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.