This talk will be devoted to two main topics. First, we discuss multi-agent models where the microscopic state of each individual is described by a spatial position and a probability measure (interpreted as a mixed strategy) over a compact metric space. The evolution is governed by a non-local interaction mechanism and by stochastic effects acting on the spatial component of the state. We illustrate well-posedness and propagation of chaos results for these systems.

Second, we present an alternative technique to prove the well-posedness of the mean-field equation describing the Consensus-Based Optimization (CBO) dynamics. The method is based on a truncation argument in the space of probability measures, and allows us to study the well-posedness problem in the classical framework of Sznitman.

Finally, we conclude by describing how these techniques can be combined to obtain well-posedness results for CBO-type models in which the state of each agent is augmented by an additional variable: a stochastic rate of information modeling the individual's knowledge of the environment.

This talk will explore the mean-field limit of non-exchangeable particle models. Starting with a model that describes the trajectories of N particles in d-dimensional space, the goal is to derive a model that describes the evolution of the distributions of the positions and velocities of these particles. Mathematically, this amounts to proving that the solution to an ordinary differential equation in R^{2dN} converges, as N tends to infinity, to the weak solution of a partial differential equation in the set of probability measures on R^{2d}. The classical interpretation of this limit requires working with an exchangeable model, that is where the interaction between two particles depends only on their respective positions and velocities. However, this assumption is unlikely to be valid in many biological systems, such as herds of cattle or sheep, where other parameters, such as age or sex, can influence how two individuals interact. First, we will present a framework, based on the notions of graphons and fibered probability spaces, that allows us to interpret the mean-field limit of non-exchangeable particle models. We will then apply this framework to derive a non-exchangeable Vlasov-type equation from the Cucker-Dong model, which incorporates the three main interactions used to describe large groups of animals: alignment, long-range attraction, and short-range repulsion.

The model of oscillating random walks introduced by Kempermann is one of the simplest example of a Markov chain with discontinuous statistics. We consider the situation when this chain converges, after proper rescaling, towards a skew Brownian motion. In the talk I will discuss the corresponding local limit theorem.

We study the mixing time of the simple random walk on the giant component of supercritical $d$-dimensional random geometric graphs generated by the unit intensity Poisson Point Process in a $d$-dimensional cube of volume $n$. With $r_g$ denoting the threshold for having a giant component, we show that for every $\epsilon>0$ and any $r\geq(1+\epsilon)r_g$, the mixing time of the giant component is with high probability $\Theta(n^{2/d}/r^2)$, thereby closing a gap in the literature. Our analysis also implies that the relaxation time is of the same order. Joint work with M. Kiwi and C. Martinez.

Mathematical modelling frameworks are becoming the foundation of in silico studies of the evolutionary dynamics of cell-to-cell communication in bacterial populations living in biofilms or growing in nutrient media. A significant trend in recent decades has been the development of hybrid models for describing complex, difficult to formalize microbial systems to predict and control their states. The present work continues these efforts by extending the hybrid framework into three advanced directions that complement the previously established models. The first direction addresses the development of a reaction-diffusion model that integrates nutrient-dependent bacterial growth, quorum-sensing-mediated population regulation, and dynamic resistance factor production during antibiotic treatment. Simulation results suggest the possibility of propagating wave fronts under certain parameter conditions, and the analysis indicates that the wave velocity and spatial profile may be influenced by the nutrient diffusion coefficient and the quorum activation threshold. The results indicate that the efficacy of antibiotic therapy depends non-monotonically on quorum-sensing intensity. Weak communication fails to trigger the diffusion barrier, whereas excessive signaling undermines bacterial protection through negative feedback. The second direction introduces an advanced hybrid computational framework for the discrete-in-space dynamical modeling of bacterial biofilms. The approach combines a cellular automaton on a hexagonal lattice with discrete analogues of reaction-diffusion equations governing nutrient and signaling molecule distribution, incorporating a quorum sensing feedback mechanism that links local signal concentration to biofilm spreading. A two-parameter analysis reveals a curved transition boundary in the nutrient-threshold plane, demonstrating that the effective quorum sensing activation threshold depends on nutrient availability. The third direction explores the application of physics-informed neural networks for solving direct and inverse problems in quorum-sensing dynamics. For a basic model of bacterial quorum sensing that combines the signaling molecule concentration, the degradation enzyme activity, and the bacterial population size, the PINN is set up to predict the time course of these variables and to recover the parameters governing the influence of natural enzymatic degradation.

