BMS-BGSMath Junior Meeting

Bridging scales to learn cell–cell adhesion from partial trajectory data

This work presents a mathematical and computational framework for estimating parameters in cell–cell adhesion models. The study builds upon a
macroscopic, nonlocal PDE model involving two interacting cell populations influenced by attractive and repulsive forces, incorporating both adhesion dynamics and volume exclusion effects. Starting from a stochastic particle system, the nonlinear PDE system for the cell densities is formally derived through mean-field limits. The parameter estimation problem is addressed by minimising an error functional that combines macroscopic densities and individual trajectories.

The model parameters are estimated using a Bayesian approach, which involves sampling from the posterior distribution of the parameters given observed data. To efficiently sample from this distribution, the pre-conditioned Crank–Nicolson (pCN) Markov Chain Monte Carlo method is used. This algorithm is derivative-free, well-suited for high-dimensional spaces, and has a single tunable parameter to optimize performance.

Geometric Learning & Inference on Manifolds

The success of modern machine learning has renewed interest in the role of geometry for statistical inference. While classical learning algorithms assume Euclidean data representations, many applications naturally involve nonlinear spaces such as Riemannian manifolds, Lie groups, and shape spaces. Developing learning algorithms that respect these structures requires combining ideas from differential geometry, statistics, and machine learning.

This talk surveys recent developments in geometric learning, including intrinsic regression models for manifold-valued data and graph neural networks on nonlinear feature spaces. A common theme is the use of geometric structure as an inductive bias, enabling learning algorithms that are mathematically consistent, computationally efficient, and applicable to challenging problems in medical imaging, computational anatomy, and archaeology. I will discuss current perspectives on the interplay between geometric modeling and data-driven learning, a rapidly developing research area that may shape the next generation of learning methods.

How can we build graphs more efficiently in random environments?

For several decades now, researchers have considered different randomised processes for generating denser and denser graphs. Studying these processes allows us to understand how the characteristics of the resulting graphs evolve as they become denser. Crucially, these processes generate graphs autonomously, with only randomness dictating the outcome of the process. For this reason, it could well be that, when the resulting random graphs satisfy a property which we are interested in, many of their edges are unnecessary for this property. In what ways can we interfere with the process in order to improve the resulting graphs in this sense?

In this talk, I will present the classical random graph process and discuss what happens when we introduce an intelligent agent who wants to optimise the resources used throughout the process. We will discuss several optimal results about the necessary resources required to achieve different properties and present a host of open problems.

Lyapunov exponents in complex dynamics

Lyapunov exponents measure the average exponential rate of expansion of a map with respect to a given measure, making them a fundamental tool for understanding the long-term behaviour of dynamical systems. In this talk, I will introduce the notion of Lyapunov exponent in the setting of complex dynamics, discuss some of its key properties, and explain how it can be used to study the dependence of the dynamics on parameters.

This talk is based on joint work with Matthieu Astorg.

Intersection patterns via homotopy colimits

Given a family of subsets of a fixed space, it is often of importance to describe the collection of subfamilies having a nonempty intersection. Both the Nerve Theorem and the well-known results about convex sets due to Helly, Berge, and Breen are concerned with such questions. Among others, the work of Meshulam, Montejano, and Meunier & Montejano shows that these theorems have homological analogues, where the assumptions and sometimes also the conclusions are replaced by the statements they would imply about homology groups.

We provide a systematic approach to these latter results, leading to further generalizations with weaker assumptions than previously. For this, we use the toolkit of homotopy colimits, which we extend by homologically flavored lemmas, relating homotopy colimits to order- and nerve complexes. This is joint work with Imre Bárány and Pavle Blagojević.

Chaotic Phenomena in the Restricted Planar Circular Three-Body Problem Beyond the Routh Mass Ratio

The Lagrangian equilibrium points L₄ and L₅ of the restricted planar circular three-body problem (PCR3BP) undergo a bifurcation at the critical value of the mass ratio known as the Routh mass ratio. For smaller values of the mass ratio, these equilibria are elliptic, whereas for larger values they become complex saddles. This is an example of a Hamiltonian–Hopf bifurcation.

In this setting, the existence of transverse homoclinic orbits to L₄ (or L₅) yields highly intricate dynamics, including a plethora of horseshoes and the so-called blue-sky catastrophe.

We prove that, for every mass ratio sufficiently close to and larger than the Routh mass ratio, the stable and unstable manifolds of L₄ intersect transversely along a homoclinic orbit, provided that an explicit non-degeneracy constant does not vanish.

This is a joint work with I. Baldomá and P. Martín.

