A central question in Markov chain mixing is the occurrence of cutoff, a phenomenon according to which a Markov chain converges abruptly to the stationary measure. The focus of this talk is the limit profile of a Markov chain that exhibits cutoff, which captures the exact shape of the distance of the Markov chain from stationarity. We will discuss techniques for determining the limit profile and its continuity properties under appropriate conditions.

Thanks to their high degree of control and tunability, ultracold atomic gases provide a rich platform for the study of quantum many-body effects. Ultracold gases of highly magnetic atoms exhibit unique interaction properties that lead to striking behaviors, both at the mean-field level and beyond [1]. A decade ago, a universal stabilization mechanism driven by quantum fluctuations was discovered in these gases. This mechanism prevents the systems from collapsing when the mean-field interactions become attractive, and instead allows exotic states of matter to arise, including ultradilute quantum droplets, crystallized quantum states, and especially the so-called supersolids [2]. In my colloquium, I will present the seminal observations of these states and how they emerged from the long-standing progress in the field. I will discuss the theoretical description of these systems via an effective mean-field theory, including the effect of quantum fluctuations via a higher-order effective interaction. I will outline our current understanding of the properties of these states and highlight open questions.

[1] L. Chomaz & al, Dipolar physics: a review of experiments with magnetic quantum gases, Reports on Progress in Physics 86, 026401 (2023).

[2] L. Chomaz, Quantum-stabilized states in magnetic dipolar quantum gases, arXiv preprint 2504.06221 (2025)

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Invited by Prof. Arnaud Triay

The problem of simultaneously estimating multiple means from independent samples has a long history in statistics, from the seminal works of Stein, Robbins in the 50s, Efron and Morris in the 70s and up to the present day. This setting can be also seen as an (extremely stylized) instance of "personalized federated learning" problem, where each user has their own data and target (the mean of their personal distribution), but potentially want to share some relevant information with "similar" users (though there is no information available a priori about which users are "similar"). In this talk I will concentrate on contributions to the high-dimensional case, where the samples and their means belong to R^d with "large" d. \[ \] We consider a weighted aggregation scheme of empirical means of each sample, and study the possible improvement in quadratic risk over the simple empirical means. To make the stylized problem closer to challenges encountered in practice, we allow (a) full heterogeneity of sample sizes (b) zero a priori knowledge of the structure of the mean vectors (c) unknown and possibly heterogeneous sample covariances. \[ \] We focus on the role of the effective dimension of the data in a "dimensional asymptotics'' point of view, highlighting that the risk improvement of the proposed method satisfies an oracle inequality approaching an adaptive (minimax in a suitable sense) improvement as the effective dimension grows large. \[ \] (This is joint work with Jean-Baptiste Fermanian and Hannah Marienwald)

Gradient descent (GD) and its variants are vital ingredients in neural network training. It is widely believed that the impressive generalization performance of trained models is partially due to some form of implicit bias of GD towards specific minima of the loss landscape. In this talk, we will review and discuss approaches to rigorously identify and analyze implicit regularization of GD in simplified training settings. We furthermore provide evidence suggesting that a single implicit bias is not sufficient to explain the effectiveness of GD in training tasks.

The theory of `Balanced Neural Networks’ is a very popular explanation for the high degree of variability and stochasticity in the brain’s

activity. We determine equations for the hydrodynamic limit of a balanced all-to-all network of $2n$ neurons for asymptotically large

$n$. The neurons are divided into two classes (excitatory and inhibitory). Each excitatory neuron excites every other neuron, and each

inhibitory neuron inhibits all of the other neurons. The model is of a stochastic hybrid nature, such that the synaptic response of each

neuron is governed by an ordinary differential equation. The effect of neuron j on neuron k is dictated by a spiking Poisson Process, with

intensity given by a sigmoidal function of the synaptic potentiation of neuron j. The interactions are scaled by O(n^{-1/2}) , which is

much stronger than the O(n^{-1}) scaling of classical interacting particle systems ( the most common scaling used in mathematical

neuroscience). We demonstrate that, under suitable conditions, the system does not blow up as n tends to infinity because the network

activity is balanced between excitatory and inhibitory inputs. If the synaptic dynamics is linear, then the limiting population dynamics is proved to be Gaussian: with the mean

determined by the balanced between excitation and inhibition, and the variance determined by the Central Limit Theorem for

inhomogeneous Poisson Processes. The limiting equations can thus be expressed as autonomous Ordinary Differential Equations for

the means and variances. We search for conditions under which spatially-distributed patterns exist, as well as oscillations.

