise bounds for hard-squares and related problems.

This is work together with Yao-ban Chan.

Peter Frankl's union-closed sets conjecture, which dates back to (at least) 1979, states that for every finite family of sets which is closed under taking unions there is an element contained in at least half of the sets. Despite considerable efforts the general conjecture is still open, and the latest polymath project is an attempt to make progress. I will give an overview of equivalent variants of the conjecture and discuss known special cases and partial results.

Model sets, which go back to Yves Meyer (1972), are a versatile class of structures with amazing harmonic properties. They are particularly relevant for mathematical quasicrystals. More recently, also systems such as the square-free integers or the visible lattice points have been studied in this context, leading to the theory of weak model sets. This talk will review some of the development, and introduce some of the concepts in the field.

We will review the (now classical) scheme of basic ($q$-) hypergeometric orthogonal polynomials. It contains more than twenty families; for each family there exists at least one positive weight with respect to which the polynomials are orthogonal provided the parameter $q$ is real and lies between 0 and 1. In the talk we will describe how to reduce the scheme allowing the parameters in the families to be complex. The construction leads to new orthogonality properties or to generalization of known ones to the complex plane.

The Degree/Diameter Problem for graphs has its motivation in the efficient design of interconnection networks. It seeks to find the maximum possible order of a graph with a given (maximum) degree and diameter. It is known that graphs attaining the maximum possible value (the Moore bound) are extremely rare, but much activity is focussed on finding new examples of graphs or families of graph with orders approaching the bound as closely as possible. This problem was first mention in 1964 and has its motivation in the efficient design of interconnection networks. A lot of great mathematician studied this problem and obtained some results but there still remain a lot of unsolved problems about this subject. Our regretted professor Mirka Miller has given a great expansion to this problems and a lot of new results were given by her and her students. One of the problem she was recently interested in, was the Degree/Diameter problem for mixed graphs i.e. graphs in which we allow undirected edges and arcs (directed edges).

Some new result about the upper bound of this Moore mixed graph has been obtained in 2015. So this talk consists on giving the main known results about those graphs.

In this presentation we address the issues and challenges for Future of Education and how Maplesoft is committed to offers Tools such as Möbius™ to handle these challenges. Möbius is a comprehensive online courseware environment that focuses on science, technology, engineering, and mathematics (STEM). It is built on the notion that people learn by doing. With Möbius, your students can explore important concepts using engaging, interactive applications, visualize problems and solutions, and test their understanding by answering questions that are graded instantly. Throughout the entire lesson, students remain actively engaged with the material and receive constant feedback that solidifies their understanding.

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An order picking system in a distribution center (DC) owned by Pep Stores Ltd. (PEP) the largest single brand retailer in South Africa is investigated. Twelve independent unidirectional picking lines situated in the center of the DC are used to process all piece picking. Each picking line consists of a number of locations situated in a cyclical formation around a central conveyor belt and are serviced by multiple pickers walking in a clockwise direction.

On a daily planning level three sequential decisions tiers exist and are described as:

These sub-problems are too complex to solve together and are addressed independently and in reverse sequence using mathematical programming and heuristic techniques. It is shown that the total walking distances of pickers may be significantly reduced when solving sub-problems 1 and 3 and that there is no significant impact when solving sub-problem 2. Moreover, by introducing additional work balance and small carton minimisation objectives into sub-problem 1 better trade-offs between objectives are achieved when compared to the current practice.

A diophantine m-tuple is a set of m-positive integers {a_1, . . . , a_m} such that the product of any two of them plus 1 is a square. For example, {1, 3, 8, 120} is a Diophantine quadruple found by Fermat. It is known that there are infinitely many such examples with m = 4 and none with m = 6. No example is known with m = 5 but if there exist, then there are only finitely many such. In my talk, I will survey what is known about this problem, as well as its variations, where one replaces the ring of integers by the ring of integers in some finite extension of Q, or by the field of rational numbers, or one looks at a variant of this problem in the ring of polynomials with coefficients in a field of characteristic zero, or when one replaces the squares by perfect powers of a larger exponent, or by members of some other interesting sequence like the sequence of Fibonacci numbers and so on.

