The parameterized function classes used in modern deep learning are highly redundant, meaning that many different parameter values can correspond to the same function. These redundancies, or parameter space symmetries, shape the geometry of the loss landscape and thereby govern optimization dynamics, generalization behavior, and computational efficiency. Focusing on fully connected multilayer perceptrons (MLPs) with ReLU activations, I will explain how the degree of this redundancy varies in highly inhomogeneous ways across parameter space. I will describe how this structure influences the topology of loss level sets and discuss its implications for optimization dynamics and model identifiability. Finally, I will present experimental evidence suggesting that the functional dimension of a network tracks the intrinsic complexity of the learning task.

We introduce a model of random walks on Z^3 with random orientations of lines. This model can be seen as a 3D-version of a model of diffusion in inhomogeneous porous medium that has been introduced by Matheron and de Marsily. This 3D-model is related to the new process of iterated random walk in random sceneries (PAPAPA in french). We establish a joint limit theorem for the random walk (PA in french), the random walk in random sceneries (PAPA in french), and the iterated random walk (PAPAPA). This result is a joint work with Nadine Guillotin-Plantard and Frédérique Watbled. We will explain the relation between this work and previous developments for random walks in random sceneries. We will also present a conjecture about iterated random walks of higher order, and discuss about the difficulties to establish this conjecture.

The talk is divided into two parts. In the first part, I will introduce structural equation processes as a model for causal inference in discrete-time stationary processes. A structural equation process (SEP) consists of a directed graph, an independent stationary (zero-mean) process for every vertex of the graph, and a filter (i.e., an absolutely summable sequence) for every link on the graph. Every structural vector autoregressive (SVAR) process, a commonly used linear time series model, admits a representation as a SEP. Furthermore, the Fourier-transformed SEP representation of an SVAR process is parameterized over the field of rational functions with real coefficients. Using this frequency domain parameterization, we will see that d- and t- separation statements about the causal graph (associated with the SVAR process) are generically characterized by rank conditions on the spectral density of the SVAR process. Here, the spectral density is considered as a matrix over the field of rational functions with real coefficients. Additionally, we will see that the Fourier-transformed SEP parameterization of an SVAR process comes with a notion of rational identifiability for the Fourier transformed link filters. This notion allows to reason about identifiability in the presence of latent confounding processes. For instance, the recent latent factor half-trek criterion can be used to determine if the effect (i.e., the associated link function) between two potentially confounded processes is a rational function of the spectral density of the observed processes. \[ \] In the second part of the talk, I will expand the SEP framework to include a specific class of non-stationary linear processes. This class of non-stationary SEPs includes SVAR processes with periodically changing coefficients. I will also demonstrate how this framework can be used to reason about identifiability in subsampled processes, i.e., when observations are gathered at a lower frequency than the frequency at which causal effects occur.

We present a multi-agent and mean-field formulation of a game between investors who receive private signals informing their investment decisions and who interact through relative performance concerns. A key tool in our model is a Poisson random measure which drives jumps in both market prices and signal processes and thus captures common and idiosyncratic noise. Upon receiving a jump signal, an investor evaluates not only the signal's implications for stock price movements but also its implications for the signals received by her peers and for their subsequent investment decisions. A crucial aspect of this assessment is the distribution of investor types in the economy. These types determine their risk aversion, performance concerns, and the quality and quantity of their signals. We demonstrate how these factors are reflected in the corresponding HJB equations, characterizing an agent's optimal response to her peers' signal-based strategies. The existence of equilibria in both the multi-agent and mean-field game is established using Schauder's Fixed Point Theorem under suitable conditions on investor characteristics, particularly their signal processes. Finally, we present numerical case studies that illustrate these equilibria from a financial-economic perspective. This allows us to address questions such as how much investors should care about the information known by their peers.

