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.