Beauty in Mathematics

Teaching Philosophy

Talks

Curriculum Vitae

Home

Mathematician, Doctoral Candidate

CARMA Priority Research Centre

University of Newcastle

About Me

I study at the *Computer-Assisted Research Mathematics and its Applications* Priority Research Centre. It was founded by my adviser, Jon Borwein, who invited me to come to Australia to work with him. Jon passed away in August
of 2016; he was a wonderful man, and he is greatly missed.

A memorial page has been set up in his memory, and my personal reflections on my time with him are located
here .

I am continuing his work together with Heinz Bauschke, Brailey Sims, and Bishnu Lamichhane.

My principal area of focus is optimisation.

Beauty in Mathematics

Teaching Philosophy

Talks

Curriculum Vitae

Home

Scott B. Lindstrom

______________________________________________________________________________

M.S. in Mathematics, Portland State University, Portland, OR. June 2015.

Advisor: N.M. Nam

B.A. in Psychology, Central Washington University, Ellensburg, WA. June 2008 .

Studied at University of Stellenbosch, Stellenbosch, Western Cape, South Africa, 2007.

PAST POSITIONS

Instructor at Fariborz Maseeh Department of Mathematics and Statistics. Portland State University: 2014-2015.

I was the instructor of record for classes ranging from college algebra to the Calculus level for which I created all lecture notes, assignments, and exams. I assessed all student performance and assigned all marks and final grades. I also held regular office hours, conducted special review sessions outside of regular class time, and assisted students at the math department tutoring station.

RESEARCH

My main area of research is optimisation. In particular, I am studying non-expansive maps through the lens of experimental mathematics. I work on both non-convex feasibility problems - in particular the Douglas-Rachford method - and in the convex setting also. Other areas I have published in - or am presently working in - include differential equations, number theory, special functions, differential geometry, and math education. I particularly enjoy developing methods for visualizing mathematics, both for experimental and for educational ends.

PUBLICATIONS

Jonathan M. Borwein, Neil J. Calkin, Scott B. Lindstrom, and Andrew Mattingly
.” *Journal of Experimental Mathematics*, (2016), issue 4 (T&F Online).

Jonathan M. Borwein and Scott B. Lindstrom. “Meetings With Lambert W and Other Special Functions in Optimization and Analysis.”
*Pure and Applied Functional Analysis*, (2017) 1(3), p.361. //www.ybook.co.jp/online2/oppafa/vol1/p361.html.

Scott B. Lindstrom, Brailey Sims, Matthew Skerritt. "Computing Intersections of Implicitly Specified Plane Curves." To appear in *Journal of Nonlinear and Convex Analysis*.

Jonathan M. Borwein, Scott B. Lindstrom, Brailey Sims, Matthew Skerritt, Anna Schneider. " Dynamics of the Douglas-Rachford Method for Ellipses and p-Spheres." Submitted to Set Valued and Variational Analysis.

WORKS SUBMITTED FOR PUBLICATION

Scott B. Lindstrom and Paul Vrbik. "*Phase Portraits of Hyperbolic Geometry*," submitted
.

Heinz Bauschke, Scott B. Lindstrom, and Minh N. Dao. "Regularizing with Bregman-Moreau Envelopes." Submitted
.

Bishnu Lamichhane, Scott B. Lindstrom, and Brailey Sims. "Application of Projection Algorithms to Differential Equations: Boundary Value Problems." Submitted to *ANZIAM Journal*.

WORKS IN PREPARATION

With Reinier Diaz-Millan and Vera Roschina, “Approximate Douglas-Rachford Method and Quasi-Nonexpansive Operators.”

With Heinz Bauschke and Minh Dao, “On the Douglas-Rachford Algorithm."

With Heinz Bauschke, “Meetings with Lambert W and other special functions in Optimization and Analysis II: Proximal Averages.”

With Brailey Sims, “A Survey of Douglas-Rachford Method for Nonconvex Curves and Surfaces.”

TALKS AND PRESENTATIONS

FURTHER INFORMATION

Languages : English and Spanish.

Computation : C, Maple, Cinderella, Geogebra, LaTex, Sonar, FL Studio.

Extra-Academic Interests: Composition (science fiction and music), audio production.

Country of Citizenship: United States.

Beauty in Mathematics

Teaching Philosophy

Talks

Curriculum Vitae

Home

*"A heavy warning used to be given [by lecturers] that
pictures are not rigorous; this has never had its bluff called and has
permanently frightened its victims into playing for safety. Some pictures, of
course, are not rigorous, but I should say most are (and I use them whenever
possible myself)."* - J. E. Littlewood (1885-1977)

'From Littlewood's Miscellany (p 35 in 1953 edition). Said long before the current graphic, visualization and geometric tools were available.' Accessed at Jonathan Borwein's webpage, comment added by Jon.

