Sketchy moduli spaces

• Hugo Parlier

Room: Amphithéâtre Hermite This talk has three parts, all centered around illustration. First, some puzzle spaces that offer a hands-on route into the geometry of moduli spaces. Then the setting moves to hyperbolic surfaces and what it means to illustrate on them. Finally, the talk turns meta: how a collection of drawings can itself be organized into a moduli space. Parts of the talk are based on joint projects with Peter Buser, Paul Turner, Bruno Teheux, Mario Gutiérrez and Reyna Juárez.

Illustration in Mathematics and CS — Made Broadly Accessible by AI: From Opportunity to Practice

• Érika Roldán

Room: Amphithéâtre Hermite Illustration is a central tool in my research, outreach, and educational practice, encompassing high-quality illustrations, video game development, 3D modeling, laser-cut physical realizations of discrete structures, mixed and virtual reality, short videos, and performances. These illustration practices have supported the formulation, pursuit, and resolution of conjectures in discrete and stochastic computational geometry and topology, the establishment of algorithmic complexity results, and the exploration of configuration spaces of rich combinatorial systems. They have also played a central role in widening access to, understanding of, and sustained engagement with contemporary mathematical research well beyond specialist communities. I will present concrete examples from this trajectory and discuss how AI is transforming—and will continue to do so at an accelerating pace—this ecosystem, reconfiguring how illustrations in mathematics and computer science are conceived, generated, and shared. I will argue that—across knowledge-based human endeavors—by drastically lowering technical and skill barriers, AI has the potential to democratize who can produce, explore, and communicate complex ideas and structures by means of illustration.

Embedding high-dimensional data into (non-)Euclidean spaces (fast)

• Martin Skrodzki

Room: Amphithéâtre Hermite The field of explorative data analysis provides methods to investigate large, potentially high-dimensional, data sets. Such exploration is best done visually, to engage the human in the loop with all pattern recognition built into our visual cortex. In my talk, I will give a brief introduction to dimensionality reduction for this purpose, especially about the t-SNE method. I will then show how embedding into non-Euclidean spaces can provide embeddings that help with visual inspection and how to compute such embeddings quickly.

Exhibition session Room: Amphithéâtre Hermite

Phase Transitions in Loewner Evolution: A Mathematical Proof of Concept

• Claire David

Room: Amphithéâtre Hermite See .pdf abstract below (click)

Pictures to get some intuition about a space of complex dimension 6

• Aurélien Alvarez

Room: Amphithéâtre Hermite Given an algebraic foliation on a complex algebraic surface, what can we say about the dynamics of the foliation? In the case of the complex projective plane, the space of modules of foliations of degree 2 is of dimension 6 and we know two open sets where the dynamics are radically different. In this talk, we will explain how with the help of numerical experimentations and pictures, it is possible to explore this space of dimension 6.

If you give a mathematician a surface...

• Samantha Fairchild

Room: Amphithéâtre Hermite If you give a mathematician a surface, their neighbor will give you another one. If their neighbor gives you another surface, you may start to wonder if they are the same. If you want to determine if two surfaces are the same, you will have algebraist, geometers, topologist and number theorists knocking at your door, asking them to show you how they think about surfaces. If you give me some time to talk, I’ll tell you about some fun ways to visualize and move between different representations of surfaces and the research inspired by these visualizations. In this talk, I will describe some fun and intuitive ways to visualize Riemann surfaces and to move between their many representations. I will also discuss how these visual perspectives inspire new questions and directions in my research.

What explicit constructions really give us

• Mélanie Theillière

Room: Amphithéâtre Hermite In 1954, Nash imagined an explicit way to build isometric embeddings in codimension 1 of regularity only $C^1$. In the 1970s, Gromov generalized this technique to one which allows to solve a large family of non linear PDE, and giving a geometrical understanding of the construction. Even if the consctruction is explicit, it's done taking the limit of an induction, and each step involves a lot of parameters. We finally have a first complete explicit construction in 2012 by the Hevea team. What was the contribution of the explicit constructions that followed? And which questions arose during the construction process?

Visualisation of minimal and constant mean curvature surfaces

• Martin Traizet

Room: Amphithéâtre Hermite Minimal and constant mean curvature (CMC) surfaces in space forms -- $\mathbb R^3$, $\mathbb S^3$ or $\mathbb H^3$ -- can be represented in terms of holomorphic data. For minimal surfaces in $\mathbb R^3$, this is the classical Weierstrass Representation. In the other cases, this is usually called the Dorfmeister Pedit Wu (DPW) method. These representations can be implemented to produce pictures. In this talk, I will present two theorems and the pictures that led to them. The first one is the existence of an embedded minimal surface in euclidean surface with finite topology and no symmetry at all. The second one is a recent counterexample to a conjecture about the isoperimetric problem. A recurent question is the following: is a picture a proof?

