Integrating Research and Illustration in Number Theory

23 mars 2026, 09:00 → 27 mars 2026, 18:00 Europe/Paris

Amphithéâtre Hermite (Institut Henri Poincaré)

Illustration as a Mathematical Research Technique

Workshop 3: Integrating Research and Illustration in Number Theory

March 23 to 27, 2026 - IHP, Paris

Summary

This workshop will highlight recent research developments in algebra and number theory informed by illustration.  It will also provide structured opportunities for participants to engage in new research collaborations that employ illustration.  The workshop schedule will include research talks as well as time set aside to establish these collaborations.

Registration is free but mandatory

Participants should register using the "Registration" link the the left menu. (The registration form is the same for the trimester and for the workshops.)

List of speakers:

• Jayadev Athreya
• Pierre Arnoux
• Claire Burrin
• Francesco Cellarosi
• Arnaud Chéritat
• Robert M. Corless
• Gabe Dorfsman-Hopkins
• Bernat Espigule
• Anna Felikson
• Claire Frechette
• Cruz Godar
• Sean Gonzalez
• Edmund Harriss
• Sally Koutsoliotas
• Sébastien Labbé
• Jonathan Love
• Daniel Martin
• Roice Nelson
• Ivan Rasskin
• Ian Short
• Drew Sutherland
• Steve Trettel

 

The full schedule can be found here.

Exceptionally on Tuesday March 24th, the live streaming will be accessible by Zoom with the folliwing link : https://zoom.us/j/88905935990

 

Organising Committee:

• Xavier Caruso (CNRS / Université de Bordeaux)
• Ellen Eischen (University of Oregon)
• Catherine Hsu (Swarthmore College)
• Katherine E. Stange (University of Colorado Boulder)

lun. 23 mars

09:30 Welcome Coffee

1 Welcome Info

Welcome & important info & announcements.

2 The power of pictures

Orateur: Andrew Sutherland (Massachusetts Institute of Technology)

3 Lift and patterns in the Spectre tilings

Orateur: Arnaud Chéritat (CNRS/Institut de Mathéamtiques de Toulouse)

11:00 Coffee & Treats

4 What Color Should This Pixel Be?

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 wants pixels to be Gaussian integers!

Orateur: Roice Nelson

5 Entangling Numbers and Knots via Sphere Packings

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 sphere packings. These perspectives provide useful tools for deriving upper bounds on geometric invariants of rational knots and links.

Orateur: Iván Rasskin (Aix-Marseille Université)

12:30 Lunch Break

Organization of Afternoon Activities

Président de session: Katherine Stange (University of Colorado Boulder)

Afternoon Activities

15:45 Coffee & Treats

Afternoon Activities

mar. 24 mars

6 Interactivity, Fidelity, and Polish: Building Modern Math Illustration

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 directly. Over the past eight years, I have created nearly 50 web applets for visualizing math — ranging from art to teaching aids to tools for mathematicians and hobbyists — and I have been fortunate to see my work meet with a wide audience. In this talk, I share the most important lessons I have learned for creating and maintaining beautiful, engaging, and functional mathematical renderings, with the goal of helping people already in the space improve their work and convincing many more to join.

Orateur: Cruz Godar (Yale University)

7 Visualizing Elliptic Curves over Finite Fields

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 elliptic curve is a torus, and the group law is simply addition.
In this talk, we explore a way to bridge the gap between these worlds. Using ideas from lattices with complex multiplication, we construct a way to "lift" any elliptic curve over a finite field to a (subset of a) complex torus, in a way that makes the group structure, the action of Frobenius, and the points over all field extensions simultaneously visible in a single picture. We will begin with an introduction to elliptic curves aimed at a general mathematical audience, building from curves over the reals and complex numbers to the finite field setting, before describing the lifting construction and the pictures it produces. This is joint work with Nadir Hajouji.

Orateur: Stephen Trettel (University of San Francisco)

8 Sonification in number theory: listening to the Riemann zeta function

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 we would never be able to see.

This talk will explore two examples of this ability to hear properties of functions, using the Riemann zeta function as a case study. First, we will try to identify the analytic continuation of the zeta function within the sound of a divergent sum. Second, we will listen to what the primes might sound like if the Riemann hypothesis were false.

Orateur: Jonathan Love (Leiden University)

11:00 Coffee & Treats

9 Rational points on the sphere

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

Orateur: Claire Burrin

Lightning Talks

12:30 Group Photo

12:40 Lunch Break

10 Connectivity of Markoff and Nielsen graphs

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 (bounding runtime of the Product Replacement Algorithm for SL_2(F_p)). In this talk, we discuss recent progress toward proving connectivity of Markoff graphs and related results on Nielsen graphs of matrix pairs from SL_2(F_p).