Phase-field models across biology and physics

We study diffuse-interface phase-field systems of Cahn-Hilliard and Allen-Cahn type. These models provide a flexible framework for describing a wide range of physical and biological phenomena. In particular, we discuss applications to tumor growth and the dynamics of saline droplets.

Our first aim is the thermodynamically consistent derivation of the governing PDE systems, for which we tailor the strategy to the specific features of each problem. We then address the mathematical analysis of these models, focusing on well-posedness. The main ingredients of the analysis are the construction of suitable discretization schemes, the derivation of a priori estimates, and the passage to the limit to establish the existence of solutions.

Shape uncertainty quantification via conditional LDDMM flow matching

Along with conditional flow matching and rectifying flows, stochastic interpolants have emerged as a versatile framework for generative modelling. While these methodologies demonstrate exceptional performance in applications involving images or snapshots on Cartesian grids, their extension to more complex domains, such as shapes or surfaces, poses significant challenges. Furthermore, the implementation of mini-batch optimal transport for kinetic energy density minimisation, while relatively straightforward on Cartesian grids, becomes computationally prohibitive and less practical when applied to general three-dimensional meshes with varying geometries. To address these limitations, we propose a novel approach based on graph neural networks and Large Deformation Diffeomorphic Metric Mapping (LDDMM) for conditional flow matching, building upon prior work. We present applications in the context of time-dependent cardiovascular simulations and shape-informed reduced-order modelling. This methodology facilitates data augmentation for three-dimensional biomedical shapes and also enables the generation of random perturbations of prescribed magnitudes for given shapes. Such capabilities are pivotal for quantifying the impact of uncertainties in the definition of computational domains from medical images on the estimation of relevant biomarkers. Inter-patient surrogate modelling for time-dependent cardiovascular simulations is also demonstrated through autoregressive conditional stochastic interpolants.

Navigating Random Games: Convergence and Cycles in Best Response Dynamics

Best response dynamics (BRD) represent arguably the most intuitive heuristic for repeated play in multi-agent systems. Players independently and sequentially update their strategies to maximize individual utility based on the immediate past action of their opponent. It is considered a fundamental mechanism for modeling strategic updates and learning in game theory. However, when applied to random games, the behavior of these dynamics reveals a rich, probabilistic landscape. This talk provides an overview of the research area surrounding BRD in random games, combining established literature with some of our most recent theoretical results. We begin by going through the established results on the converging behavior of BRD, e.g. the expected number of pure Nash equilibria. Because convergence is not always guaranteed, we will also discuss the non-converging behavior of these dynamics. We characterize the absorbing cycles (sink equilibria) that BRD frequently collapses into, discussing structural insights into their expected size, number etc. In the presence of n players, we introduce two distinct variants: Random and Cyclic BRD, and we compare their respective efficiencies and limitations in identifying Nash equilibria.

Algebraic Statistics of Discrete Statistical Models

Maximum likelihood (ML) estimation is an important method in statistics and data analysis. In this talk, we study the associated optimization problem from an algebraic perspective, with particular emphasis on the ML degree, an algebraic complexity measure of the estimation problem. We focus on discrete statistical models described by toric varieties. In this setting, the ML degree depends on the embedding of the underlying variety. For generic embeddings, the ML degree coincides with the degree of the variety, whereas the locus of non-generic embeddings is given by the principal A-determinant. We present several illustrative examples and discuss recent developments and current research directions.

Spectral problems for many body systems: energy of one- and two-component Bose gases

In a famous 1947 paper, Bogoliubov developed a theory for the spectral analysis of Hamiltonians describing the microscopic behavior of dilute quantum Bose gases. Since then, a major line of research in mathematical physics has been the rigorous justification for, and refinement of, Bogoliubov’s theory. In this talk, we review historical milestones regarding the research on thermodynamic energy of dilute Bose gases and present recent developments. Specifically, we derive a two-term energy expansion for a two-component Bose gas subject to repulsive, pairwise intra- and inter-species interactions. We compare this result to the single-component case, demonstrating its correspondence with the celebrated Lee-Huang-Yang formula and highlighting its universality with respect to the interaction potentials.

Equidistribution of Random Normal Matrices’ eigenvalues

Determinantal Point Processes are a class of point processes characterized by a built-in repulsion among the points. Such point processes are used by physicists to model fermionic systems, and to produce uniformly distributed points. An example is the spectrum of the Random Normal Matrix model.

As the number of eigenvalues increases, they accumulate in a certain compact subset called the droplet, making the empirical measure converge weakly to the equilibrium measure. This talk will focus on quantifying the equidistribution using the expected 2-Wasserstein distance, a metric on the space of finite measures defined via optimal transport. The main tool developed is a regularization using the heat equation with Neumann boundary conditions on a suitable subset of the droplet. This scheme can also be applied to Stationary point processes.