In our uncertain and ever-changing world, many systems face the danger of crossing tipping thresholds in the future. Therefore, there is a growing interest in developing swift and reliable early warning methods to signal such crossings ahead of time. Until now, most approaches have relied on critical slowing down, typically assuming white noise and neglecting spatial effects.

We introduce a data-driven method that reconstructs the linearised reaction–diffusion dynamics directly from spatio-temporal data. From the inferred model, we compute the dispersion relation and analyse the stability of Fourier modes, allowing early detection of both homogeneous and spatial instabilities.

By framing early detection as a data-driven stability analysis, this approach provides a unified and quantitative way to indicate whether and when a system is approaching a tipping point or a Turing-type transition.

In this talk I’ll describe the use of Design Based Research in an ongoing project to develop and improve a set of digital simulations that seek to develop mathematics teachers’ capacities to teach geometry through problems and classroom discussions. The intervention is inscribed in the approach known as practice-based teacher education and the simulations are conceived as approximations of practice—where the teacher-learner practices key tasks of teaching. The first simulation acquaints teachers with the problem space: using a problem about constructing a circle tangent to two lines to teach the tangent segments theorem where they act as observers of an avatar teacher. Teachers then simulate teaching the whole lesson to a class of student avatars for the first time, having the opportunity to select and sequence students’ work for classroom discussion and eliciting and responding comments from students. The following two simulations give them opportunity to notice students’ work and to respond to students’ contribution respectively. Finally, they have another opportunity at teaching the whole lesson. I’ll share considerations for the design of these 4 simulations and how they developed as a result of waves of data collection and analysis. Initial observation of the performance differences when simulating the teaching of the whole lesson (before and after the two learning simulations) suggest these are associated with performance gains. I’ll share how we are using participants’ responses during the learning activities to reveal learning traces that might help explain those gains. _______________________ Invited by Prof. Stefan Ufer

We provide a unified framework to obtain numerically certain quantities, such as the distribution function, absolute moments and prices of financial options, from the characteristic function of some (unknown) probability density function using the Fourier-cosine expansion (COS) method. The classical COS method is numerically very efficient in one-dimension, but it cannot deal very well with certain integrands in general dimensions. Therefore, we introduce the damped COS method, which can handle a large class of integrands very efficiently. We prove the convergence of the (damped) COS method and study its order of convergence. The method converges exponentially if the characteristic function decays exponentially. To apply the (damped) COS method, one has to specify two parameters: a truncation range for the multivariate density and the number of terms to approximate the truncated density by a cosine series. We provide an explicit formula for the truncation range and an implicit formula for the number of terms. Numerical experiments up to five dimensions confirm the theoretical results.

A graph is perfect if and only if its stable set polytope is equal to its associated QSTAB polytope, defined via clique inequalities. In 1975, Chvátal defined the analogous class of t-perfect graphs, which replaces the clique inequalities with certain odd-circuit inequalities. In 1994, Shepherd conjectured that the chromatic number of a t-perfect graph is at most 4. In 2024, Chudnovsky, Cook, Davies, Oum, and Tan have shown the first finite upper bound of 199,053.

This talk discusses their proof, how their bound arises, and why they believe that it can be considerably improved.

There is a well-known discrepancy in mathematical fluid mechanics between phenomena that we can observe and phenomena on which we have theorems. The challenge for the mathematician is then to formulate an existence theory of solutions to the equations of hydrodynamics which is able to reflect observation. The most important such observation, forming the backbone of turbulence theory, is anomalous dissipation. In the talk, we survey some of the recent developments concerning weak solutions in this context.

The Wasserstein distance W 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 AW distance 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 AW and W are well understood, their differences as metrics remain largely unexplored beyond the trivial bound W<AW . This talk closes this gap by providing upper bounds of AW in terms of W through investigation of the smooth adapted Wasserstein distance. Our upper bounds are explicit and are given by a sum of W, Eder's modulus of continuity and a term characterizing the tail behavior of measures. As a consequence, upper bounds on W automatically hold for AW under mild regularity assumptions on the measures considered. A particular instance of our findings is the inequality AW < CW^(1/2) 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.