This talk is devoted to three basic forms of the inverse function theorem. The classical inverse function theorem identify a smooth single-valued localization of the inverse on the condition of nonsingularity of the Jacobian.

I will explain what groups are and give some examples and applications.

At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. We extend the work of Bringmann and Ono, and Ahlgren and Treneer on answering this question.

Firstly, from [1] we consider a mixed formulation for an elliptic obstacle problem for a 2nd order operator and present an hp-FE interior penalty discontinous Galerkin (IPDG) method. The primal variable is approximated by a linear combination of Gauss-Lobatto-Lagrange(GLL)-basis functions, whereas the discrete Lagrangian multiplier is a linear combination of biorthogonal basis functions. A residual based a posteriori error estimate is derived. For its construction the approximation error is split into a discretization error of a linear variational equality problem and additional consistency and obstacle condition terms.

Secondly, an hp-adaptive $C^0$-interior penalty method for the bi-Laplace obstacle problem is presented from [2]. Again we take a mixed formulation using GLL-basis functions for the primal variable and biorthogonal basis functions for the Lagrangian multiplier and present also a residual a posteriori error estimate. For both cases (2nd and 4th order obstacle problems) our numerical experiments clearly demonstrate the superior convergence of the hp-adaptive schemes compared with uniform and h-adaptive schemes.

References
[1] L.Banz, E.P.Stephan, A posteriori error estimates of hp-adaptive IPDG-FEM for elliptic obstacle problems, Applied Numerical Mathematics 76,(2014) 76-92
[2] L.Banz, B.P.Lamichhane, E.P.Stephan, An hp-adaptive $C^0$-interior penalty method for the obstacle problem of clamped Kirchhoff plates, preprint (2015)

(Joint work with Lothar Banz, University Salzburg, Austria)

We'll answer the question "What's a wavelet?" and discuss continuous wavelet transforms on the line and connections with representation theory and singular integrals. The focus will then turn to discretization techniques, including multiresolution analysis. Matrix completion problems arising from higher-dimensional wavelet constructions will also be described.

I am going to discuss a construction of functional calculus $$f\mapsto f(A,B),$$ where $A$ and $B$ are noncommuting self-adjoint operators. I am going to discuss the problem of estimating the norms $\|f(A_1,B_1)-f(A_2,B_2)\|$, where the pair $(A_2,B_2)$ is a perturbation of the pair $(A_1,B_1)$.

The use of GPUs for scientific computation has undergone phenomenal growth over the past decade, as hardware originally designed with limited instruction sets for image generation and processing has become fully programmable and massively parallel. This talk discusses the classes of problem that can be attacked with such tools, as well as some practical aspects of implementation. A direction for future research by the speaker is also discussed.

We consider identities satisfied by discrete analogues of Mehta-like integrals. The integrals are related to Selberg’s integral and the Macdonald conjectures. Our discrete analogues have the form

where $\alpha,\beta,\delta,r,n$ are non-negative integers subject to certain restrictions.

In the cases that we consider, it is possible to express $S_{\alpha,\beta,\delta} (r,n)$ as a product of Gamma functions and simple functions such as powers of two. For example, if $1 \leq r \leq n$, then $$S_{2,2,3} (r,n) = \prod_{j=1}^r \frac{(2n)!j!^2}{(n-j)!^2}.$$

The emphasis of the talk will be on how such identities can be obtained, with a high degree of certainty, using numerical computation. In other cases the existence of such identities can be ruled out, again with a high degree of certainty. We shall not give any proofs in detail, but will outline the ideas behind some of our proofs. These involve $q$-series identities and arguments based on non-intersecting lattice paths.

This is joint work with Christian Krattenthaler and Ole Warnaar.

We consider the stability of a class of abstract positive systems originating from the recurrence analysis of stochastic systems, such as multiclass queueing networks and semimartingale reflected Brownian motions. We outline that this class of systems can also be described by differential inclusions in a natural way. We will point out that because of the positivity of the systems the set-valued map defining the differential inclusion is not upper semicontinuous in general and, thus, well-known characterizations of asymptotic stability in terms of the existence of a (smooth) Lyapunov function cannot be applied to this class of positive systems. Following an abstract approach, based on common properties of the positive systems under consideration, we show that asymptotic stability is equivalent to the existence of a Lyapunov function. Moreover, we examine the existence of smooth Lyapunov functions. Putting an assumption on the trajectories of the positive systems which demands for any trajectory the existence of a neighboring trajectory such that their difference grows linearly in time and distance of the starting points, we prove the existence of a $C^\infty$-smooth Lyapunov function. Looking at this hypothesis from the differential inclusions perspective it turns out that differential inclusions defined by Lipschitz continuous set-valued maps taking nonempty, compact and convex values have this property.

