Conditional-independence-based discovery uses statistical tests to identify a graphical model that represents the independence structure of variables in a dataset. These tests, however, can be unreliable, and algorithms are sensitive to errors and violated assumptions. Often, there are tests that were not used in the construction of the graph. In this talk, we will see that these redundant tests have the potential to detect or sometimes correct errors in the learned model. But we will further see that not all tests contain this additional information and that such redundant tests have to be applied with care. Precisely, we argue that the conditional (in)dependence statements that hold for every probability distribution are unlikely to detect and correct errors - in contrast to those that follow only from graphical assumptions.

Generative modeling builds on and substantially extends the classical idea of generating synthetic data from observed samples. In this talk, I will show that this principle provides a natural and theoretically well-founded foundation for bootstrap inference. The resulting method yields statistically valid confidence intervals for both regular and irregular estimators, including settings in which Efron’s bootstrap fails. From this perspective, the generative-modeling bootstrap could be viewed as a modern extension of the smoothed bootstrap: it has the potential to mitigate the curse of dimensionality and remain effective in challenging regimes where estimators may not be root-n consistent or admit a Gaussian limit.

Iron acquisition is essential for microbial survival, biofilm formation, and pathogenicity. Many bacteria use siderophores, such as pyoverdines produced by fluorescent pseudomonads, to chelate iron under limiting conditions, and studies show that siderophore production and cross-feeding confer a fitness advantage over species that lack these abilities. This phenomenon has been experimentally studied in species such as rhizosphere bacterium Pseudomonas protegens Pf-5 and Pseudomonas aeruginosa PAO1 using chemostat and batch culture systems. However, most previous studies do not account for spatial heterogeneity. In contrast to well-mixed, homogeneous environments, bacterial populations typically grow in spatially structured communities such as colonies. In this study, we address this gap by extending a previously developed ordinary differential equation model of iron chelation and cross-feeding into a reaction-diffusion model for two related species in two-dimensional space. This approach leads to a highly non-linear system of partial differential equations. To solve it numerically, we use a finite difference method with semi-implicit discretization in both space and time. Our findings indicate that siderophore production and cross-feeding confers a significant growth advantage even in spatially structured environments. This modeling approach provides a quantitative framework for understanding complex microbial interactions in heterogeneous environments.

Joint work with Andrew Dennehy, Ramchandran Muthukumar and Rebecca Willett. Score-based generative models exhibit a puzzling behavior: they often appear to cover all modes of a target multimodal distribution and yet may fail to learn the correct relative mode amplitudes, which can be interpreted as mixture weights. We resolve this apparent paradox by relating the diffusion score matching (DSM) loss to the error in estimating mixture weights from generated samples. We show that, even when the target score is insensitive to mixture weights, generated samples can recover the weights accurately if the scores at intermediate noise levels are informative about the weights. Accordingly, we define the diffusion score sensitivity index (DSSI) as the variation in the DSM loss relative to changes in a parameter. We then show that the DSSI governs the accuracy with which the parameter of the target distribution can be estimated from generated samples. For Gaussian mixtures in arbitrary dimensions, we prove that the mixture weight estimation errors are on the same order as the DSM loss under mild conditions. Empirically, we show the emergence of sensitivity during the noising process of benchmark data distributions under typical noise schedules, and that these sensitivity values predict how well a well-trained model recovers mixture weights. Furthermore, we show that the choice of noise schedule can reduce diffusion sensitivity, leading to mode amplification. Although we focus on mixture weights, the proposed sensitivity framework governs the recovery of any qualitative parameter of the target distribution.

The quantity of interest in the classical Cramér–Rao theory of unbiased estimation (i.e., the Cramér–Rao lower bound, exact efficiency in exponential families, and asymptotic efficiency of maximum likelihood estimation) is the variance, which represents the instability of an estimator when its value is compared to the value for an independently sampled data set from the same distribution. In this paper, we study a different quantity that captures the instability of an estimator when its value is compared to that obtained under an infinitesimal additive perturbation of the original data set; we refer to this as the sensitivity of an estimator.

The resulting theory of sensitivity is based on Wasserstein geometry in much the same way that the classical theory of variance is based on Fisher–Rao (equivalently, Hellinger) geometry. This perspective yields several results paralleling the classical case: a Wasserstein–Cramér–Rao lower bound for the sensitivity of any unbiased estimator, a characterization of models admitting unbiased estimators that attain this bound exactly, and a guarantee that Wasserstein projection estimators achieve the bound asymptotically. We illustrate the theory through a range of statistical examples, in some cases revealing new optimality properties of existing estimators and in others introducing new ones.

This is joint work with Nicolas Garcia Trillos (University of Wisconsin) and Adam Jaffe (Columbia), based on the paper: https://arxiv.org/pdf/2511.07414.