Stochastic optimal control problems naturally arise in contexts such as optimal investment, optimal consumption, and economic growth. Moreover, many fundamental models in robust finance - such as G-Brownian motion, G-diffusions, or G-semimartingales - can be translated to frameworks of stochastic control. A central aspect of these problems is the connection between value functions and Hamilton-Jacobi-Bellman (HJB) equations. For controlled diffusions with sufficiently regular coefficients, this link is typically established either through the comparison method, relying on a comparison principle for discontinuous viscosity solutions, or via the verification approach, which requires the existence of classical or Sobolev solutions. In this talk, we consider a general class of controlled diffusions for which these traditional methods break down. We present a new approach that connects stochastic control problems and HJB equations by combining probabilistic and analytic techniques. Furthermore, we discuss uniqueness results, leading to stochastic representations of HJB equations in terms of control problems, and provide stability results for associated value functions.

In this talk, we study dependence uncertainty and the resulting effects on tail risk measures, which play a fundamental role in modern risk management. We introduce the notion of a regular dependence measure, defined on multimarginal couplings, as a generalization of well-known correlation statistics such as the Pearson correlation. The first main result states that even an arbitrarily small positive dependence between losses can result in perfectly correlated tails beyond a certain threshold and seemingly complete independence before this threshold. In a second step, we focus on the aggregation of individual risks with known marginal distributions by means of arbitrary nondecreasing left-continuous aggregation functions. In this context, we show that under an arbitrarily small positive dependence, the tail risk of the aggregate loss might coincide with the one of perfectly correlated losses. A similar result is derived for expectiles under mild conditions. In a last step, we discuss our results in the context of credit risk, analyzing the potential effects on the value at risk for weighted sums of Bernoulli distributed losses.

Martingales associated with path-dependent payoff functions are intrinsically linked to path-dependent PDEs. While this connection is typically established via a functional Itô formula, in this talk we present a semigroup-theoretic framework for the analytic characterization of martingales with path-dependent terminal conditions. Specifically, we show that a measurable adapted process of the form V(t) - ∫_0^t Ψ(s)ds is a martingale if and only if a time-shifted version of V is a mild solution to a final value problem (FVP) involving a path-dependent differential operator. We establish existence and uniqueness of solutions to such FVPs using the concept of evolutionary semigroups on path space. We also discuss the relationship between semigroups on path space, nonlinear expectations and their penalty functions. The talk is based on joint work with David Criens, Robert Denk and Markus Kunze.

Given the growing prevalence of authoritarian regimes and the emergence of anti-liberal tendencies in certain established democracies, gaining insight into the dynamic and statistical characteristics of political regimes is crucial. Despite their relevance, a comprehensive quantitative assessment of these dynamics on a historical scale remains largely unexplored. I will present a rigorous and quantitative analysis of the dynamic and statistical properties of political regimes worldwide by examining changes in freedoms of expression, association, and electoral quality throughout the 20th century. To do so, we used the multidimensional and disaggregated V-Dem dataset, which covers over 170 countries across more than a century, and we analyzed it with tools from statistical physics. We explored these dynamics using the Diffusion Map dimensionality-reduction technique applied to V-Dem data (1900-2021). Our analysis reveals that changes in freedoms of expression, association, and electoral quality follow a power-law distribution, indicating that political regimes exhibit scale-free dynamics. Through the lens of anomalous diffusion, we identified three distinct classes of dynamics in the data which are clearly dependent on the regime type. I will also present some work in progress. A stability analysis of the country dynamics reveals a sharp decline in stability around the year 2010. I will discuss potential explanations for this observation. This work presents a novel approach to the study of political systems. Our quantitative methodology promotes a bridge between political science and the physics of complex systems.

