CRM-CNRS Internal Seminar

The CRM-CNRS will hold its internal seminar on November 12, 2024, from 9 a.m. to 1 p.m. in room 4336 of the Aisenstadt Pavilion (Université de Montréal).

Four scientific talks will be given by:

• Denis Grebenkov
• Sophie Dabo-Niang
• Antoine Zurek
• Nicolas Bousquet.

Program

9:30 a.m.: Antoine Zurek

Some theoretical and numerical results for local and nonlocal diffusion systems

In this talk, I will provide a brief overview of my research on parabolic PDEs. First, I will present some results related to the numerical analysis of the local and nonlocal Shigesada-Kawasaki-Teramoto (SKT) cross-diffusion system. Next, I will introduce and explain the main ideas behind the Computer-Assisted Proofs (CAP) approach. These methods can be powerful tools to study some PDEs when more classical techniques do not yield results. As an example, I will briefly explain how these techniques allow us to obtain the first existence result for the so-called Diffusion Poisson Coupled Model (DPCM).

10:15 a.m.: Denis Grebenkov

The Steklov spectral problem: recent advances and open questions

In this overview talk, I will present the encounter-based approach to diffusive processes in Euclidean domains and highlight its fundamental relation to the Steklov spectral problem. Steklov eigenfunctions turn out to be particularly useful for representing heat kernels with Robin boundary condition and disentangling diffusive dynamics from reaction events on the boundary. I will also discuss applications of this approach to diffusion-controlled reactions in physical chemistry and to first-passage-time statistics in statistical physics. Some open questions concerning the spectral, probabilistic, and numerical aspects of this problem will also be mentioned.

11 a.m.–11:30 a.m.: Break

11:30 a.m.: Sophie Dabo-Niang

Functional data analysis: a PCA approach for learning models

Functional data, arising from random variables with values in a functional space, present major challenges for modeling curves, patterns, images, and other complex structures. This presentation focuses on a Principal Component Analysis (PCA) approach specifically designed for functional data sets. We will define functional data, highlighting their frequent occurrence in areas such as environmental monitoring and biomedical research. Emphasis will be placed on theoretical aspects of functional PCA. We will discuss practical implementations of functional PCA to capture variability and reduce dimensionality. Concrete examples will illustrate the effectiveness of these methods in various contexts. Finally, we will address challenges related to applying PCA to functional data analysis, including handling infinite-sample attributes, large data collections, and computational requirements.

12:15 p.m.: Nicolas Bousquet

A journey through configuration graphs

What do a Rubik’s cube, genetic distance between species, and the Hirsch conjecture in combinatorial optimization have in common? All three are reconfiguration problems. A reconfiguration problem consists, given two solutions to the same problem (e.g. a configuration of the Rubik’s cube), in determining whether it is possible to transform one into a target solution (the Rubik’s cube with monochromatic faces) via a sequence of elementary transformations (rotations) that maintain a solution throughout the process.

The aim of the presentation is to introduce configuration graphs, present the central questions in combinatorial reconfiguration, and briefly mention some recent results.

Hosted on long-term CNRS delegation at the CRM-CNRS, Antoine Zurek is an associate professor at the Université de Technologie de Compiègne.

Denis Grebenkov is a CNRS research director assigned to the CRM-CNRS.

Hosted on long-term CNRS delegation at the CRM-CNRS, Sophie Dabo-Niang is a professor at the Université de Lille.

Nicolas Bousquet is a CNRS research scientist assigned to the CRM-CNRS.