Séminaire Doctorants
organisé par l'équipe DOCT
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Anna Marduel
An introduction to barcodes
9 janvier 2025 - 16:30Salle de conférences IRMA
Barcodes are simple combinatorial objects that detect topological features. They first appeared as a tool for topological data analysis and have found new applications in Morse and Floer theories. We will use examples to illustrate the information they can provide (particularly quantitative information) and set out the basic definitions. This presentation is based on "Topological Persistence in Geometry and Analysis" by Polterovich, Rosen, Samvelyan, and Zhang. -
Arthur Douay
Drawing the p-adic unit disc
16 janvier 2025 - 16:30Salle de conférences IRMA
In algebraic geometry, we study geometric objects given by zeros of polynomial equations. However, in this context, we only have a very coarse topology, and we are jealous of the powerful tools of real analysis. A common object in algebraic geometry, the field of p-adic numbers, has both an arithmetic and analytic nature. One might then be tempted to try and develop a theory of p-adic analytic geometry. I will go through the various constructions of p-adic analytic geometry theories (from Tate’s rigid geometry to Huber’s adic spaces) and give motivations for these theories through the example of the p-adic unit disc. -
Claire Schnoebelen
An introduction to Nambu structures
23 janvier 2025 - 16:30Salle de séminaires IRMA
Nambu structures on a smooth manifold are a special case of Poisson structures. This notion can be seen as a generalization of symplectic structures on odd-dimensional manifolds. We will present what a Nambu structure is and try to give some elements of comparison between Nambu mechanics and Hamiltonian mechanics.
References :
Y. Nambu, Generalized Hamiltonian mechanics, Phys. Rev. D7 (1973), 2405-2412
L. Takhtajan, On foundation of the generalized Nambu mechanics, Comm. Math. Phys. 160 (1994), 295-315
J.-P. Dufour, N. T. Zung, Poisson structures and their normal forms, coll. Progress in Mathematics, Birkhauser, 2005 -
Nicolas Stutz
An introduction to Engel structures with a split through examples
30 janvier 2025 - 16:30Salle de conférences IRMA
In this talk, we will discuss Engel structures with a split through three examples. These geometric structures appear on differential manifolds of dimension four as rank-two distributions together with a decomposition into two preferred line fields. We will first consider the configuration space of a car moving in a plane, then we will look at third-order differential equations and finally we will present the geometry of pointed lightlike geodesics in three-dimensional Lorentzian manifolds. This talk is based on the article https://arxiv.org/pdf/1908.01169 by C. D. Hill and P. Nurowski. -
Vincent Ferrari-Dominguez
What is entropy ?
6 février 2025 - 16:30Salle de conférences IRMA
Entropy is a concept that was first introduced in thermodynamics and later famously used in information theory by Shannon. It then permeated many domains of mathematics, from statistics to dynamical systems. The goal of this seminar is to give some intuitions about this notion and more specifically about the Kolmogorov-Sinai entropy which is an invariant of dynamical systems that quantifies the degree of 'chaos' of a system. -
Florent Dupont
Quantum computing and Shor's algorithm
13 février 2025 - 16:30Salle de conférences IRMA
The RSA algorithm is one of the oldest widely used public-key encryption algorithm. It relies on the difficulty of factorizing primes, with the best-known classical algorithms performing in sub-exponential time. Shor's algorithm, which relies on the laws of quantum mechanics, drastically reduces this time complexity to polynomial time, making RSA vulnerable if implemented. The goal of this talk is to introduce quantum computing with Shor's algorithm as a guiding example. -
Louise Martineau
The ToMATo clustering algorithm for neuronal spike sorting
27 février 2025 - 16:30Salle de conférences IRMA
Recording and decoding the activity of multiple neurons is a major subject in contemporary neuroscience. The raw data produced by these recordings are almost systematically a mixture of activities from several neurons. In order to find the number of neurons which contributed to the recording and identify which neuron generated each of the visible spikes, a pre-processing step called spike sorting is required. Spike sorting is nowadays a semi-automatic process that involves many steps. We explain how the ToMATo (Topological Mode Analysis Tool) clustering algorithm, a method that combines graph mode-seeking and persistent homology, can simplify and streamline the usual spike sorting procedure carried out by most neuroscientists. -
Lauriane Turelier
Key characteristic of the Landau-Lifschitz-Gilbert equation
6 mars 2025 - 16:30Salle de conférences IRMA
The magnetization in a nanowire is described and follows the Landau-Lifshitz-Gilbert equation. As with any PDE, it is useful to know some of its properties. Here, I will give a brief overview of what I know about this equation: the stability of certain solutions, the asymptotic decomposition of solutions, and also a numerical examination of how the solutions behave. -
Aurélien Minguella
A brief introduction to stochastic partial differential equations
13 mars 2025 - 16:30Salle de conférences IRMA
Stochastic partial differential equations (SPDEs) are the mathematical objects used to describe the random dynamics of infinite-dimensional objects. They take applications in a broad range of areas, from statistical and theoretical physics, to fluid mechanics. These objects display very rich mathematical behaviour and have known a gain of interest since Martin Hairer was awarded the Fields medal in 2014 for constructing a solution theory for a very broad class of singular equations.