The Wasserstein distance $\mathcal{W}_p$ is an important instance of an optimal transport cost. Its numerous mathematical properties as well as applications to various fields such as mathematical finance and statistics have been well studied in recent years. The adapted Wasserstein distance $\mathcal{A}\mathcal{W}_p$ extends this theory to laws of discrete time stochastic processes in their natural filtrations, making it particularly well suited for analyzing time-dependent stochastic optimization problems.

While the topological differences between $\mathcal{A}\mathcal{W}_p$ and $\mathcal{W}_p$ are well understood, their differences as metrics remain largely unexplored beyond the trivial bound $\mathcal{W}_p\lesssim \mathcal{A}\mathcal{W}_p$. This paper closes this gap by providing upper bounds of $\mathcal{A}\mathcal{W}_p$ in terms of $\mathcal{W}_p$ through investigation of the smooth adapted Wasserstein distance. Our upper bounds are explicit and are given by a sum of $\mathcal{W}_p$, Eder's modulus of continuity and a term characterizing the tail behavior of measures. As a consequence, upper bounds on $\mathcal{W}_p$ automatically hold for $\mathcal{AW}_p$ under mild regularity assumptions on the measures considered. A particular instance of our findings is the inequality $\mathcal{A}\mathcal{W}_1\le C\sqrt{\mathcal{W}_1}$ on the set of measures that have Lipschitz kernels.

Our work also reveals how smoothing of measures affects the adapted weak topology. In fact, we find that the topology induced by the smooth adapted Wasserstein distance exhibits a non-trivial interpolation property, which we characterize explicitly: it lies in between the adapted weak topology and the weak topology, and the inclusion is governed by the decay of the smoothing parameter.

This talk is based on joint work with Jose Blanchet, Martin Larsson and Jonghwa Park.

Lattice gauge theories are probabilistic models from statistical mechanics, introduced in the 1970s as a discrete model of gauge theories from physics. From a probabilistic point of view, they can be seen as a higher-dimensional analogue of classical spin systems such as the Ising model: spins live on vertices and interact along edges in the Ising model; while spins live on edges and interact around plaquettes (2-cells) in lattice gauge theories.

In this talk, we will introduce the $\mathbb{Z}_2$ lattice gauge theory on the lattice $\mathbb{Z}^m, ~ m \geq 2$, and discuss some of its main questions, such as phase transitions and correlation inequalities.

We will then focus on Ursell functions, or connected correlation functions, which are higher-order analogues of covariance. While Shlosman's theorem (1986) shows that Ursell functions on spins in the Ising model have alternating signs, depending only on the number of spins considered, we will see that this picture breaks down for lattice gauge theory. In particular, at sufficiently low temperature and in dimension $m \geq 3$, one can prove that for any number of Wilson loops, there is a choice of Wilson loop observables whose Ursell function is positive.

Graph alignment asks which vertex of a given graph corresponds to which vertex of a second, unlabelled graph. We formalise this question as a statistical inference problem: two random graphs are correlated through a latent vertex correspondence, and the goal is to recover this correspondence with high probability in polynomial time.

In this talk, we introduce a model of correlated Erdős–Rényi graphs that may have different numbers of vertices as well as varying edge densities. In the sparse setting with constant average degrees, these graphs are locally tree-like. Our alignment algorithm compares local tree neighbourhoods, which leads to a tree correlation testing problem. The feasibility of this testing problem exhibits a sharp phase transition, which we quantify in our main result. This result is surprising in two ways. First, the density of one tree can compensate for the sparsity of the other. Second, the proof revolves around a diagonalisation formula for the likelihood ratio over the space of unlabelled trees.

