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

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 → 10:00 Welcome Coffee

10:00 → 10:25 TBA

Orateur: Andrew Sutherland (Massachusetts Institute of Technology)

10:30 → 10:55 TBA

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

11:00 → 11:30 Coffee & Treats

11:30 → 11:55 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

12:00 → 12:25 TBA

Orateur: Iván Rasskin (Aix-Marseille Université)

14:30 → 15:00 Organization of Afternoon Activities

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

15:00 → 15:45 Afternoon Activities

15:45 → 16:15 Coffee & Treats

16:15 → 17:15 Afternoon Activities

mar. 24 mars

09:30 → 09:55 TBA

Orateur: Cruz Godar (Yale University)

10:00 → 10:25 TBA

Orateur: Stephen Trettel (University of San Francisco)

10:30 → 10:55 TBA

Orateur: Jonathan Love (Leiden University)

11:00 → 11:30 Coffee & Treats

11:30 → 11:55 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

12:00 → 12:30 Lightning Talks

14:30 → 14:55 TBA

Orateur: Daniel Martin (Clemson University)

15:00 → 15:45 Afternoon Activities

15:45 → 16:15 Coffee & Treats

16:15 → 17:15 Afternoon Activities

mer. 25 mars

09:30 → 09:55 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)

10:00 → 10:25 TBA

Orateur: Edmund Harriss

10:30 → 10:55 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 → 11:30 Coffee & Treats

11:30 → 11:55 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)

12:00 → 12:30 Exhibition

jeu. 26 mars

09:30 → 09:55 TBA

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

10:00 → 10:25 TBA

Orateur: Robert Corless (Western University)

10:30 → 10:55 TBA

Orateur: Sébastien Labbé

11:00 → 11:30 Coffee & Treats

11:30 → 11:55 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)

12:00 → 12:30 Lightning Talks

14:30 → 14:55 TBA

Orateur: Sally Koutsoliotas (Bucknell University)

15:00 → 15:45 Afternoon Activities

15:45 → 16:15 Coffee & Treats

16:15 → 17:15 Afternoon Activities

ven. 27 mars

09:30 → 09:55 TBA

Orateur: Claire Frechette

10:00 → 10:25 TBA

Orateur: Bernat Espigule (Universitat de Girona)

10:30 → 10:55 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 → 11:30 Coffee & Treats

11:30 → 11:55 TBA

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

12:00 → 12:30 Lightning Talks