Integrating Research and Illustration in Number Theory

Contribution List

23. Welcome Info

3/23/26, 9:50 AM

Welcome & important info & announcements.

1. The power of pictures

Andrew Sutherland (Massachusetts Institute of Technology)

3/23/26, 10:00 AM

3. Lift and patterns in the Spectre tilings

Arnaud Chéritat (CNRS/Institut de Mathéamtiques de Toulouse)

3/23/26, 10:30 AM

2. What Color Should This Pixel Be?

Roice Nelson

3/23/26, 11:30 AM

Often mathematical illustration boils down to drawing digital images, which itself is ultimately setting colors of individual pixels. I'm going to share escapades in the context of this reductionistic perspective. We'll discuss how the particular color choice for a pixel can have big effects, how much image sizes limit us, and weirder questions like choosing pixel shapes. Technology really...

4. Entangling Numbers and Knots via Sphere Packings

Iván Rasskin (Aix-Marseille Université)

3/23/26, 12:00 PM

Rational tangles were introduced by Conway as a family of tangles that are in bijection to rational numbers. This correspondence can be described algebraically by relating the construction of rational tangles to the operations appearing in continued fraction expansions. In this talk, we will explore different ways to visualize this connection through some generalizations of integral Apollonian...

5. Interactivity, Fidelity, and Polish: Building Modern Math Illustration

Cruz Godar (Yale University)

3/24/26, 9:30 AM

Visual representations of mathematical concepts are as old or older than the subject itself, but the use of technology as a medium remains nascent in comparison. The rapidly changing environment of screen sizes, input methods, and computational abilities presents a challenging space to enter, but also offers one of the most compelling and effective ways to communicate mathematical concepts...

6. Visualizing Elliptic Curves over Finite Fields

Stephen Trettel (University of San Francisco)

3/24/26, 10:00 AM

Elliptic curves over finite fields are central to modern number theory and cryptography, yet they are rather difficult to visualize — their points form finite sets without obvious geometric structure, and the group law that makes them so useful is obscured in most pictures. This is quite different from the more familiar story over the complex numbers, where geometry runs the show: every...

7. Sonification in number theory: listening to the Riemann zeta function

Jonathan Love (Leiden University)

3/24/26, 10:30 AM

When given a complicated function, one standard technique is to compute a Fourier transform, decomposing the function into simple components that are easier to analyze. On the other hand, the structure of the human ear allows it to mechanistically identify Fourier coefficients of incoming sound waves. We can use this ability to our advantage: we may be able to hear properties of functions that...

8. Rational points on the sphere

Claire Burrin

3/24/26, 11:30 AM

I will discuss some recent work on the distribution of rational points on the unit sphere and related conjectures.

9. Connectivity of Markoff and Nielsen graphs

Daniel Martin (Clemson University)

3/24/26, 2:30 PM

The generalized Markoff equation gives rise to a dynamical system via the Markoff group action on its solution set. Over finite fields, the action produces graphs that are conjectured by Bourgain, Gamburd, and Sarnak to form an expander family. This conjecture has implications in both number theory (strengthening the affine linear sieve for Markoff numbers) and computational group theory...

10. Dots and Laurent Series

Jayadev Athreya (University of Washington)

3/25/26, 9:30 AM

We'll describe some ongoing discussions and visuals around a beautiful example of Francois Ledrappier, known as the 3-dots example. We'll explain the connection between this example and Laurent series over finite fields, which we learned from Doug Lind, and explain some ongoing work on understanding periodic objects with Aaron Abrams, Edmund Harriss, and Glen Whitney.

11. The mathematics of making pictures of lattices

Edmund Harriss

3/25/26, 10:00 AM

In engineering a problem is considered and then tools lined up to solve it. In the artisan tradition of craft the tool is asked what it can do. Just as mathematics is often a curiosity driven research, different ways of experiencing mathematics can reveal different aspects of it, and we might not know what might be revealed until we have explored the ideas. Mathematics is often involved in...

12. A dichotomy in the tail behaviour of quadratic Weyl sums

Francesco Cellarosi (Queen's University), Francesco Cellarosi (Queen's University)

3/25/26, 10:30 AM

Jointly with Tariq Osman, we completed the classification of the tail behaviour of the limiting distributions of all quadratic Weyl sums of the form 1/\sqrt{N} \sum_{n=1}^N e( ((1/2)n^2+\beta n)x+\alpha n).
When \alpha and \beta are both rational, while trying to understand the contribution of certain orbits to the heavy tails, we discovered that some pairs actually lead to a compactly...