In this talk, Prof. Fanghui Liu will discuss how to improve the performance of Low-Rank Adaption (LoRA) for fine-tuning in large language models (LLMs) guided by theory. The theoretical results show that LoRA will align to the certain singular subspace of one-step gradient of full fine-tuning. Hence, the subspace alignment and generalization guarantees can be directly achieved by a well-designed spectral initialization strategy for both linear and nonlinear models. The analysis leads to the LoRA-One algorithm, a theoretically grounded algorithm that achieves significant empirical improvement over vanilla LoRA and its variants on several benchmarks by fine-tuning Llama 2. Interestingly, the results can also demonstrate that one-step Muon in full fine-tuning is equivalent to LoRA under spectral initializations.

Consider the problem of minimizing, over a space of probability measures, the sum of an energy and the entropy, which arises in many situations (models from statistical physics, high-dimensional algorithms...). The associated Wasserstein gradient flow can be interpreted as a nonlinear Langevin process, with the entropy cost leading to Brownian noise. A natural variation of this process with momentum is the underdamped Langevin process, which corresponds to the Vlasov-Fokker-Planck equation. We will see that, for displacement-convex energies, this process achieves a Nesterov acceleration with respect to the gradient flow, meaning that its convergence rate is of the order of the square root of the Polyak-Lojasiewicz constant of the objective function.

We investigate a family of radially symmetric Coulomb gas systems at inverse temperature $\beta=2$ coming from the normal matrix model. The family is characterised by the property that the density of the equilibrium measure vanishes on a ring at radius $r_∗$, which lies strictly inside the droplet. The large n expansion of the free energy is obtained up to a novel $n^{1/4}$ term. Next, we perform a scaling limit of the correlation kernel at the $n^{1/4}$ scale and obtain a new limiting kernel in the bulk, which differs from the well-known Ginibre kernel.

Joint work with Matthias Allard based on arXiv:2509.24529

We shed some elementary light on the Hambly-Lyons uniqueness theorem and provide new results on the reconstruction of reduced path from their signature transform. (joint work with Walter Schachermayer and Valentin Tissot Daguette).

This paper develops a framework for cost-sensitive training of probabilistic machine learning models that predict the direction of aggregate stock returns. We introduce a bi-objective loss function that augments conventional log-loss optimization with an expected error-cost objective, using the option-implied conditional value-at-risk as a forward-looking measure of misclassification cost. We illustrate how cost-sensitive learning produces economically sensible predictions. Market timing strategies constructed using forecasts from elastic-net logistic regression and gradient boosted decision trees trained using our framework improve out-of-sample risk-adjusted returns and substantially reduce downside risk.

The (Non-Preemptive) Throughput Maximization problem is a natural and fundamental scheduling problem. We are given n jobs, where each job j is characterized by a processing time and a time window, contained in a global interval [0,T), during which j can be scheduled. Our goal is to schedule the maximum possible number of jobs non-preemptively on a single machine, so that no two scheduled jobs are processed at the same time. This problem is known to be strongly NP-hard. The best-known approximation algorithm for it has an approximation ratio of ~1.551+eps [Im, Li, Moseley IPCO'17], improving on an earlier result in [Chuzhoy, Ostrovsky, Rabani FOCS'01]. In this paper we substantially improve the approximation factor for the problem to 4/3+eps for any constant eps>0. Using pseudo-polynomial time (nT)^{O(1)}, we improve the factor even further to 5/4+eps. Our results extend to the setting in which we are given an arbitrary number of (identical) machines.

Effectively describing the long-time dynamical behavior of quantum systems is a long-standing open problem. When the coupling constant $\lambda$ of the interaction is small, the dynamics of the system up to kinetic time $t \sim \lambda^{-2}$ is conjectured to be effectively governed by the Boltzmann-Nordheim kinetic equation. In this talk, we consider a system of weakly interacting lattice fermions at thermal equilibrium, and I will describe how the two-point time correlation function of the system depends on the equilibrium solution of the Boltzmann-Nordheim equation. We approach the problem using a perturbation expansion and represent the appearing terms by Feynman diagrams.