Visualizing tilings, packings and fullerenes

• Thomas Fernique

Room: Amphithéâtre Hermite In this talk, I will illustrate the important role of visualization—both in generating ideas and in illustrating them—in three of my research topics: * Tilings, particularly tilings of the Euclidean plane obtained by digitizing planes in high-dimensional spaces (e.g., Penrose tilings). * Packings, specifically disk and sphere packings that aim to maximize density. * Fullerenes, in connection with the edge-unfolding conjecture and the approximation of isoperimetric quotients.

Some instances where topological illustration induced new mathematics

• Sofia Lambropoulou

Room: Amphithéâtre Hermite We shall present instances from generalised knot theory, braid theory and their interactions, where illustration promoted understanding and inspired new mathematics.

Video session Room: Amphithéâtre Hermite

Video session Room: Amphithéâtre Hermite

Temperley-Lieb Algebra - Visualizing Meanders and Idempotents

• Lou Kauffman

Room: Amphithéâtre Hermite The Temperley-Lieb algebra first arose as a matrix algebra describing transfer functions in statistical mechanics models such as the Potts and Ising models. The algebra acquired a formal definition in terms of generators and relations that allowed its representations to be identified in multiple contexts. In the early 1980's Vaughan Jones found the algebra once again in a context between mathematics and physics as an algebra of projectors that arose in a tower construction of von Neumann algebras. For this context, Jones investigated the formally defined algebra and its matrix representations and he constructed a trace function on the Temperley-Lieb (TL) algebra (a function $tr$ to a commutative ring such that $tr(ab) = tr(ba)$ for $ab$ a product in the (non-commutative) Temperley-Lieb algebra) and he discovered a representation of the Artin Braid group to the TL algebra. By composing this representation with the trace $tr$, Jones defined an invariant of braids that could be modified via the Markov Theorem for braids, knots and links to produce a polynomial invariant of knots that is now known as the Jones polynomial. The speaker discovered knot diagrammatic and combinatorial interpretations of the Jones polynomial and the Temperley-Lieb algebra that allow the polynomial to be seen as part of a generalized Potts model partition function defined on planar link diagrams and planar graphs. The combinatorial interpretation of the Temperley-Lieb algebra allows the Jones trace to be interpreted as a loop count for closures of Connection Monoid representations of the Temperley-Lieb algebra. The multiplicative structure of the Temperley-Lieb algebra is represented in the speaker's work by a Connection Monoid whose elements are families of planar connections between two rows of points where the connections can go from row to row or from one row to the other. The talk will begin with the formal definition of the TL monoid and will show how it is modeled by the Connection Monoid and similarly with the TL Category and a Connection Category. This interpretation allows us to see answers to algebra questions about the Temperley-Lieb Monoid that would be invisible without the combinatorial interpretation. In particular we will show how the structure of repeated powers of elements in TL appears and how idempotents correspond to generalized meanders. A meander is a Jordan curve in the plane cut through transversely by a straight line. The fascinating and highly visual combinatorics of the meanders informs the structure of the TL algebra via the way meanders correspond to factorizations of the identity in the Temperley-Lieb Category.

Losing dimension

• Arnaud Chéritat

Room: Amphithéâtre Hermite TBA

How discrete differential geometry and visualization helped to solve a classical problem in differential geometry

• Alexander Bobenko

Room: Amphithéâtre Hermite We consider a classical problem in differential geometry, known as the Bonnet problem, whether a surface is characterized by a metric and mean curvature function. We explicitly construct a pair of immersed tori that are related by a mean curvature preserving isometry. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique compact surface. Visualizations based on structure preserving discretizations of Discrete Differential Geometry were used to find crucial geometric properties of surfaces. This is a joint work with Tim Hoffmann and Andrew Sageman-Furnas.

Film Screening: Solving the Bonnet problem Room: Amphithéâtre Hermite The documentary "Solving the Bonnet Problem" follows the work of three mathematicians as they collaborate to solve a long-standing geometrical problem proposed by Pierre Ossian-Bonnet. Alongside their work, the film explores the lives and contributions of Bonnet and his contemporary Gaston Darboux, as well as the history of French mathematics in the 19th century. Captivating computer graphics help unravel intricate concepts and make them accessible to all. [Trailer][1] [1]: https://www.youtube.com/watch?v=iQvsKbw-ksg

Discussion / Q&A: Solving the Bonnet problem Room: Amphithéâtre Hermite Discussion and Q&A session with the film director Ekaterina Eremenko and the mathematician Alexander Bobenko.