We present a framework for determining effectively the spectrum and stability of traveling waves on networks with symmetries, such as rings and lattices, by computing master stability curves (MSCs). Unlike traditional methods, MSCs are independent of system size and can be readily used to assess wave destabilization and multistability in small and large networks.

In this talk, I will present selected developments from the past decade on the geometry of random polytopes. Particular emphasis will be placed on fluctuation results, both those obtained by means of Stein’s method and those derived through alternative approaches. I will also highlight recent progress in the planar setting, where techniques from analytic combinatorics have opened up new perspectives.

The circuit diameter of a polyhedron is the maximum length (number of steps) of a shortest circuit walk between any two vertices of the polyhedron. Introduced by Borgwardt, Finhold and Hemmecke (SIDMA 2015), it is a relaxation of the combinatorial diameter of a polyhedron. These two notions of diameter lower bound the number of iterations taken by circuit augmentation algorithms and the simplex method respectively for solving linear programs. Recently, an analogous lower bound for path-following interior point methods was introduced by Allamigeon, Dadush, Loho, Natura and Végh (SICOMP 2025). Termed straight line complexity, it refers to the minimum number of pieces of any piecewise linear curve that traverses a specified neighborhood of the central path. In this paper, we study the relationship between circuit diameter and straight line complexity. For a n-dimensional polyhedron P, we show that its circuit diameter is up to a poly(n) factor upper bounded by the straight line complexity of linear programs defined over P. This yields a strongly polynomial circuit diameter bound for polyhedra with at most 2 variables per inequality.

This is joint work with Daniel Dadush and Zhuan Khye Koh.

Representing and processing data in spherical domains presents unique challenges, primarily due to the curvature of the domain, which complicates the application of classical Euclidean techniques. Implicit neural representations (INRs) have emerged as a promising alternative for high-fidelity data representation; however, to effectively handle spherical domains, these methods must be adapted to the inherent geometry of the sphere to maintain both accuracy and stability. In this context, we propose Herglotz-NET (HNET), a novel INR architecture that employs a harmonic positional encoding based on complex Herglotz mappings. This encoding yields a well-posed representation on the sphere with interpretable and robust spectral properties. Moreover, we present a unified expressivity analysis showing that any spherical-based INR satisfying a mild condition exhibits a predictable spectral expansion that scales with network depth. Our results establish HNET as a scalable and flexible framework for accurate modeling of spherical data. Finally, inspired by the HNET analysis, we propose a common framework to understand the expressivity of the positional encodings used in Euclidean and spherical INRs. ____________________ Invited by Prof. Johannes Maly

Diffusion models are one of the key architectures of generative AI. Their main drawback, however, is the high computational cost. The first part of the talk introduces diffusion models, outlining their main ideas and their role in modern generative AI. The second part of the talk will focus on our recent research showing how the concept of sparsity, well known especially in statistics, can provide a pathway to more efficient diffusion pipelines. Our mathematical guarantees prove that sparsity can reduce the influence of the input dimension on computational complexity to that of a much smaller intrinsic dimension of the data. Our empirical findings further confirm that inducing sparsity can indeed lead to better samples at a lower cost.

We present the relativistic quantum physics model hierarchy from Dirac-Maxwell to Vlasov/Euler-Poisson that models fast moving charges and their self-consistent electro-magnetic field. Our main interest is (asymptotic) analysis of these nonlinear time-dependent PDE, with focus on the Pauli-Poisswell/Darwin system which is the consistent model at first/second order in 1/c (c = speed of light) that keeps both relativistic effects "spin" and "magnetism". Emphasis is on the (semi)classicial limit for vanishing Planck constant. We use both WKB methods and Wigner functions where we extend the 1993 results of P. L. Lions & Paul and Markowich & Mauser on the limit from Schrödinger-Poisson to Vlasov-Poisson, with similar subtilities of pure quantum states vs mixed states. In the hope of taming the mathematical complications stemming from the magnetic field, we are interested also in developing "quantum/semiclassical velocity averaging lemmata" building on the 1988 ideas of Golse, Perthame, Sentis and P.L.Lions. This talk aims to explain the models, the new results & ideas of proofs/techniques, as worked out in joint works mainly with Jakob Möller (X) and also Pierre Germain (ICL), Changhe Yang (Caltech), François Golse (X).