The classification of mixed-state phases requires criteria beyond two-point correlation functions, such as the decay of the mutual information (MI) and the conditional mutual information(CMI), with the latter encapsulated in the notion of Markov length. In this talk, we show how such static properties of the fixed point of a Lindbladian follow from natural dynamical features of its generator: rapid mixing and frustration-freeness. We focus on systems with long-range interactions, and prove i)that local Lindbladians satisfying (global) rapid mixing and frustration-freeness have fixed-points whose CMI decays with the shielding distance, and ii) that (local) rapid mixing together with primitivity and regularity implies global decay of MI. For long-range interactions both quantities decay polynomially rather than exponentially, in contrast to the finite- and short-range regimes where exponential decay (a finite Markov length) is expected within a phase. We further show that Gibbs states of long-range, non-commuting Hamiltonians satisfy a local Markov property at any temperature.

Wir untersuchen eine Darstellung berechenbarer Funktionen als totale Funktionen über der Menge der endlichen und unendlichen Folgen über {0,1}. In diesem Modell werden unendliche Folgen als nicht-terminierende Berechnungen interpretiert, während endliche Folgen die Summe ihrer Ziffern darstellen. Wir führen ein neues Definitionsprinzip ein, die Funktionsraum-Korekursion (function space corecursion), das gleichzeitig die Minimierung und die primitive Rekursion verallgemeinert. Dies definiert die Klasse der berechenbaren korekursiven Funktionen, die unter Komposition und Funktionsraum-Korekursion abgeschlossen ist. Wir beweisen, dass berechenbare korekursive Funktionen alle partiell-rekursiven Funktionen darstellen, und zeigen durch Übersetzung in den untypisierten Lambda-Kalkül, dass alle berechenbaren korekursiven Funktionen tatsächlich berechenbar sind. Abschliessend skizzieren wir, wie diese Darstellung berechenbarer Funktionen zu einem algebraischen Ansatz zur Berechenbarkeit und Komplexität im Begriffe sind, weiter zu entwickeln. ____________________________________ Invited by Prof. Helmut Schwichtenberg

Once diagnosed, cancer requires a fast, inexpensive and reliable assessment of the current state and potential progression of the disease. A new method for estimating tumor cell diffusivity 𝐷 and proliferation rate 𝛾 from single-point-in-time routine biopsies aims to deliver just that, and the ratio of its estimates 𝐷/𝛾 is a promising candidate for a new biomarker for risk-stratification in radiotherapy. Here, we extend the findings of the researchers at MD Anderson Cancer Center [1]<https://ecmtb2026.org/contributions/ms71-12#ref-Pasetto-2024>, who developed the method, by providing a simulation-based validation. The method is applied to in silico biopsies which are generated by solving the three-dimensional reaction-diffusion (RD) equation for different growth terms (exponential and logistic) with a Dirac-Delta initial condition, and transforming the continuous results into spatial point patterns via a form of reverse coarse-graining. References: [1] Pasetto, S., Montejo, M., Zahid, M. U., Rosa, M., Gatenby, R., Schlicke, P., Diaz, R., & Enderling, H. (2024). Calibrating tumor growth and invasion parameters with spectral spatial analysis of cancer biopsy tissues. Npj Systems Biology and Applications, 10(1), 1–9. https://doi.org/10.1038/s41540-024-00439-0 ----- -

Submarine missions represent a unique operational environment characterized by restricted natural time cues, shift work, and high cognitive demands. Understanding how individuals adapt their sleep, caffeine consumption, and circadian timing under these conditions is essential for maintaining performance and wellbeing. We analyzed data from 30 participants before, during, and after a submarine mission. During the mission, participants were assigned to one of two operational shifts.

We investigated caffeine consumption patterns, sleep behavior, and circadian rhythmicity using wearable sensor data and mathematical models. Individual caffeine intake records were incorporated into a pharmacokinetic model to estimate caffeine concentration profiles over time. These estimates enabled us to examine whether caffeine consumption was primarily socially driven, habitually driven, or used strategically as a countermeasure against sleepiness.

Sleep timing was analyzed relative to operational shift schedules. Differences between shifts were observed, indicating shift-specific adaptations of sleep behavior during the mission. Furthermore, post-mission assessments revealed considerable variability in circadian timing across participants. We hypothesize that differences in sleep timing during the mission contribute to this variability in circadian phase following mission completion.

Circadian timing was estimated using established mathematical models driven by environmental and behavioral inputs. However, model predictions did not consistently align with physiological measurements, indicating an incomplete understanding of the effective zeitgeber input in the submarine environment. Ongoing work focuses on identifying the factors responsible for these discrepancies and improving the characterization of circadian entrainment under conditions of limited natural time cues.