This is joint work with Phan Thành Nam, Herbert Spohn, and Minh-Binh Tran.

Consider an unknown random vector X, taking values in R^d. Is it possible to "guess" its mean accurately if the only information one is given consists of N independent copies of X? More accurately, given an arbitrary norm on R^d, the goal is to find a mean estimation procedure: upon receiving a wanted confidence parameter \delta and N independent copies X_1,...,X_N of an unknown random vector X - that has a finite mean and covariance -, the procedure returns \hat{\mu} for which the error \| \hat{\mu} - E X\| is as small as possible with probability at least 1-\delta (with respect to the product measure).

The mean estimation problem has been studied extensively over the years and I will present some of the ideas that have led to its solution (and to the solution of other questions of a similar flavour that I will outline). Two surprising facts are that in all these problems the obvious choices fail miserably (for mean estimation, that choice is N^{-1}\sum_{i=1}^N X_i); and, that the solution behaves as if the (arbitrary) random vector X were gaussian. ________________________________________ Invited by Prof. Holger Rauhut

In this talk, we introduce a new class of kinetic models that overcome the standard assumption in kinetic transport theory that collisions occur instantaneously. In particular, this modeling approach is interesting for applications in the life sciences, where the interaction time between biological agents cannot be meaningfully neglected. At the level of the underlying stochastic processes, this amounts to replacing the jump processes that define collisions with continuous stochastic processes. As an example, we will investigate a kinetic model with non-instantaneous alignment collisions between particles. The collisions are described by a transport process in the joint state space of the colliding particles, where the states of the particles approach their midpoint. Moreover, we will elaborate on the question of which model can be used as an accurate first-order non-instantaneous correction in the regime where the collision time is very small, implying that the collisions are almost instantaneous. Lastly, the instantaneous limit will be considered, which leads to standard collisional kinetic models of the Boltzmann type.

This is joint work with Carmela Moschella, Christian Schmeiser and Veronica Tora.

Loss underreporting in insurance refers to the practice that insureds intentionally hide the full extent of losses to insurance companies (insurers). This behavior is driven by the “hunger for bonus” embedded in experience rating systems for pricing insurance policies, which reward insureds who do not report a claim but penalize those who do. In this talk, we first formulate optimal loss reporting problems under both discrete-time and continuous-time models and offer a rigorous analysis of such problems. Next, we discuss how strategic underreporting affects the insured’s demand for insurance contracts. Finally, we extend the analysis to a game framework to study how insureds’ loss reporting impacts the insurer’s pricing strategies

In this talk, I propose a gradient-flow interpretation for a class of nonlinear kinetic Fokker—Planck equations (among which the linear case of the Vlasov—Fokker—Planck). The challenge lies in the fact that such dynamics are driven by an interplay of conservative and dissipative effects. Namely, free transport and diffusion in the velocity variables only. Thus, we rely on the geometry of “kinetic optimal transport”, which is based on Newton’s equations. In this framework, regular curves of measures can be rewritten as solutions to Vlasov’s equations, for a certain force field.

The main result of the talk is a chain rule for a family of free energy functionals along Vlasov’s equations, and the computation of the slope of the energy. Curves of maximal slope correspond to solutions to the kinetic Fokker—Planck equation. I will then discuss convergence for JKO-like schemes, and further analytical properties of the non-linear kinetic equations, together with a few open questions. My presentation is based on joint work with Guillaume Carlier, Jean Dolbeault, Jan Maas, and Filippo Quattrocchi.

Bistable planar fronts play a fundamental role across a wide range of applications as propagating or stationary transition layers between two stable homogeneous states. In multidimensional dissipative systems, such as reaction-diffusion models, these fronts are known to be asymptotically stable against localized perturbations under natural spectral assumptions. However, this localization requirement is not entirely satisfactory. Not only does it exclude initial perturbations corresponding to a spatial translation of the front, but it also prevents certain interesting dynamics of the perturbed front from being observed. In particular, the front interface may undergo persistent oscillations without ever settling to a steady translate, precluding asymptotic (orbital) stability. In this talk, we establish Lyapunov stability of bistable planar fronts in general multi-component reaction-diffusion systems under fully nonlocalized perturbations. Such perturbations could previously be treated only for scalar equations via comparison principles. Moreover, we show that the leading-order dynamics of the front interface are governed by a viscous Hamilton-Jacobi equation. This effective description reveals that asymptotic stability can be recovered by imposing localization of perturbations in the transverse spatial directions. This is joint work with Joris van Winden (Leiden University).

