Stochastic differential equations (SDE) of McKean-Vlasov type or mean-field type are SDE where the solution's law appears inside the equation's coefficients making them more complicated to solve. On the other hand, such equations have been prominently applied in finance, agent dynamics and machine learning. In this mini course we offer an introduction to this framework covering wellposedness of McKean-Vlasov SDE and properties alongside the associated approximating interacting SDE particle systems and corresponding propagation of chaos.
Stochastic differential equations (SDE) of McKean-Vlasov type or mean-field type are SDE where the solution's law appears inside the equation's coefficients making them more complicated to solve. On the other hand, such equations have been prominently applied in finance, agent dynamics and machine learning. In this mini course we offer an introduction to this framework covering wellposedness of McKean-Vlasov SDE and properties alongside the associated approximating interacting SDE particle systems and corresponding propagation of chaos.
The phenomenon of dispersion in a physical system occurs whenever the elementary building blocks of the system, whether they are particles or waves, overall move away from each other, because each evolves according to a distinct momentum. This physical process limits the superposition of particles or waves, and leads to remarkable mathematical properties of the densities or amplitudes, including local and global decay, Strichartz estimates, and smoothing.
In kinetic theory, the effects of dispersion in the whole space were notably well captured by the estimates developed by Castella and Perthame in 1996, which, for instance, are particularly useful in the analysis of the Boltzmann equation to construct global solutions. However, these estimates are based on the transfer of integrability of particle densities in mixed Lebesgue spaces, which fails to apply to general settings of kinetic dynamics.
Therefore, we are now interested in characterizing the kinetic dispersive effects in the whole space in cases where only natural principles of conservation of mass, momentum and energy, and decay of entropy seem to hold. Such general settings correspond to degenerate endpoint cases of the Castella–Perthame estimates where no dispersion is effectively measured. However, by introducing a suitable kinetic uncertainty principle, we will see how it is possible to extract some amount of entropic dispersion and, in essence, measure how particles tend to move away from each other, at least when they are not restricted by a spatial boundary.
A simple application of entropic dispersion will then show us how kinetic dynamics in the whole space inevitably leads, in infinite time, to an asymptotic thermodynamic equilibrium state with no particle interaction and no available heat to sustain thermodynamic processes, thereby providing a provocative interpretation of the heat death of the universe.
The gene regulatory network models all the biochemical reactions between the different species (proteins, mRNA, etc.) present in a cell. A stochastic approach to this network, using jump processes, has been studied during the '70s (Kurtz), and it was established that in large populations, the rescaled process (thought as a concentrations process) converges towards the solution of an EDO, which is entirely deterministic. In order to obtain a random limit, Crudu, Debussche and Radulescu proposed a multiscale model. In this talk, I will propose a relevant stochastic representation of the multiscale model, for which it is possible to obtain the uniform convergence towards a PDMP and a CLT for the fluctuations around this limit.
We investigate the topology and local geometry of various models of random cubical complexes. In the first part, we consider two types of random subcomplexes of the regular cubical grid: percolation clusters and the Eden cell growth model, analyzing their geometric and topological features. In the second part, we study the fundamental group of random 2-dimensional subcomplexes of an n-dimensional cube, identifying interesting threshold phenomena. This latter model serves as a cubical analogue of the Linial–Meshulam model for random simplicial complexes.
We present the theory of local and nonlocal minimal surfaces in relation to models of phase coexistence, with special attention to regularity and geometric properties.
I will discuss joint work with Agustina Czenky. We introduce a -dimensional TQFTs which is generated, in some sense, by the derived category of quantum group representations. This TQFT is valued in the -category of dg vector spaces, and the value on a genus surface is a -th iterate of the Hochschild cohomology for the aforementioned category. I will explain how this TQFT arises as a derived variant of the usual Reshetikhin–Turaev theory and, if time allows, I will discuss the possibility of introducing local systems into the theory. Our interest in local systems comes from proposed relationships with 4-dimensional non-topological QFT.
We present the theory of local and nonlocal minimal surfaces in relation to models of phase coexistence, with special attention to regularity and geometric properties.
This lecture first provides an introduction to classical variational inference (VI), a key technique for approximating complex posterior distributions in Bayesian methods, typically by minimizing the Kullback-Leibler (KL) divergence. We'll discuss its principles and common uses.
Building on this, the lecture introduces Fenchel-Young variational inference (FYVI), a novel generalization that enhances flexibility. FYVI replaces the KL divergence with broader Fenchel-Young (FY) regularizers, with a special focus on those derived from Tsallis entropies. This approach enables learning posterior distributions with significantly smaller, or sparser, support than the prior, offering advantages in model interpretability and performance.
This lecture first provides an introduction to classical variational inference (VI), a key technique for approximating complex posterior distributions in Bayesian methods, typically by minimizing the Kullback-Leibler (KL) divergence. We'll discuss its principles and common uses.
Building on this, the lecture introduces Fenchel-Young variational inference (FYVI), a novel generalization that enhances flexibility.FYVI replaces the KL divergence with broader Fenchel-Young (FY) regularizers, with a special focus on those derived from Tsallisentropies. This approach enables learning posterior distributions with significantly smaller, or sparser, support than the prior, offering advantages in model interpretability and performance.
We construct an action of sl(2) on equivariant Khovanov–Rozansky link homology. We will discuss some topological applications and show how the construction simplifies in characteristic p. This is joint with You Qi, Louis-Hadrien Robert, and Emmanuel Wagner.
We define a nonlinear Fourier transform which maps sequences of contractive matrices to -valued functions on the circle . We characterize the image of compactly supported sequences and square-summable sequences on the half-line, and prove that the inverse map is well-defined on -valued functions whose diagonal blocks are outer matrix functions. As an application, we prove infinite generalized quantum signal processing in the fully coherent regime.
We define a nonlinear Fourier transform which maps sequences of contractive matrices to -valued functions on the circle . We characterize the image of compactly supported sequences and square-summable sequences on the half-line, and prove that the inverse map is well-defined on -valued functions whose diagonal blocks are outer matrix functions. As an application, we prove infinite generalized quantum signal processing in the fully coherent regime.
We first introduce a brief review of the history of Brownian Motion up to the modern experiments where isolated Brownian particles are observed. Later, we introduce a one-space-dimensional wavefunction model of a heavy particle and a collection of light particles that might generate “Brownian-Motion-Like” trajectories as well as diffusive motion (displacement proportional to the square-root of time). This model satisfies two conditions that grant, for the temporal motion of the heavy particle:
An oscillating series with properties similar to those of the Ornstein-Uhlenbeck process;
A best quadratic fit with an “average” non-positive curvature in a proper time interval.
We note that Planck’s constant and the molecular mass enter into the diffusion coefficient, while they also recently appeared in experimental estimates; to our knowledge, this is the first microscopic derivation in which they contribute directly to the diffusion coefficient. Finally, we discuss whether cat states are present in the thermodynamic ensembles.