These findings highlight the complex interactions between caffeine use, sleep scheduling, and circadian regulation in operational settings and underscore the need for improved modeling approaches to understand human adaptation in isolated and shift-work environments.

In this talk, we discuss the long-time behavior of three of the most commonly used piecewise deterministic samplers: Randomized Hamiltonian Monte Carlo (RHMC), Zigzag process (ZZP), and Bouncy Particle Sampler (BPS). All of these, alongside kinetic Langevin dynamics, are second-order lifts of the overdamped Langevin dynamics. The kinetic samplers are advantageous due to their potentially accelerated long-time convergence rates and high accuracy in numerical implementation. We discuss the long-time behavior of these samplers in both L^2 energy and relative entropy, and showcase the differences between these dynamics, as well as between L^2 energy and entropy, explaining why the convergence results in entropy cannot be generalized to entropy. Joint work with Jianfeng Lu (Duke) and Pierre Monmarché (U Gustave Eiffel).

Rigorously enclosing Lyapunov exponents of stochastic flows is a long-standing and notoriously difficult problem: even determining the sign of the top exponent is often out of reach outside special structures or perturbative regimes. In this paper, we develop a computer-assisted method to obtain rigorous upper and lower bounds on Lyapunov exponents of stochastic flows under mild hypoellipticity assumptions.

Our approach starts from the Furstenberg-Khasminskii representation of the top Lyapunov exponent as an ergodic average over the invariant measure of the associated projective process, but crucially avoids any rigorous computation of that invariant measure. Instead, we introduce a numerical adjoint method, amenable to rigorous numerics, and show that an approximate solution of the associated Poisson equation yields a certified enclosure of the Lyapunov exponent via an explicit bound on the residual. This converts a non-rigorous numerical approximation into a rigorous quantitative estimate.

The method applies to systems on both compact and non-compact state spaces, does not require special geometric structure, and is not restricted to small-noise or other perturbative settings. We use it to prove positivity of the top Lyapunov exponent for several stochastic systems, including examples exhibiting noise-induced chaos and parameter-dependent sign changes, and we combine it with continuation methods to obtain rigorous bounds over large parameter regions.

A central theme in combinatorial optimization is to represent a set of feasible solutions as a set of integer points X in an appropriate space and to optimize a linear function over them. To treat this problem algorithmically, a common approach is to devise a system of linear inequalities whose feasible integer points are precisely X. The smallest number of inequalities in such a system that does not use auxiliary variables is called the relaxation complexity rc(X). In this talk, we discuss the relaxation complexity of the arguably simplest full-dimensional set of integer points, the discrete standard simplex. Surprisingly, the number of inequalities needed in a relaxation depends strongly on the coefficients that are allowed: While it is known that a linear number of inequalities is needed when only using rational coefficients, we present an explicit and elementary construction showing that a logarithmic number of inequalities suffices when irrational coefficients are allowed.

Nonautonomous saddle node bifurcations commonly appear in mathematical models; however, an optimal control theory tailored to such systems is not yet well established. In this talk, we present conceptual work on the nonautonomous saddle node normal form. We develop an approach that reformulates the

optimal control problem into an unconstrained minimization problem by exploiting a connection between bifurcation and rate–induced overshoots. The key component is a rate function that assigns the critical rate to every nonautonomous forcing. We show that in the saddle node normal form, the critical rate is

unique, and hence the rate function is well defined.

While this reformulation is valid in any case, the rate function is often unknown. To resolve this, we additionally present an approximation of the rate function that leads to an analytic expression of the approximated control. This makes the approach appealing in situations in which we need repeated solves of the optimal control problem such as in cases that induce a sampling-based approach.

Finally, we show that the reformulated problem can be used to recover a generalization of the Euler-Lagrange equations.

TBA

Continuous-time stochastic systems are often used to model causal relationships between variables that evolve over time. In many applications, however, the full time evolution is not available. Instead, we only observe the system after it has reached stationarity. This leads to the question of which causal information can be recovered from such stationary data.

In this talk, I will consider this question for linear stochastic differential equations when the causal structure is known. I will explain why identifying the exact strength of a direct causal effect is generally not possible without additional assumptions. Instead, I will focus on whether a direct causal effect is positive or negative. This leads to the notion of edge-sign identifiability, which asks when the sign of a direct causal effect is uniquely determined by the stationary covariance matrix. I will present criteria for determining whether an edge sign is identifiable, non-identifiable, or partially identifiable, and illustrate these criteria using examples from both classical causal structures and cyclic systems.

