Recent breakthroughs in Mathematics and General Assembly 2024
About
The Belgian Mathematical Society is happy to invite you to its "Recent breakthroughs in Mathematics" symposium which will take on Wednesday December 18 2024 at ULB, Campus Solbosch (room Somville, building S). On this occasion we will also organise the society's general assembly.
Participation is free for BMS members and PhD students, but registration is mandatory, see below. Non PhD students who are not BMS members will be invited to become members to enjoy the free drinks :-).
Programme
10h30-11h00 : Welcome coffee
11h00-12h00 : Jean-Michel Roquejoffre about the work of Luis Cafarelli (Abel prize 2023)
Free boundary problems for diffusion equations: the work of L. Caffarelli.
Diffusion equations are partial differential equations arising in various contexts of physics, chemistry or biology. They are usually posed in multidimensional domains, and specifying the value of the solutions at the boundary of the domain suffices, in general, to determine it uniquely.Sometimes one needs to impose additional conditions on a part of the boundary, which forces it to adjust to the new conditions, and to become an unknown of the problem. This explains the term "free boundary". Little was known on the differentiability properties of free boundaries, until L. Caffarelli devised, at the beginning of the 1980's, new methods of investigation that have changed the landscape. Remarkably, they have proved to be fruitful until now, and in the understanding of problems quite remote from those for which they had initially been conceived. The goal of the talk is to explain, in a pedestrian way, the issues and some of the ideas introduced by L. Caffarelli.
12h00-14h00 : lunch
13h00-13h30 : BMS Board meeting
13h30-14h00 : BMS General assembly
14h00-15h00 : David Gontier about the work of Maryna Viazovska (Fields Medal 2022)
Sphere packing and the work of Maryna Viazovska (slides)
In this talk, we review what is currently known, conjectured and open about the sphere packing. We will prove that the triangular lattice is optimal in dimension d=2. Then, we will present the method by Cohn-Elkies, and explain how it was solved by Viazovska in dimension d=8.
15h00-16h00 : Jozefien D'haeseleer, winner of the Young Scholar Award 2024
Intersecting Families, Spectral Insights and Sunflowers in Finite Geometry
In this talk I will discuss three distinct but interconnected areas of research that I have investigated in recent years, all linked by the theme of finite geometry.
In the first part, we delve into Erdos-Ko-Rado (EKR) problems, a classical topic in combinatorics that explores intersection properties within mathematical structures. The foundational question concerns the maximum size of a family of k-subsets of an n-set, where every pair of subsets have at least one element in common. Erdos, Ko, and Rado proved that for n >=2k, the optimal construction involves subsets containing a fixed element. This result, known as the EKR theorem, extends to various other settings, including multisets, permutations, and finite geometries. In this talk, I will discuss both classical and recent results in the context of finite geometries.
The second part will be about generalized Johnson and Grassmann graphs, within the field of spectral graph theory. A central question is whether a graph is uniquely determined by its spectrum (the eigenvalues of its adjacency matrix). While many graphs are known to have this property, numerous well-known graphs possess cospectral mates (non-isomorphic graphs with identical spectra). This phenomenon appears prominently in highly symmetrical graphs, including Johnson and Grassmann graphs. Using tech- niques such as switching, I will present our construction of cospectral mates for several specific generalized Johnson and Grassmann graphs.
In the final part, we make the link with coding theory, specifically sub- space codes, which can be used in random network coding. A t-intersecting constant dimension code (SCID) is a family of k-spaces pairwise intersecting in precisely a t-space. Sunflowers, which are families of subspaces all containing a fixed t-space, represent a classical example of SCIDs. It is known that large SCIDs must be sunflowers, governed by the so-called sunflower bound. I will present recent improvements to this bound for k-spaces pairswise intersecting in a projective point.
16h00-16h30 : coffee break
16h30-17h30 : Ronan Herry about the work of Michel Talagrand (Abel prize 2024)
Celebrating the Work of Michel Talagrand
In this talk, I will recount some of the groundbreaking contributions of Michel Talagrand. Following his Abel Prize award citation, I will focus on three areas where his work has been particularly impactful: the concentration of measure phenomenon, the study of suprema of Gaussian processes, and spin glasses. Rather than delving into technical details, I will emphasize the key ideas and the historical context that shaped these remarkable achievements.
17h30-18h30 : drink
Documents: attendance list
Registration is now closed.