In this talk, we will try to give a very quick overview of some simple SPDEs, but where some essential phenomena already arise. Some emphasis will be given on the invariant measures for such equations. We will first give a review of basic stochastic calculus and continue with an example of a linear SPDE: the stochastic heat equation. If time permits, we will give a more advanced example where a non-linear SPDE shows up as the scaling limit of the dynamics of a particle system. We will finish with an informal overview of the most recent theories and current challenges in the field.
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Taichi Katayama
An Introduction to elliptic multiple zeta values and regularization
20 mars 2025 - 16:30Salle de conférences IRMA
In 2016, Enriquez introduced elliptic multiple zeta values (eMZVs) as analogues of multiple zeta values in genus 1 curves. In this talk, I will first give a brief introduction to eMZVs. Then, I will present my result showing that eMZVs with non-admissible indices can be expressed as polynomial combinations of eMZVs with admissible indices and eMZVs with specific indices consisting of 0 and 1. -
William Sarem
The Poincaré disk, its complex generalization and holomorphic functions
27 mars 2025 - 16:30Salle de conférences IRMA
The main aim of this talk is to present the complex hyperbolic space, and give a few reasons why people from different mathematical horizons could be interested in it. I will start with a nice non-technical description of the "Poincaré disk" borrowed from Poincaré's book La Science et l'Hypothèse. Then I will discuss some features of the higher-dimensional complex hyperbolic space. Depending on the time, I may also speak about the theory of holomorphic functions on quotients of this space. -
Jean-Pierre Noot
Study of engine component lifespan.
3 avril 2025 - 16:30Salle de conférences IRMA
This talk will present two methods for estimating the Remaining Useful Lifetime (RUL) of oil filters in Liebherr diesel engine. The first method is based on transformers (deep learning) and utilizes all available sensors, while the second is a parametric approach that only relies on oil differential pressure for RUL estimation. -
Vincent Ferrari-Dominguez
Pontryagin duality
10 avril 2025 - 16:30None
What are the properties of the real numbers or of the torus that allow the construction of the Fourier transform ? In other words, in which context can we define an analogue of the Fourier transform ? A beautiful theory exists in the case of locally compact abelian groups, based on the notion of dual of such a group. I will present the main theorems of the theory (Pontryagin duality, Fourier inversion and Plancherel formula) and, if time permits, give some applications (Wiener tauberian theorem, additive combinatorics). Salle C41 -
Lucas Noël
Hausdorff dimension of self-similar sets
24 avril 2025 - 16:30Salle de conférences IRMA
Among the fractal sets there are special sets known as self-similar sets. A more precise analysis of these sets leads to a surprising notion: they can be assigned a dimension, which, unlike the dimension we can find in the vector spaces theory, is (very often) not an integer. The aim of this talk will be to explain the construction of this dimension. We'll be doing a bit of topology and measure theory to get to the point. Once the theory has been worked out, depending on the time available, one or more applications will be made to determine the Hausdorff dimension of certain well-known self-similar sets. -
Ludovic Felder
Elliptic curves and modular curves
15 mai 2025 - 16:30Salle de conférences IRMA
Elliptic curves are widely studied in several areas of mathematics and have had a significant impact on modern mathematics. In this talk I will give a very brief introduction to elliptic curves, explaining the basic definitions and properties that are necessary to understand them ; then I will explain the construction of modular curves and their link with elliptic curves. Finally, my main goal is to explain, in a very informal way, how they were used to prove a very important result from the end of the 20th century : Fermat's last theorem. -
Nicolas Pailliez
Neural Methods for Solving Partial Differential Equations
22 mai 2025 - 16:30Salle de séminaires IRMA
Partial Differential Equations (PDEs) are fundamental to modeling a wide range of physical phenomena. Traditional numerical methods for solving PDEs can be computationally intensive and may struggle with high-dimensional problems. In this talk, I will explore innovative neural network-based methods that have emerged as powerful tools for solving PDEs. -
