Presentation of the day and the Bézout Research Federation and Graduate Program Room: Amphi Cauchy

Ecology in the Age of Randomness : How Random Matrix Theory Explains Species Coexistence

• Mohammed-Younes Gueddari (LIGM)

Room: Amphi Cauchy We study the stability of large ecosystems by modeling species interactions with random matrices within the Lotka–Volterra framework. Since real interaction data are difficult and costly to collect, we adopt a probabilistic approach. I will introduce key concepts from Random Matrix Theory to analyze the typical behaviour of ecological equilibria. Unlike classical spectral questions, our focus lies on a nonlinear property of the random matrix. To study it, we use Approximate Message Passing, a tool from high-dimensional statistics.

The computational and mathematical challenges of simulating non-equilibrium molecular systems

• Shiva Darshan (CERMICS)

Room: Amphi Cauchy Statistical physics gives a probabilistic description of the microscopic dynamics of a system allowing one to deduce its macroscopic properties. The numerical realization of this idea: molecular dynamics, i.e the simulation of the dynamics of molecular and atomistic systems, provides scientists a "numerical microscopic" to conduct computer experiments allowing them to test physical theories and to make precise quantitative measurements of simulated systems. The static or equilibrium case is well understood and consequently powerful methods permit computing quantities of interest to high precision in the matter of hours in most cases. Furthermore, we have strong theoretical guarantees on the convergence. The dynamic or non-equilibrium case is much less well understood in contrast. Standard simulation methods of take weeks if not months of computation time and practitioner mostly have to guess if their simulation converges.        We give a brief introduction to non--equilibrium molecular dynamics with special focus on the computation of transport coefficients. Our talk will highlight why the non--equilibrium case is so much more difficult than the equilibrium case as well as some attempts at accelerating certain non--equilibrium methods.

The primitive equations of the ocean and the atmosphere

• Valentin Lemarié (LAMA)

Room: Amphi Cauchy With the aim of predicting meteorological phenomena, in 1922 the mathematician and meteorologist Richardson proposed and used a simplified version of the Navier-Stokes equations: the primitive atmospheric equations. These equations proved to be a good model for studying large-scale flows where the vertical component of motion is, in this case, much weaker than the horizontal component. Bryan then applied them to oceanographic models (1969), noting that the ocean layer on Earth is very thin compared with the planet's dimensions. In this purely introductory presentation, we are going to show the physical origin of this model and justify the mathematical study that I carried out in one of my thesis works.

Braid normal forms and combinatorics.

• June Roupin (LIGM)

Room: Amphi Cauchy A mathematical braid is a set of threads whose topological structure we wish to study. These objects can also be seen as an equivalence class of words, where each word corresponds to a different way of producing the same braid. A very natural problem is the choice of a single representative word for each braid, in order to be able to compare efficiently and make quick computations on the braids: this is called a normal form. I'm going to present a few classic normal forms and explain their strengths.

Enhancing Sampling in Molecular Dynamics: Integrating Autoencoders and Linear Discriminant Analysis for Efficient Collective Variables

• Charlotte Chapelier (CERMICS)

Room: Amphi Cauchy In molecular dynamics, transitions between conformations are rare events, posing significant challenges for sampling. Enhanced sampling methods, such as the extended Adapted Biasing Force (eABF), utilize collective variables (CVs) to capture the slow components of these transitions. While intuitive selection of CVs can sometimes be effective, it often fails to capture critical transitions. Building on the work of my predecessor, this study explores a novel approach using autoencoders combined with Linear Discriminant Analysis (LDA) to identify optimal CVs. By validating our methods on semi-supervised datasets of a protein or an RNA, we aim to maximize the likelihood of obtaining effective CVs for various therapeutic targets, thereby improving the efficiency and accuracy of enhanced sampling methods in molecular dynamics

Disc flow on discrete sweepouts of a sphere by closed curves.

• Jean Chartier (LAMA)

Room: Amphi Cauchy Simple closed geodesics on Riemannian spheres have been the subject of many incomplete existence proofs throughout the 20th century, since Birkhoff's first paper in 1908. One stumbling block is preserving the simplicity of the curves by using shortening flows. In 1994, Hass and Scott described a new flow for shortening curves: the disc flow, which solves the simplicity issue, but brings other difficulties. We present an adaptation of this flow in a discrete setting, as well as a new proof of Birkhoff's theorem.