Functional data analysis (FDA) provides statistical methods for analyzing samples of time-continuous stochastic processes. Measurements often arise in the form of sensor data for a scientific variable. The practical problem of irregular sensor disruptions has fostered interest in analyzing partially observed random functions. To allow statistical analysis, we develop the first inference methods for dependent, partially observed functional time series. Mathematically, we model data on the space of bounded functions equipped with the supremum norm. This allows simultaneous inference across the entire functional domain, including simultaneous confidence bands -- something existing Hilbert-space-based methods cannot provide. To study non-stationary trends along the time series, we extend state-of-the-art multiscale inference methods (originally developed for scalar data) to partially observed functions. The key application of the latter methods is testing for excessive pollution levels in inner cities. Interestingly, our results also improve on existing results for fully observed functional time series by avoiding a functional CLT.

The goal of this event is to bring together researchers working on structural aspects of convex geometry, such as Helly-type phenomena, optimal configurations in geometric inequalities, or combinatorial / discrete aspects. It is a 3-day Workshop, taking place from 9 a.m. until 5 p.m. on July 20 and 21, and until 1 p.m. on July 22. If you have any questions or would like to register, please write to: lozb@cit.tum.de.

List of Speakers: Gergely Ambrus (University of Szeged); Eleon Bach (Technical University of Munich); Zhang Chen (Tongji University); Katherina v. Dichter (Brandenburg University of Technology); Paolo Dulio (Politecnico di Milano); Ferenc Fodor (University of Szeged); Ansgar Freyer (Free University of Berlin); Ilias Ftouhi (Université de Nîmes); Bernardo González Merino (Universidad de Murcia); Florian Grundbacher (Technical University of Munich); Mei Han (Technical University of Berlin); Maria A. Hernández Cifre (Universidad de Murcia); Tomasz Kobos (Jagiellonian University); Alexander Litvak (University of Alberta); Márton Naszódi (Alfréd Rényi Institute of Mathematics); Matthias Schymura (University of Rostock); Konrad Swanepoel (London School of Economics and Political Science); Jesús Yepes Nicolás (Universidad de Murcia); Stefan Weltge (Technical University of Munich)

We present FLOP (Fast Learning of Order and Parents), a score-based causal discovery algorithm for linear models. It pairs fast parent selection with iterative Cholesky-based score updates, cutting run-times over prior algorithms. This makes it feasible to fully embrace discrete search, enabling iterated local search with principled order initialization to find graphs with scores at or close to the global optimum. The resulting structures are highly accurate across benchmarks, with near-perfect recovery in standard settings. This performance calls for revisiting discrete search over graphs as a reasonable approach to causal discovery.

Proving is a fundamental activity in mathematics, and its teaching has been widely discussed in mathematics education research. A central theme in this body of research concerns the transition from making or evaluating a conjecture to proving it, with proof construction often viewed as the ultimate goal of mathematical activity. However, proving also involves ongoing processes that extend beyond proof construction, including the revision and generalisation of proved statements. In this talk, I discuss these relatively understudied, explorative aspects of proving through two illustrative cases: one relates to Lakatos-style mathematical activity involving proofs and refutations in a secondary school context, and the other focuses on proof by mathematical induction at the undergraduate level. I also discuss implications for task design aimed at introducing explorative proving to students. ____________________ Invited by Prof. Stefan Ufer

In this talk, I will discuss new results on large deviations for stochastic evolution equations. Starting with the variational setting, I will present a large deviation principle (LDP) for a class of SPDEs that includes many new examples with gradient noise and unbounded spatial domains. We will then move to non-variational settings and explore LDPs for the stochastic 3D primitive equations and reaction-diffusion equations. While our method is based on the well-known weak convergence approach, main novel ingredients come from the theory of critical spaces and maximal regularity techniques. Based on joint work with Antonio Agresti and Mark Veraar.

This talk develops mean-field limits for heterogeneous stochastic interacting particle systems with states in separable Hilbert spaces and interaction structures described by digraph measures. The finite system is formulated in mild form, and its well-posedness is established using infinite-dimensional stochastic-analysis and semigroup methods.

The main result extends the digraph-measure mean-field framework to Hilbert-space-valued dynamics. The limit is a label-dependent family of path-space laws rather than a single exchangeable McKean--Vlasov law. We show existence and uniqueness of this limiting process, weak convergence in probability of empirical path measures to its averaged law, and Lipschitz regularity of in the label variable with respect to the Wasserstein distance. Under analytic-semigroup assumptions, the convergence further improves to stronger fractional-domain path topologies.

Finally, the theory is applied to two large heterogeneous learning systems, namely, an RKHS-valued diffusion-KLMS model and a delayed recurrent-neural-network model formulated on an infinite-dimensional Sobolev phase space. These examples provide continuum descriptions of distributed kernel learning and recurrent neural dynamics on large networks.