We will be answering the following question raised by Christopher Bishop:

'Suppose we stand in a forest with tree trunks of radius $r > 0$ and no two trees centered closer than unit distance apart. Can the trees be arranged so that we can never see further than some distance $V < \infty$, no matter where we stand and what direction we look in? What is the size of $V$ in terms of $r$?'

The methods used to study this problem involve Fourier analysis and sharp estimates of exponential sums.

A dimension adaptive algorithm for sparse grid quadrature in reproducing kernel Hilbert spaces on products of spheres uses a greedy algorithm to approximately solve a down-set constrained binary knapsack problem. The talk will describe the quadrature problem, the knapsack problem and the algorithm, and will include some numerical examples.

We discuss problems of approximation of an irrational by rationals whose numerators and denominators lie in prescribed arithmetic progressions. Results are both, on the one hand, from a metrical and a non-metrical point of view, and on the other, from an asymptotic and also a uniform point of view. The principal novelty of this theory is a Khintchine-type theorem for uniform approximation in this setup. Time permitting some applications of this work will be discussed.

I will talk a bit about the benefits of a regular outlook.

We study the family of self-inversive polynomials of degree $n$ whose $j$th coefficient is $\gcd(n,j)^k$, for a fixed integer $k \geq 1$. We prove that these polynomials have all of their roots on the unit circle, with uniform angular distribution. In the process we prove some new results on Jordan's totient function. We also show that these polynomials are irreducible, apart from an obvious linear factor, whenever $n$ is a power of a prime, and conjecture that this holds for all $n$. Finally we use some of these methods to obtain general results on the zero distribution of self-inversive polynomials and of their "duals" obtained from the discrete Fourier transforms of the coefficients sequence. (Joint work with Sinai Robins).

A motion which is periodic may be considered symmetric under a transformation in time. A measure of the phase relationship these motions have with respect to a geometric figure which is symmetric under some transformation in space is presented. The implications this has on discretised patterns generated is discussed. The talk focuses on theoretical formalisms, such as those which display the fractal patterns of 'strange attractors', rather than group theory for symmetric transformations.

We consider monotone systems defined by ODEs on the positive orthant in $\mathbb{R}^n$. These systems appear in various areas of application, and we will discuss in some level of detail one of these applications related to large-scale systems stability analysis.

Lyapunov functions are frequently used in stability analysis of dynamical systems. For monotone systems so called sum- and max-separable Lyapunov functions have proven very successful. One can be written as a sum, the other as a maximum of functions of scalar arguments.

We will discuss several constructive existence results for both types of Lyapunov function. To some degree, these functions can be associated with left- and right eigenvectors of an appropriate mapping. However, and perhaps surprisingly, examples will demonstrate that stable systems may admit only one or even neither type of separable Lyapunov function.

In scanning ptychography, an unknown specimen is illuminated by a localised illumination function resulting in an exit-wave whose intensity is observed in the far-field. A ptychography dataset is a series of these observations, each of which is obtained by shifting the illumination function to a different position relative to the specimen with neighbouring illumination regions overlapping. Given a ptychographic data set, the blind ptychography problem is to simultaneously reconstruct the specimen, illumination function, and relative phase of the exit-wave. In this talk I will discuss an optimisation framework which reveals current state-of-the-art reconstruction methods in ptychography as (non-convex) alternating minimization-type algorithms. Within this framework, we provide a proof of global convergence to critical points using the Kurdyka-Łojasiewicz property.

We use random walks to experimentally compute the first few terms of the cogrowth series for a finitely presented group. We propose candidates for the amenable radical of any non-amenable group, and a Følner sequence for any amenable group, based on convergence properties of random walks.