13. A Few Attempts at Visualizing Shimura Varieties

Sean Gonzales (UC Berkeley)

3/25/26, 11:30 AM

Shimura varieties are notoriously complex objects; the very definition of a Shimura variety is typically avoided in a research presentation, lest the entire talk is eaten up by the details. In this talk, I will share some of my attempts at visualizing Shimura varieties, ranging from the modular curve to higher dimensional Shimura varieties in characteristic p. Prior exposure to Shimura...

16. Basic illustrations associated with continued fractions

Mr PIERRE ARNOUX (Université d'Aix-Marseille)

3/26/26, 9:30 AM

I will propose a number of illustrations, many of them very elementary, and many of them of a dynamical nature, which have guided my research on continued fractions for many years.

15. A Selection from Forty Years of Illustrating Mathematical Results

Robert Corless (Western University)

3/26/26, 10:00 AM

I will touch on several topics that have interested me over the years, including illustrations of Carlsson's 1907 theorem and Baker & Rippon's resolution of the gap remaining, which together solve Condorcet's problem about the convergence of the iterated exponential. I will mention Mandelbrot polynomials and matrices, fractal eigenvectors, and Bohemian matrices. I will give one small theorem...

14. Aperiodic tilings and polygonal partitions of the torus

Sébastien Labbé

3/26/26, 10:30 AM

De Bruijn proved in the early 1980's that Penrose aperiodic
tilings can be constructed from a method based on multigrids. As
observed by Moody and Lagarias in the 1990's, this method, now known as
cut and project scheme, was originally formalized by Meyer in 1970's. A
cut and project scheme includes a physical space (the space we want to
tile) and an internal space (an additional helpful...

17. Classifying integer hypertilings

Ian Short (The Open University)

3/26/26, 11:30 AM

In 2018 Demonet et al observed that there is essentially only one three-dimensional positive integer tiling with the property that each cross-section is an SL2-tiling (in which all squares have determinant 1). Inspired by Bhargava’s work on binary quadratic forms, we describe a model for the more general class of three-dimensional integer tilings, which we call hypertilings, with the property...

18. Initial regularity in the landscapes of L-functions

Sally Koutsoliotas (Bucknell University)

3/26/26, 2:30 PM

The collection of all L-functions can be visualized as
points in euclidean spaces called landscapes. The first
few points in each landscape appear to lie close to an
arithmetic progression. Visualizations of the L-functions
and their zeros led to an explanation for this initial regularity.

19. Lattice Models: How Physics Illustrates/Inspires Number Theory

Claire Frechette (Umeå University)

3/27/26, 9:30 AM

Lattice models were originally used to study molecule interactions like temperature and potential energy, but in their abstract form as colored labelled graphs, they can be used to study many functions arising from algebra and number theory and suggest interesting identities through diagrammatic proofs. In this talk, we'll introduce lattice models and see how they can be used to investigate...

21. Finite capture and the closure of roots of restricted polynomials

Bernat Espigule (Universitat de Girona)

3/27/26, 10:00 AM

Fix n and consider the roots of polynomials whose coefficients are integers with absolute value at most n−1. Taken over all degrees, these roots form a countable set of algebraic numbers, but their closure has a striking fractal geometry. I will explain how, after a reciprocal-power-series reformulation, the problem becomes a connectedness question for a family of self-similar sets. The key...

20. Unintentional illustration

Anna Felikson (Durham University)

3/27/26, 10:30 AM

I will report on two stories when illustration happened without my conscious participation. The first story is elementary and concerns tilings on the plane. The second one is about friezes on surfaces (a generalization of Conway-Coxeter's frieze patterns) and hyperbolic geometry. This second story is based on the joined work with Pavel Tumarkin (see arXiv:2410.13511).

22. Implicitization in Real Time

Gabriel Dorfsman-Hopkins (St. Lawrence University)

3/27/26, 11:30 AM

A cubic surface can be realized as the projective plane blown up at six points, and the twenty-seven lines on the surface can be deduced from the coordinates of those six points. This relationship is practically begging for a dynamic visualization, allowing a user to drag around the six points and watch the associated cubic surface (and lines) deform. Building this visualization requires...