Equidistribution: from points to cycles

The equidistribution of points of small height is a strong result in Arakelov geometry that predicts the behaviour of a sequence of arithmetically simple points in a variety X defined over a number field. Applications can be found in diophantine geometry and dynamics, among others.

In this short talk, after recalling the classical statement for points, we will explore the higher dimensional situation. In particular, given a sequence of d-dimensional cycles in X with prescribed arithmetical features, we describe what is true and what can fail for the asymptotic of the corresponding integration currents.

Optimal Metrics in Odd-Dimensional Geometry

We consider an energy functional on the space of metrics compatible with certain odd-dimensional geometric structures (including contact, cosymplectic, and stable Hamiltonian structures). In dimension 3, or under additional assumptions, we show that structures with optimal metrics either have energy 0 or they are covered by left-invariant structures on certain Lie groups.

To prove this, we first describe the local structure (essentially only using calculus and linear algebra) and then use hyperbolic dynamics and Lie theory to obtain global results.

Joint work with Ángel González-Prieto, Eva Miranda, and Daniel Peralta-Salas.

On Payne-Weinberger inequalities

In this talk, I will first review the classical Payne–Weinberger inequality, which provides an upper bound on the difference between the first Dirichlet eigenvalue of a domain and that of the disk with the same volume. The bound depends only on the isoperimetric deficit of the domain. I will briefly discuss the main ideas behind its proof and explain how the same approach yields analogous inequalities for other operators, including Robin Laplacians and quantum dot Dirac operators.

Statistical Inference Under Privacy Constraints

Our research investigates the trade-off between preserving privacy and retaining statistical accuracy when working with multivariate data subject to componentwise local differential privacy (CLDP). Under CLDP, each component of the private data is released through an individual privacy channel, allowing for different levels of privacy protection across components or enabling each to be privatized by different entities with distinct privacy policies.

This framework also reflects practical scenarios in which joint privatization of all components is infeasible. We develop general techniques to establish minimax bounds that characterize the statistical cost of privacy in terms of the componentwise privacy levels α₁, …, α_d for d-dimensional data.

We showcase the power and flexibility of these methods through several statistical applications. In particular, the talk will focus on nonparametric density estimation, with and without privacy constraints. We derive matching upper and lower bounds (up to constants) and propose a corresponding adaptive, data-driven procedure. Additionally, we present an in-depth analysis of the effective privacy level, examining how private characteristics of individuals may be inferred from publicly released features of the same individuals.

Maximum likelihood thresholds for colored Gaussian graphical models

Colored Gaussian graphical models are statistical models arising from graphs with a coloring in its vertices and edges. We study maximum likelihood thresholds (MLT) for these models: the minimum number of observations that ensure existence of the maximum likelihood estimator. We extend known results for MLT to the colored setting, determine thresholds for certain families graphs and implement algorithms that exploit the underlying algebraic geometry.

Bayesian inference in stochastic reaction-diffusion equations using spatial information

We consider the Bayesian nonparametric estimation of the reaction term in a stochastic reaction-diffusion equation. Posterior contraction rates are proven by making use of the spatial ergodicity of the stochastic reaction-diffusion equation while the time horizon is fixed. We additionally prove a non-parametric Bernstein- von Mises Theorem for the posterior distribution. The analysis of the posterior requires new concentration results for spatial averages of transformation of the stochastic reaction-diffusion equation, which are based the combination of the Clark-Ocone formula with bounds on the marginal densities.

Minimal Covering Bodies

We study convex bodies that are inclusion minimal with the property that their integer translates cover ℝ^d, which we call minimal covering bodies. It was observed by Xue and Zong (2017) that such convex bodies are not necessarily translative tiles.

In this talk, we will see that minimal covering bodies are polytopes with at least 2d facets, which further implies that every convex covering body contains a polytopal (but not necessarily convex) translative tile. Further, we extend the method of Xue and Zong to obtain a large family of examples of minimal covering bodies which are not tiles, relating this class of bodies to covering properties of Minkowski combinations of convex bodies. This talk is based on joint work with Giulia Codenotti and Ansgar Freyer.

The temporal stochastic block model

Motivated by the need to understand infection spreading in inhomogeneous populations, we consider a temporal version of the stochastic block model, where each edge is equipped with a random label, interpreted as a timestamp. We study the size and structure of reachable sets via increasing paths (that is, paths whose edge timestamps are strictly increasing) where the connections per node are of the order of log n. We prove tight results for the first order asymptotics of the size of the reachable set from a typical vertex, and the proportions of reached vertices in each community.