The Hardy and Paley-Wiener Spaces are defined due to important structural theorems relating the support of a function's Fourier transform to the growth rate of the analytic extension of a function. In this talk we show that analogues of these spaces exist for Clifford-valued functions in n dimensions, using the Clifford-Fourier Transform of Brackx et al and the monogenic ($n+1$ dimensional) extension of these functions.

Given a finite presentation of a group, proving properties of the group can be difficult. Indeed, many questions about finitely presented groups are unsolvable in general. Algorithms exist for answering some questions while for other questions algorithms exist for verifying the truth of positive answers. An important tool in this regard is the Todd-Coxeter coset enumeration procedure. It is possible to extract formal proofs from the internal working of coset enumerations. We give examples of how this works, and show how the proofs produced can be mechanically verified and how they can be converted to alternative forms. We discuss these automatically produced proofs in terms of their size and the insights they offer. We compare them to hand proofs and to the simplest possible proofs. We point out that this technique has been used to help solve a longstanding conjecture about an infinite class of finitely presented groups.

We survey the literature on orthogonal polynomials in several variables starting from Hermite's work in the late 19th century to the works of Zernike (1920's) and Ito (1950's). We explore combinatorial and analytic properties of the Ito polynomials and offer a general class in 2 dimensions which as interesting structural properties. Connections with certain PDE's will be mentioned.

We propose new path-following predictor-corrector algorithms for solving convex optimization problems in conic form. The main structural properties used in our design and analysis of the algorithms hinge on some key properties of a special class of very smooth, strictly convex barrier functions. Even though our analysis has primal and dual components, our algorithms work with the dual iterates only, in the dual space. Our algorithms converge globally at the same worst-case rate as the current best polynomial-time interior-point methods. In addition, our algorithm have the local superlinear convergence property under some mild assumptions. The algorithms are based on an easily computable gradient proximity measure, which ensures an automatic transformation of the global linear rate of convergence to the locally superlinear one under some mild assumptions. Our step-size procedure for the predictor step is related to the maximum step size (the one that takes us to the boundary).

This talk is based on joint work with Yu. Nesterov.

Lift-and-Project operators (which map compact convex sets to compact convex sets in a certain contractive way, via higher dimensional convex representations of these sets) provide an automatic way for constructing all facets of the convex hull of 0,1 vectors in a polytope given by linear or polynomial inequalities. They also yield tractable approximations provided that the input polytope is tractable and that we only apply the operators O(1) times. There are many generalizations of the theory of these operators which can be used, in theory, to generate (eventually, in the limit) arbitrarily tight, convex relaxations of essentially arbitrary nonconvex sets. Moreover, Lift-and-Project methods provide universal ways of applying Semidefinite Programming techniques to Combinatorial Optimization problems, and in general, to nonconvex optimization problems.

I will survey some of the developments (some recent, some not so recent) that I have been involved in, especially those utilizing Lift-and-Project methods and Semidefinite Optimization. I will touch upon the connections to Convex Algebraic Geometry and present various open problems.

In this talk we will begin with a brief history of the mathematics of aperiodic tilings of Euclidean space, highlighting their relevance to the theory of quasicrystals. Next we will focus on an important collection of point sets, cut and project sets, which come from a dynamical construction and provide us with a mathematical model for quasicrystals. After giving definitions and examples of these sets, we will discuss their relationship with Diophantine approximation, and show how the interplay between these two subjects has recently led to new results in both of them.

I will complete the proof of the Kemnitz conjecture and make some remarks about related zero-sum problems.

Supervisors: Mirka Miller, Joe Ryan and Andrea Semanicova-Fenovcikova

We give some background to the labeling schemes like graceful, harmonious, magic, antimagic and irregular total labeling. Then we will describe why study of graph labeling is important by narrating some applications of graph labeling. Next we will briefly describe the methodology like Robert's construction to obtain completing separating systems (CSS) which will help us to determine the antimagic labeling of graphs and Alon's Combinatorial Nullstellensatz. We will illustrate an example from many applications of graphs labelling. Finally we will introduce reflexive irregular total labelling and explain its importance. To conclude, we add research plan and time line during candidature of research.

After briefly describing a few more simple applications of Alon's Nullstellensatz, I will present in detail Reiher's amazing proof of the Kemnitz conjecture regarding lattice points in the plane.