We study finite-player dynamic stochastic games with heterogeneous interactions and non-Markovian linear-quadratic objective functionals. We derive the Nash equilibrium explicitly by converting the first-order conditions into a coupled system of stochastic Fredholm equations, which we solve in terms of operator resolvents. When the agents' interactions are modeled by a weighted graph, we formulate the corresponding non-Markovian continuum-agent game, where interactions are modeled by a graphon. We also derive the Nash equilibrium of the graphon game explicitly by first reducing the first-order conditions to an infinite-dimensional coupled system of stochastic Fredholm equations, then decoupling it using the spectral decomposition of the graphon operator, and finally solving it in terms of operator resolvents. Moreover, we show that the Nash equilibria of finite-player games on graphs converge to those of the graphon game as the number of agents increases. This holds both when a given graph sequence converges to the graphon in the cut norm and when the graph sequence is sampled from the graphon. We also bound the convergence rate, which depends on the cut norm in the former case and on the sampling method in the latter. Finally, we apply our results to various stochastic games with heterogeneous interactions, including systemic risk models with delays and stochastic network games. ______________________________

Invited by Prof. Alexander Kalinin

Credibility methods in insurance provide a linear approximation, formulated as a weighted average of claim history, making them highly interpretable for estimating the predictive mean of the a posteriori rate. In this presentation, we extend the credibility method to a generalized coefficient regression model, where credibility factors—interpreted as regression coefficients—are modeled as flexible functions of claim history. This extension, structurally similar to the attention mechanism, enhances both predictive accuracy and interpretability. A key challenge in such models is the potential issue of non-identifiability, where credibility factors may not be uniquely determined. Without ensuring the identifiability of the generalized coefficients, their interpretability remains uncertain. To address this, we first introduce a state-space model (SSM) whose predictive mean has a closed-form expression. We then extend this framework by incorporating neural networks, allowing the predictive mean to be expressed in a closed-form representation of generalized coefficients. We demonstrate that this model guarantees the identifiability of the generalized coefficients. As a result, the proposed model not only offers flexible estimates of future risk—matching the expressive power of neural networks—but also ensures an interpretable representation of credibility factors, with identifiability rigorously established. This presentation is based on joint work with Mario Wuethrich (ETH Zurich) and Hong Beng Lim (Chinese University of Hong Kong).

Self-interacting Markov chains arise in a range of models and applications. For example, they can be used to approximate the quasi-stationary distributions of irreducible Markov chains and to model random walks with edge or vertex reinforcement. The term self-interacting Markov chain is something of a misnomer, as such processes interact with their full path history at each time instant, and therefore are non-Markovian. Under conditions on the self-interaction mechanism, we establish a large deviation principle for the empirical measure of self-interacting chains on finite spaces. In this setting, the rate function takes a strikingly different form than the classical Donsker-Varadhan rate function associated with the empirical measure of a Markov chain; the rate function for self-interacting chains is typically non-convex and is given through a dynamical variational formula with an infinite horizon discounted objective function.

We start the talk by presenting general results of strong density of sub-algebras of bounded Lipschitz functions in metric Sobolev spaces. We apply such results to show the density of smooth cylinder functions in Sobolev spaces of functions on the Wasserstein space $\mathcal P_2$ endowed with a finite positive Borel measure. As a byproduct, we obtain the infinitesimal Hilbertianity of Wassertein Sobolev spaces. By taking advantage of these results, we further address the challenging problem of the numerical approximation of Wassertein Sobolev functions defined on probability spaces. Our particular focus centers on the Wasserstein distance function, which serves as a relevant example. In contrast to the existing body of literature focused on approximating efficiently pointwise evaluations, we chart a new course to define functional approximants by adopting three machine learning-based approaches: 1. Solving a finite number of optimal transport problems and computing the corresponding Wasserstein potentials. 2. Employing empirical risk minimization with Tikhonov regularization in Wasserstein Sobolev spaces. 3. Addressing the problem through the saddle point formulation that characterizes the weak form of the Tikhonov functional's Euler-Lagrange equation. As a theoretical contribution, we furnish explicit and quantitative bounds on generalization errors for each of these solutions. In the proofs, we leverage the theory of metric Sobolev spaces introduced above and we combine it with techniques of optimal transport, variational calculus, and large deviation bounds. In our numerical implementation, we harness appropriately designed neural networks to serve as basis functions. Consequently, our constructive solutions significantly enhance at equal accuracy the evaluation speed, surpassing that of state-of-the-art methods by several orders of magnitude.

We will discuss our work on the convergence of iterative hard threshold algorithms for sparse signal recovery problems. For classification problems with nonseparable data this algorithm can be thought of minimizing the so-called ReLU loss. It seems to be very effective (statistically optimal, simple iterative method) for a large class of models of nonseparable data - sparse generalized linear models. It is also robust to adversarial perturbation.

Based on joint work with Namiko Matsumoto.