Hermann Weyl changed his views on the foundations of mathematics several times, with commentators often distinguishing four different phases. These changes give us precious insights into Weyl’s philosophical positions, but also into the foundations of mathematics more generally.

Notwithstanding Weyl’s evolving views on foundations, there are also significant elements of continuity throughout his philosophical reflection. In this talk, I will focus on a constructive attitude that pervades Weyl’s philosophy of mathematics. One aspect I will elaborate on, are the important connections between Weyl’s constructive attitude and his changing views on quantification and logic.

Mean field limits are an important tool in the context of large-scale dynamical systems, in particular, when studying multiagent and interacting particle systems. While the continuous-time theory is well-developed, few works have considered mean field limits for deterministic discrete-time systems, which are relevant for the analysis and control of large-scale discrete-time multiagent system. We prove existence results for the mean field limit of very general discrete-time dynamical systems, which in particular encompass typical multiagent systems. As a technical novelty, we utilize kernel mean embeddings, which are an established tool in machine learning and statistics, but have been rarely used in the context of multiagent systems and kinetic theory. Our results can serve as a rigorous foundation for many applications of mean field approaches for discrete-time dynamical systems, from analysis, simulation and control to learning.

In this talk, we present two novel variants of the contact process. In the first variant individuals carry a viral load. An individual with viral load zero is classified as healthy and otherwise infected. If an individual becomes infected it begins with a viral load of one, which then evolves according to a Birth-Death process. In this model, viral load indicates severity of the infection such that individuals with a higher load can be more infectious. Moreover, the recovery times of individual is not necessarily exponentially distributed and can even be chosen to follow a power-law distribution.

In the second variant individuals are permanently infected albeit in two states: actively infected or dormant. The dynamics of these individual states are again governed by a Birth-Death process. Dormant infections do not interact with neighbouring individuals but may reactivate spontaneously. Active infections reactivate dormant neighbours at a constant rate and may become dormant themselves.

We present a Poisson construction for both variants. For the first model, we study the phase transition of survival and discuss existence of a non-trivial upper invariant law. Additionally, we derive a duality relationship between the two variant, which we use to uncover a phase transition regarding invariant distributions in the second variant.

The initial-boundary value problem for a system of nonlinear parabolic equations modeling the growth and migration of glioma cells is studied. New a priori estimates of the solution are obtained, based on which the non-local in time unique solvability of the initial-boundary value problem is proved. The conducted computational experiments demonstrate the ability of the mathematical model to reflect the desired properties like development of hypoxia in the tumor microenvironment, the switching of proliferative tumor cells to invasive migratory ones due to hypoxia, the regression of the vasculature in the area occupied by the tumor and simultaneous angiogenesis at the periphery of the tumor.

We develop and analyse a finite volume scheme for a nonlocal active matter system known to exhibit a rich array of complex behaviours. The model under investigation was derived from a stochastic system of interacting particles describing a foraging ant colony coupled to pheromone dynamics. In this work, we prove that the unique numerical solution converges to the unique weak solution as the mesh size and the time step go to zero. We also show discrete long-time estimates, which prove that certain norms are preserved for all times, uniformly in the mesh size and time step. In particular, we prove higher regularity estimates which provide an analogue of continuum parabolic higher regularity estimates. Finally, we numerically study the rate of convergence of the scheme, and we provide examples of the existence of multiple metastable steady states.

Joint work with Maria Bruna and Markus Schmidtchen.