Orateur: Daniel Martin (Clemson University)

Afternoon Activities

15:45 Coffee & Treats

Afternoon Activities

mer. 25 mars

11 Dots and Laurent Series

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.

Orateur: Jayadev Athreya (University of Washington)

12 The mathematics of making pictures of lattices

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 the details of understanding what we are seeing. In this talk I will show illustrations of lattices and flows on lattices using shaders, some of the math that goes into creating and understanding them and how this is leading to faster images of algebraic numbers.

Orateur: Edmund Harriss

13 A dichotomy in the tail behaviour of quadratic Weyl sums

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 supported limiting distribution. I will especially emphasise the role of mathematical illustration in our understanding of the geometry of the relevant orbits, as well the importance of numerical simulations to validate our results and prompt new questions.

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

11:00 Coffee & Treats

14 A Few Attempts at Visualizing Shimura Varieties

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 varieties is not required.

Orateur: Sean Gonzales (UC Berkeley)

Exhibition

jeu. 26 mars

15 Basic illustrations associated with continued fractions

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.

Orateur: M. PIERRE ARNOUX (Université d'Aix-Marseille)

16 A Selection from Forty Years of Illustrating Mathematical Results

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 (about Lebesgue functions for "blends" in numerical analysis) which has a pretty number-theoretic proof and a simple but satisfying illustration with a (not-so-surprising) appearance by Fibonacci numbers. I will (for sure) run out of time, but I will at least give several links to interesting illustrations, not all of which are mine. I think that you will agree that some of those are, in fact, truly beautiful.

This is joint work with a very large number of people.

Orateur: Robert Corless (Western University)

17 Aperiodic tilings and polygonal partitions of the torus

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 coordinate space).
Many known aperiodic tilings are 4-to-2 cut-and-project schemes, meaning
that the dimension of both spaces is 2. These include Penrose tilings,
the Ammann tilings, the Jeandel-Rao tilings and tilings by the hat
monotile. The goal of this talk is to explain and understand aperiodic
tilings coming from 4-to-2 cut and project schemes with illustrations,
experimentations, discussions and using as many senses as possible
(sight, hearing, touch, smell and taste) but mostly the first three.

Orateur: Sébastien Labbé

11:00 Coffee & Treats

18 Classifying integer hypertilings

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 that each cross-section is a two-dimensional integer tiling (in which all squares have the same determinant). This model comprises a Bhargava cube and a triple of paths in the weighted Farey graph. The hypertiling entries are encoded geometrically by lambda lengths between horocycles or arithmetically by data from the weighted Farey graph. This is joint work with Oleg Karpenkov, Matty van Son, and Andrei Zabolotskii.

Orateur: Ian Short (The Open University)

Lightning Talks

12:30 Lunch Break

19 Initial regularity in the landscapes of L-functions

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.

Orateur: Sally Koutsoliotas (Bucknell University)

Afternoon Activities

15:45 Coffee & Treats

Afternoon Activities

ven. 27 mars

20 Lattice Models: How Physics Illustrates/Inspires Number Theory

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 Whittaker functions and suggest a surprising connection to quantum groups.

Orateur: Claire Frechette (Umeå University)

21 Finite capture and the closure of roots of restricted polynomials

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 new idea is a finite-capture depth filtration built from a canonical trap-and-enclosure construction in a natural two-disk lens region of parameter space. The level Θₖ​(n) consists of parameters that can be certified by following a single marked point for at most k inverse steps. The main result shows that these layers fit together with uniform regularity: every limit of depth-k parameters already lies in depth k+2. In the lens, closing up the finite-capture locus recovers the entire non-real closure of the set of roots, and for n≥20 this yields the full non-real picture. The talk will emphasize the geometry and illustrations behind this finite organization of a fractal closure of algebraic numbers.

The corresponding preprint is available at arXiv:2603.07397
https://arxiv.org/abs/2603.07397

Orateur: Bernat Espigule (Universitat de Girona)

22 Unintentional illustration

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).

Orateur: Anna Felikson (Durham University)

11:00 Coffee & Treats

23 Implicitization in Real Time

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 finding the implicit equation of a cubic surface in real time (ideally 60 frames per second). This talk will discuss how searching for ways to quickly solve this implicitization problem led us to some interesting facts about Gröbner bases which have applications more generally to the implicitization of rational projective hypersurfaces with base points.

Orateur: Gabriel Dorfsman-Hopkins (St. Lawrence University)

Lightning Talks