Joint work with G. Lugosi and G. Perarnau

Enumeration of unlabeled graph classes

Counting graphs is a classical theme in combinatorics and graph theory. For instance, Cayley’s formula, which dates back to 1860, famously establishes that the number of labeled trees on n vertices is nⁿ⁻². However, the problem of counting the number of unlabeled trees on n vertices is much more difficult and in fact it was not solved until 1948 by Otter.

The striking contrast between the labeled and unlabeled settings stems from the presence of symmetries, which makes the latter considerably more challenging. This talk will provide an introduction to the enumeration of unlabeled graph classes, focusing on the techniques used to account for automorphisms. Central to this approach is Pólya’s enumeration theory, together with generating functions, which provide powerful tools for counting non-isomorphic combinatorial structures.

Using trees as a motivating example, we will discuss how these ideas can be extended to more general graph classes, particularly some families of chordal graphs. This is joint work with Michael Drmota and Clément Requilé.

Particle Correlations and Noise in Conservative SPDEs: Analysis and Numerics

This talk explores how correlations between particles affect the noise in equations describing their collective density. The first examples concern particles exposed to the same random motion, first through a common Brownian shift and then through a random environment that varies in space. In this setting, particles at different positions can be correlated in different ways, leading to conservative stochastic heat equations whose noise depends linearly on the density. This is contrasted with particles driven separately, where the resulting density fluctuations lead to the nonlinear noise in the Dean–Kawasaki equation. Alongside these examples, we discuss some of the analytical and numerical difficulties of conservative SPDEs, particularly whether simulations can retain basic properties of the original equations, such as conservation of total mass and nonnegativity. The talk is intended to serve as an introduction to the topic and includes numerical perspectives motivated by ongoing work.

Mean-field optimal control with stochastic leaders

We study the stochastic optimal control of populations influenced by a small number of leaders, considering both finite-agent systems and their mean-field limits. The limiting dynamics are described by a conditional McKean-Vlasov SDE coupled with the leaders’ dynamics, which we show to be equivalent to a nonlinear Fokker-Planck PDE/SDE system. Optimal controls for the mean-field system provide approximate strategies for the corresponding finite agent-based models, but their computation is challenging as the control acts on the space of probability measures. We develop a numerical method for approximating optimal controls of the limiting PDE/SDE system and demonstrate its effectiveness in steering agents towards consensus in a prototypical opinion dynamics model.

Microstructural Foundations of Rough Log-Normal Volatility Models

When modelling financial markets, we mathematicians like to use continuous-time models to approximate the discrete nature of real-world trading. In these continuous-time models, stock movements are driven by random noise. However, the behaviour of these markets is influenced by market participants buying and selling assets. This is known as market microstructure.

In this talk, we introduce such a market microstructure model, where orders to buy or sell an asset arrive according to a Poisson process and have a long-lasting impact on volatility. Under an appropriate scaling, we show that this discrete model converges to a popular continuous-time model, the so-called rough Bergomi model.

Mathematically, we discuss weak convergence in Skorokhod space, the space of right-continuous functions. In addition to establishing weak convergence, we also provide an explicit upper bound for the convergence rate and present numerical simulations and visualisations to illustrate the behaviour of the model.

Moving Mass, Decreasing Energy: Learning Probability Distributions with Wasserstein Gradient Flows

Many problems in machine learning require approximating a complicated probability distribution, for example, in Bayesian inference, variational inference, and generative modeling. In this talk, I will describe a gradient-flow perspective on this problem. A gradient flow describes how a point evolves to decrease an energy as rapidly as possible relative to a chosen geometry. But what does this mean when the evolving object is not a point in Euclidean space, but an entire probability distribution?

I will explain how optimal transport equips the space of probability measures with the so-called Wasserstein geometry, allowing notions such as velocity and steepest descent to be transferred to this infinite-dimensional setting. I will conclude by discussing how kernel regularization turns these gradient flows into interacting particle algorithms for learning complex target distributions.

The singular Φ⁴ equation on ℝ³, a simple construction

The dynamical Φ⁴ equation is a stochastic partial differential equation that is motivated by quantum field theory, as it describes the natural reversible dynamics for the Euclidean Φ⁴ quantum field model. Because the driving space-time white noise is highly irregular, the solution is expected to be distribution-valued, and the cubic nonlinearity is not classically well-defined. The equation is therefore singular and has to be interpreted through a suitable renormalisation procedure. In the past decade, there have been many developments dealing with singular SPDEs including the Φ⁴ equations. We discuss a relatively simple construction of a solution by means of an exponential transformation often called a partial Cole–Hopf transformation and explain how this construction can be adapted from the three-dimensional torus to the unbounded space. This is joint work with Oscar Junge.