I am especially interested in putting the pictures back into mathematics. I've witnessed nothing more damaging to mathematics as a discipline and no worse assault on the sensibilities of our students than the proliferation of - I would say bad - texts which teach only through symbol those concepts which were first discovered geometrically. Certainly geometry has its limitations, and certainly symbol becomes easier once the geometry is understood. Certainly the symbolic representation of concepts must not be neglected. However, it is unfair for us to expect our students to learn through a purely symbolic proof that which we ourselves learned through geometry.

When we fail to introduce our students to geometrically intuitive concepts through the visual avenues by which they were first appreciated, we condemn our students either to memorize strategies for deriving symbolic solutions or to outright failure. The consequences go deeper still. When we devalue graphical intuition in favor of purely symbolic methods, we communicate to our students that such a method of teaching is acceptable for them to employ also. Is it any wonder that we now have students entering our college math courses unable to do basic arithmetic because they never developed a conceptual understanding of what fractions represent?

The most obvious contributor to this problem is our natural tendency to forget how we first learned something, how difficult it was for us to read mathematical symbols when first we encountered them. We first encountered them coupled together with their geometric meanings or with their explicitly written expansions. When we present the shortest, cleanest version of a result to our students, we are almost certain to lose them. What seems beautifully concise to us is impossibly compact for them.

*“Mathematics, rightly viewed, possesses not only truth, but
supreme beauty—a beauty cold and austere, like that of sculpture, without
appeal to any part of our weaker nature, without the gorgeous trappings of
painting or music, yet sublimely pure, and capable of a stern perfection such
as only the greatest art can show.” -Bertrand Russel (A History of Western
Philosophy)*

This problem has been compounded by many now-deeply-rooted mathematical traditions, most notably the tendency to turn course notes into published manuscripts. Course notes have historically been devoid of pictures, and for good reason: drawing the pictures is the lecture! We create course notes with the symbols so that we may spend our class time focusing on discussion of the geometric understanding. When we then publish such notes as a textbook, we have essentially taken a the complete exploration of a mathematical subject and systematically removed all of the most important components necessary for understanding it, leaving only empty language and symbol.

If you are skeptical that a core cause of our cultural fear of math, is in fact, a dearth of pictures, consider the following exchange I once had. A student - not mine I hasten to add - once complained to me that "The university system is unfairly biased against artistic students; it requires students with artistic minds to take math courses, but it does not require students with mathematical minds to take art classes." The student was apparently frustrated about the entire one mathematics class required by most liberal arts institutions in order to receive a bachelor's degree. Could a student with even a rudimentary understanding of what mathematics is ever entertain such a false dichotomy between mathematics and artistry? More concerning is this: if a university student can believe that only some people have logical minds and are capable of reason, then the educational system is failing in its most important responsibility, that of fostering critical thinking.

*"Art challenges technology, and technology inspires the
art." - John Lasseter on the foundation of Pixar*

In this regard technology may be our saving grace. The advent of the age of personal computing and the world wide web has irreversibly changed the way we teach. It must and will continue to do so. Tools like Maple TA, endeavors by academicians to make lectures available online, and innovations from private organizations like Khan Academy have placed a multi-dimensional exploration of mathematics at our students' fingertips. Finally, of course, there is our collaboration. I've amassed a good collection of course notes with visual components and activities. Contact me for more information or to share your own.

In this regard technology may be our saving grace. The advent of the age of personal computing and the world wide web has irreversibly changed the way we teach. It must and will continue to do so. Tools like Maple TA, endeavors by academicians to make lectures available online, and innovations from private organizations like Khan Academy have placed a multi-dimensional exploration of mathematics at our students' fingertips. Finally, of course, there is our collaboration. I've amassed a good collection of course notes with visual components and activities. Contact me for more information or to share your own.

Beauty in Mathematics

Teaching Philosophy

Talks

Curriculum Vitae

Home

Beauty in Mathematics

Teaching Philosophy

Talks

Curriculum Vitae

Home

**Click on a talk for Abstract and Details**

**Upcoming Events**

I will present at South Pacific Optimization Meeting, Curtin University, Perth, Australia 8-10 December 2017.

I will present at the 61st annual meeting of Australian Mathematical Society, Sydney, Australia 12-15 December 2017.

**Past Talks**Conference Talk: "Phase Portraits of Hyperbolic Geometry," Jonathan M. Borwein Commemorative Conference, Newcastle, Australia 25-29 September 2017.

Workshop Talk: "Dynamics of the Douglas-Rachford method for Ellipses and p-Spheres," West Coast Optimization Meeting, Vancouver, BC, Canada, 16 September 2017.

Workshop: "Douglas Rachford for ODEs." WOMBAT, Melbourne, Australia, November 25, 2016.

Thesis Presentation: “Understanding the Quasi-Relative Interior.” Portland State University, Portland, OR, USA, June 2015.

Math Club: “Fenchel Duality.” Portland State University, Portland, OR, USA, April 2015.'