Virgile Bertrand
The Structural Method for Partial Differential Equations
28 mai 2025 - 16:30A confirmer
One of the main ways to numerically solve partial differential equations (PDE) is to use a finite difference method (FDM) to approximate the partial differential operator with some discrete equivalent. We will present the Structural Method, which proposes a new paradigm to conceptualize FDM. The idea is to separate the physics of the problem (the PDE in itself) and the estimation of the partial derivatives. It allows us to decouple the physics and the underlying discretization of the space. It also introduces a novel way of deriving schemes with an automated routine. -
Lucas Bourgoin
Geometric quantization
5 juin 2025 - 16:30Salle de séminaires IRMA
In classical mechanics physical systems are described using Hamiltonian or Lagrangian formulations. In this settings the classical observables are functions defined on the phase space. On the other hand, in quantum mechanics observables are operators acting on quantum states. We will briefly introduce geometric quantization which is a method to try to connect this two physics using symplectic manifolds and line bundles. -
Clarence Kineider
A magic trick, and a bit of holomorphic dynamics
11 juin 2025 - 16:30A confirmer
The "American-style" card shuffle has remarkable mathematical properties. After demonstrating it through an impressive magic trick, we'll dive into some math to find this shuffle reappearing in a completely different area: holomorphic dynamical systems. -
Livia Grammatica
Bernshtein's theorem
19 juin 2025 - 16:30Salle de conférences IRMA
Finding the solutions to a system of polynomial equations is hard, but counting the number of solutions is a surprisingly doable computational problem. This is because of Bernshtein's theorem, which relates the number of solutions to the volume of certain polyhedra in affine euclidean space. We will explain this result and maybe show how effective it is for computations. -
Roxana Sublet
Mathematical modeling of apoptosis in cell collective dynamics: microscopic and macroscopic points of view
26 juin 2025 - 16:30Salle de conférences IRMA
In this work, we are interested in the mathematical modeling of the impact of apoptosis on the dynamics of cellular tissues, and in particular on cell tissue fluidity. Indeed, cell apoptosis corresponds to the programmed cell death: when they leave the tissue, they induce local contractions but also enable cells rearrangements.
We first propose an individual-based model that provides the dynamics of the positions, velocities and polarities of the cells, idealized as hard spheres. Cells interact with each other through contact forces, smooth attraction, and polarity alignment. The model also involves a microscopic description of apoptotic and proliferation events. The present work is an extension of the model proposed in [3] and validated by experiments on cellular rings. Numerical simulations are performed and several indicators of fluidity are analyzed.
Next, we derive a macroscopic description, following the methodology proposed in [1], [2]. We start from a mean-field dynamics of the kinetic distribution function in phase-space (position, polarity, radius), where contact forces have been replaced with repulsion forces. We then introduce a specific time and space rescaling and identify the equilibria distribution functions, which are parameterized by two macroscopic quantities: the density and the mean polarity. Based on the Generalized Collision Invariant (GCI) method [2], we are then able to identify their dynamics: the resulting description can be seen as a modified Self-Organized Hydrodynamics (SOH) model. We finally discuss the obtained model and highlight the effect of the apoptotic events on the dynamics.
This is a joint work with Laurent Navoret (Université de Strasbourg) and Marcela Szopos (Université Paris Cité). It has also been carried out in collaboration with Romain Levayer (Institut Pasteur) and Daniel Riveline (IGBMC, Université de Strasbourg) in the context of the ANR project MAPEFLU.
References
[1] Degond, P., Dimarco, G., Mac, T. B. N., and Wang, N. Macroscopic models of collective motion with repulsion. Communications in Mathematical Sciences, 13(6), 1615–1638 (2015).
[2] Degond, P., and Motsch, S. Continuum limit of self-driven particles with orientation interaction. Mathematical Models and Methods in Applied Sciences, 18(supp01), 1193–1215 (2008).
[3] Vecchio, S. L., Pertz, O., Szopos, M., Navoret, L., and Riveline, D. Spontaneous rotations in epithelia as an interplay between cell polarity and boundaries. Nature Physics (2024).