Noga Alon's Combinatorial Nullstellensatz, published in 1999, is a statement about polynomials in many variables and what happens if one of these vanishes over the set of common zeros of some others. In contrast to Hilbert's Nullstellensatz, it makes strong assumptions about the polynomials it is talking about, and this leads a tool for producing short and elegant proofs for numerous old and new results in combinatorial number theory and graph theory. I will present the proof of the algebraic result and some of the combinatorial applications in the 1999 paper.

Stability analysis plays a central role in nonlinear control and systems theory. Stability is, in fact, the fundamental requirement for all the practical control systems. In this research, advanced stability analysis techniques are reviewed and developed for discrete-time dynamical systems. In particular, we study the relationships between the input-to-state stability related properties and l¬2-type stability properties. These considerations naturally lead to the study of input-output models and, further, to questions of incremental stability and convergent dynamics. Future work will also outline several applications scenario for our theories including observer analysis and secure communication.

Supervisors: A/Prof. Christopher Kellett and Dr. Björn Rüffer

Existing of perfect matchings in regular graph is a fundamental problem in graph theory, and it closely model many real world problems such as broadcasting and network management. Recently, we have studied the number of edge disjoint perfect matching in regular graph, and using some well-known results on the existence of perfect matching and operations forcing unique perfect matchings in regular graph, we are able to make some pleasant progress. In this talk, we will present the new results and briefly discuss the proof.

In either the inviscid limit of the Euler equations, or the viscously dominated limit of the Stokes equations, the determination of fluid flows can be reduced to solving singular integral equations on immersed structures and bounding surfaces. Further dimensional reduction is achieved using asymptotics when these structures are sheets or slender fibers. These reductions in dimension, and the convolutional second-kind structure of the integral equations, allows for very efficient and accurate simulations of complex fluid-structure interaction problems using solvers based on the Fast Multipole or related methods. These representations also give a natural setting for developing implicit time-stepping methods for the stiff dynamics of elastic structures moving in fluids. I'll discuss these integral formulations, their numerical treatment, and application to simulating structures moving in high-speed flows (flapping flags and flyers), and for resolving the complex interactions of many, possibly flexible, bodies moving in microscopic biological flows.

Partitioning is a basic fundamental technique in graph theory. Graph partitioning technique is used widely to solve several combinatorial problems. We will discuss the role of edge partitioning techniques on graph embedding. The graph embedding includes some combinatorial problems such as bandwidth problem, wirelength problem, forwarding index problem etc and in addition includes some cheminformatics problems such as Wiener Index, Szeged Index, PI index etc. In this seminar, we study convex partition and its characterization. In addition, we also analyze the relationship between convex partition and some other edge partitions such as Szeged edge partition and channel edge partition. The graphs that induce convex partitions are bipartite. We will discuss the difficulties in extending this technique to non-bipartite graphs.

This completion talk is in two parts. In the first part, I will present a characterisation of the cyclic Douglas-Rachford method's behaviour, generalising a result which was presented in my confirmation seminar. In the second part, I will explore non-convex regularity notions in an application arising in biochemistry.

Amenability is of interest for many reasons, not least of which is its paradoxical decomposition into so many various characterisations, each equal to the whole. Two of these are the characterisation in terms of the cogrowth rate, and the existence of a Følner sequence. In exploring a known method of computing the cogrowth rate using a random walk, and by analyzing which groups seem to be pathological for this algorithm, we discover new connections between these properties.

Mathematicians sometimes speak of the beauty of mathematics which to us is reflected in proofs and solutions for the most part. I am going to give a few proofs that I find very nice. This is stuff that post-grad discrete students certainly should know exists.

Functions that are piecewise defined are a common sight in mathematics while convexity is a property especially desired in optimization. Suppose now a piecewise-defined function is convex on each of its defining components – when can we conclude that the entire function is convex? Our main result provides sufficient conditions for a piecewise-defined function f to be convex. We also provide a sufficient condition for checking the convexity of a piecewise linear-quadratic function, which play an important role in computer-aided convex analysis.

Based on joint work with Heinz H. Bauschke (Mathematics, UBC Okanagan) and Hung M. Phan (Mathematics, University of Massachusetts Lowell).