This page contains a sampling of some of the mathematical beauty I have discovered in my research. I use the words mathematical beauty with great intention. The phenomena contained in this page were not created by me and my collaborators, although some have been colored or framed for aesthetic reasons. Rather, the phenomena were discovered. They have always existed. We merely create pictures to represent the phenomena because the pictures help us to learn about them. To that end, we can claim their discovery and framing but we cannot claim them. Much as an explorer might feel who stumbles upon a beautiful mountain and sketches it in her diary, we create representations, but the real beauty they endeavor to display is the work of some other creator. We hope you find them as lovely as we do.

This image is from a forthcoming work on the Douglas Rachford algorithm applied to non-convex sets. Differently colored dots correspond to unique sequences of iterates started at different places in real 2-space. The solutions for the feasibility problem are the two feasible points where the line intersects with the ellipse. Depending upon the starting location, sequences may converge to these solutions as the two blue sequences do on the far left and far right. However, if the algorithm starts elsewhere, sequences may be pulled into attractive instances of what we are calling basins of periodicity which prevent them from converging to the solution. As far as we are aware, the upper limit on the periodicity of points is limited only by how much we stretch out the ellipse and line. Pictures like this are extremely valuable for studying how small changes to a problem (such as stretching a sphere into an ellipse) can cause drastic changes to the behavior of a simple algorithm. They also illustrate what kinds of things can go "wrong." We find the patterns themselves to be aesthetically beautiful. Zooming in, we find lovely swirls and stars for subsequences converging to periodic points. We explore their unique and fascinating shapes in higher definition and greater detail in the forthcoming paper. This particular image was created using Cinderella. Colors for non-convergent sequences were chosen by taking an RGB color wheel with secondary and tertiary colors (12 colors total) and rotating by 5 colors for each new sequence. This method of choosing a value near to 1/3 of the total number but relatively prime to it ensures that colors are distinct enough for the eye to tell them apart. This image appears on the poster for the Australian Mathematical Society's special interest group Mathematics of Computation and Optimization (MoCaO).

These images were created as part of a collaboration with my advisor Jonathan Borwein, Brailey Sims, Matt Skerritt, and Anna Schneider.

This picture pictures the same feasibility problem but uses different methods to do so. Matt Skerritt adapted my Cindyscript algorithm to run on a graphics processing unit and output display differently. Rather than showing individual iterates, we run the Douglas Rachford algorithm starting from each individual pixel in the plane. We compute the first one thousand iterates before coloring the starting points according to which periodic point (or feasible point) their one thousandth iterate was nearest to. We used the first image - and the Cinderella app used to construct it - in order to roughly approximate where the periodic points lie and construct a list to check against. Colors were chosen by Matt Skerritt and inspired by Australian aboriginal artwork. This image illustrates, among other things, the importance of checking hypotheses with multiple methods of visualization. While the swirling patterns shown above might intuitively suggest curved basins, those which emerge in this image appear more polyhedral than one might expect. Works like this also highlight the importance of parallel computing. This picture is the main poster image for Australian Mathematical Society special interest group Mathematics of Computation and Optimization (MoCaO).

These images were created as part of a collaboration with my advisor Jonathan Borwein, Brailey Sims, Matt Skerritt, and Anna Schneider.

These images show symmetrical rotations on hyperbolic space. The intrinsic geometry of hyperbolic space is actualized in the extrinsic geometry of a pseudosphere. By slicing the pseudosphere and twisting it downwards, we obtain Dini's surface,
a representation which allows us to see the pseudosphere wrapping around more than one time. The colors all meet together beyond the pseudosphere's rim at a "point at infinity" corresponding to the preimage of another point at infinity
located at the "bottom" point of Dini's surface, infinitely far out of view down the stem of the visible surface. Unique colors highlight curves which are geodesics in hyperbolic space, the shortest paths along the surface between points,
analogous to lines. These images are from a collaborative work with Paul Vrbik and appeared on the poster voted most popular at the University of Newcastle MAPS (Mathematical and Physical Sciences) event "A Night of MAPS and Art."

The image at left exploits a different representation of hyperbolic space to reveal the behavior of a translation on the space. The representation is known as the Poincaré disc and is closely related to the upper half of the complex plane
via the conformal map of inversion in a circle centered at -i with radius root(2). We exploit these connections by mapping out to complex space, applying the function we care about, and then using a sequence of other functions to assign
unique colors to one set of geodesics in hyperbolic space while assigning unique heights to another distinct set of geodesics. Interestingly, both of these coloring techniques are examples of a more classical idea known as "phase plotting"
which has been used to visualize and teach complex analysis for some time.

The first of these final images (at left) shows another translation on hyperbolic space. However, in juxtaposition with the previous image, we have swapped the roles of the two phase plotting maps so that the coloring map now determines the
plot height while the height map determines the color. The final image shows yet another rotation on hyperbolic space.