Perel’man and Pukhov studied the ratio between successive inner and outer radii of an $n$-dimensional convex body $K$ and they showed that \[R_{n-i+1}(K)/r_i(K)\leq i+1, \qquad 2 \leq i \leq n-1,\] which is far from being the best possible. Above, $R_{n-i+1}(K)$ is the smallest radius of a solid cylinder with $(n-i+1)$-dimensional spherical cross-section containing $K$ and $r_i(K)$ means the biggest radius of an $i$-dimensional Euclidean disc contained in $K$.

In this talk we show some relations between this inner and outer measures with the diameter and minimal width of a convex body, respectively. As a consequence, we derive improved upper bounds for this quotient for several values of the parameter $i$.

This is a joint work with Bernardo Gonzalez Merino and Mia Runge.

The cover time of a Markov chain is the first time at which every state has been visited at least once. In this talk, we consider random walks that jump to stationarity every L steps; or at rate 1/L, where L is a given parameter that may diverge. We will show that the order of the expected cover time is the same in both setups and study when the cover time is concentrated. Aldous proved, in 1991, that for reversible Markov chains, the cover time is concentrated around its expectation if and only if the maximal expected hitting time is of strictly smaller order than the maximal expected cover time. We will give a similar concentration criterion and show that the concentration of the cover time is equivalent in both setups under certain conditions. Joint work with Omer Angel, Jonathan Hermon and Pietro Lavino.

Echo state networks (ESNs) are recurrent models whose stability and memory properties are often summarized by the echo state property (ESP). This thesis develops a continuous-time ESN framework in the language of non-autonomous dynamical systems using the skew-product formalism. Building on this setting, we introduce an input-wise notion of multistability through an echo index, which measures the number of attracting solutions induced by a fixed input signal. We provide sufficient conditions guaranteeing echo index one, including contraction- type regimes and large-input regimes, and study robustness under perturbations of the input signal.

The Euclidean algorithm for computing the greatest common divisor of two numbers is one of the oldest known algorithms. In this talk, we discuss a higher-dimensional generalization of this algorithm and its applications, including the computation of lattice bases and determinants of integer matrices. To optimize the bit complexity, we establish several intermediate results. These contain a structural lemma on the arrangement of integral points in affine images of the unit cube, as well as analyses for special cases of matrix multiplication and solving linear systems of equalities.

"The talk addresses the quantitative study of resilience in dynamical systems by analyzing a range of resilience indicators within a unified mathematical and computational framework. Emphasis is placed on ecological resilience and on extending existing approaches from autonomous systems to settings with time-dependent perturbations, leading to nonautonomous dynamics. To address the lack of classical invariant structures in this context, characteristic performance ranges are used as practical reference states. The developed framework is applied to case studies, demonstrating how time-dependent disturbances and rate-induced effects influence resilience and tipping behavior."

We estimate the arrival time of vertices in a uniform random recursive tree from its unlabeled structure. Using centrality-based rankings, we derive tail bounds for the relative estimation error that are uniform in the vertex and the tree size. For the ranking induced by Jordan centrality, the probability that the estimate exceeds the true arrival time by a factor $S$ decays on the order of $1/S$, while the probability that it is smaller than the true arrival time by a factor $1/S$ decays exponentially in $S$. We introduce a refined centrality measure whose overestimation probability decays on the order of $(\log S)/S^{2}$, at the cost of a heavier lower tail of order $1/S^{2}$. These results identify a tradeoff between upper- and lower-tail performance in arrival-time estimation. Joint work with Simon Briend and Joost Jorritsma