I'll give an overview of some recent developments in the theory of groups of automorphisms of trees which are discrete in the full automorphism group of the tree and are locally-transitive. I'll also mention some questions which have been provoked by this work.

We generalize the Burger-Mozes universal groups acting on regular trees by prescribing the local action on balls of a given radius, and study the basic properties of this construction. We then apply our results to prove a weak version of the Goldschmidt-Sims conjecture for certain classes of primitive permutation groups.

In recent years, there has been quite a bit of interest in generalized Fourier transforms in Clifford analysis and in particular for the so-called Clifford-Fourier transform.

In the first part of the talk I will provide some motivation for the study of this transform. In the second part we will develop a new technique to find a closed formula for its integral kernel, based on the familiar Laplace transform. As a bonus, this yields a compact and elegant formula for the generating function of all even dimensional kernels.

Managing railway in general and high speed rail in particular is a very complex task which involves many different interrelated decisions in all three strategic, tactical, and operational phases. In this research two different mixed integer linear programing models are presented which are the literature's first models of their kind. In the first model a single line with two different train types is considered. In the second model a cyclic train timetabling and platforming assignment problems are considered and solved to optimality. For this model, methods for obtaining bounds on the first objective function are presented. Some pre-processing techniques to reduce the number of decision variables and constraints are also proposed. Proposed models' objectives are to minimize (1) the cyclic length, called Interval, and (2) the total journey time of all trains dispatched from their origin in each cycle. Here we explicitly consider the minimization of the cycle length using linear constraints and linear objective function. The proposed models are different from and faster than the widely-used Period Event Scheduling Problem (PESP).

This will be an informal talk from our UoN Engineering colleague Prof Bill McBride who recently visited some "Mid-West" Universities in the USA. Prof McBride will discuss what he saw and learnt, with reference to first year maths teaching for Engineering students.

Advantages of EEG in studying brain signals include excellent temporal localization and, potentially, good spatial localization, given good models for source localization in the brain. Phase synchrony and cross-frequency coupling are two phenomena believed to indicate cooperation of different brain regions in cognition through messaging via different frequency bands. To verify these hypotheses requires ability to extract time-frequency localized components from complex multicomponent EEG data. One such method, empirical mode decompositions, shows increasing promise through engineering and we will review recent progress on this approach. Another potential method uses bases or frames of optimally time-frequency localized signals, so-called prolate spheroidal wave functions. New properties of these functions developed in joint work with Jeff Hogan will be reviewed and potential applications to EEG will be discussed.

Arising originally from the analysis of a family of compressed sensing matrices, Ian Wanless and I recently investigated a number of linear algebra problems involving complex Hadamard matrices. I will discuss our main result, which relates rank-one submatrices of Hadamard matrices to the number of non-zero terms in a representation of a fixed vector with respect to two unbiased bases of a finite dimensional vector space. Only a basic knowledge of linear algebra will be assumed.

Computers are changing the way we do mathematics, as well as introducing new research agendas. Computational methods in mathematics, including symbolic and numerical computation and simulation, are by now familiar. These lectures will explore the way that "formal methods," based on formal languages and logic, can contribute to mathematics as well.

In the 19th century, George Boole argued that if we take mathematics to be the science of calculation, then symbolic logic should be viewed as a branch of mathematics: just as number theory and analysis provide means to calculate with numbers, logic provides means to calculate with propositions. Computers are, indeed, good at calculating with propositions, and there are at least two ways that this can be mathematically useful: first, in the discovery of new proofs, and, second, in verifying the correctness of existing ones.

The first goal generally falls under the ambit of "automated theorem proving" and the second falls under the ambit of "interactive theorem proving." There is no sharp distinction between these two fields, however, and the line between them is becoming increasingly blurry. In these lectures, I will provide an overview of both fields and the interactions between them, and speculate as to the roles they can play in mainstream mathematics.

I will aim to make the lectures accessible to a broad audience. The first lecture will provide a self-contained overview. The remaining lectures are for the most part independent of one another, and will not rely on the first lecture.

In this colloquium-style presentation I will describe these combinatorial objects and how they relate to each other. Time permitting, I will also show how they can be used in other areas of Mathematics. Joint work with Sooran Kang and Samuel Webster.