View Abstract

Abstract: Statistical mechanics aims to predict the macroscopic features of a large system from a statistical description of its microscopic constituents and their interactions. The subject area has seen a lot of activity on both the physics and mathematics ends with many breakthroughs occurring over the last two decades. The workshop will focus on recent developments in statistical mechanics of lattice models, with topics to be discussed ranging from interacting spin systems, gradient models and lattice gauge theories to disordered systems such as spin models with random interaction, first-passage percolation and minimal surfaces in random environments. The workshop will bring together both senior and junior researchers, aiming to disseminate recent developments and facilitate a discussion between practitioners with different perspectives. The hope is that the workshop will inspire collaboration and initiate further progress on the outstanding problems of the field.

View Abstract

Abstract: We review the Kauffman bracket skein algebra of an oriented surface, which is a quantization of the SL2 character variety. We discuss several vector space bases of this algebra and investigate some remarkable phenomena and conjectures, in relation to various topics like quantum cluster varieties and representation theory.

View Abstract

Abstract: We present a complete classification of the indecomposable (extreme) characters and their associated II_1-factor representations for the topological full groups and approximately finite (AF) full groups of Cantor minimal systems. We show that every nontrivial indecomposable character is uniquely determined by a finite collection of invariant ergodic measures of the underlying system. This result establishes a direct link between group characters and measure-preserving actions, confirming Vershik’s conjecture for this class of groups. As a consequence, we discuss the automatic continuity of finite-type unitary representations with respect to the uniform topology.

This talk is based on recent joint work with Artem Dudko (Institute of Mathematics of the Polish Academy of Sciences).

View Abstract

Abstract: Transport-based models, such as score-based diffusion and flow-matching models, have become a leading paradigm for generative modeling: from a dataset of samples, one learns dynamics that generate new samples from the same distribution. In many scientific settings, however, the objective is not simply to reproduce a training distribution, but to adapt or infer distributions under new constraints or incomplete observations. This motivates extending the transport viewpoint beyond the standard training-and-sampling setting.

I will describe two such directions. First, inference-time adaptation: modifying the inference dynamics induced by a pre-trained model to sample from new target distributions without retraining, while preserving stability and efficiency. Second, distributional inference from limited or noisy data: constructing measure dynamics that recover target distributions when the observation process is available only through a black-box simulator. Together, these examples illustrate how transport-based methods enable flexible control and inversion at the level of probability measures, substantially broadening their role in scientific and engineering applications.

View Abstract

Abstract: We’ll describe the global topological structure of the space of subgroups of PSL2R, equipped with the Chabauty topology. The most interesting part of it is similar to the topology of the space of all pointed hyperbolic surfaces, equipped with the Gromov-Hausdorff topology. We’ll describe how to move around in this latter space, and discuss connectivity, local connectivity and simple connectivity!

View Abstract

Abstract: We propose an autoregressive framework for modelling dynamic networks with
dependent edges. It encompasses models which accommodate, for example, transitivity,
density-dependent and other stylized features often observed in real network data. By
assuming the edges of network at each time are independent conditionally on their
lagged values, the models, which exhibit a close connection with temporal ERGMs,
facilitate both simulation and maximum likelihood estimation in a straightforward manner.
Due to the possible large number of parameters in the models, the initial MLEs may
suffer from slow convergence rates. An improved estimator for each component
parameter is proposed based on an iteration employing a projection which mitigates the
impact of the other parameters. Leveraging a martingale difference structure, the
asymptotic distribution of the improved estimator is derived without a stationarity
assumption. The limiting distribution is not normal in general, and it reduces to normal
when the underlying process satisfies some mixing conditions. Illustration with a
transitivity model was carried out in both simulation and a real network data set. Joint
work with Jinyuan Chang, Qin Fang, Peter MacDonald and Qiwei Yao.

View Abstract

Abstract: Cancer causes the cells of the body to shatter their well-defined roles, proliferate, and invade other tissues, leading to premature death. As a lapsed mathematician turned bioengineer, I lead a research group that studies cancer to propose novel therapies based on the spatial structure of tumor tissue. While there is no such thing as a "topological combinatorial construct" as far as I know, there is great significance in how different cells in our body are positioned in space and relative to each other. Our modern understanding of cancer proposes that a complex network of biological signals, partitioned into various cell types, is the fabric that frays and eventually dissolves in cancer. The functions woven into this fabric include immune cell control of diseased cells, secreted signals that attract and repel cells, and even a physical meshwork of collagenous and fibrotic material that impedes tumor and immune migration. Currently, my lab uses spatial molecular tools that can measure dozens of proteins and thousands of RNA biomolecules directly within intact tissue, allowing us to reconstruct the physical cellular architecture of tumors. We can use this information to characterize tumor tissue in depth and identify structures with predictive significance, based on the spatial cellular organization. However, the tools that we use to describe tumor structures are still simplistic, and our vocabulary is still limited when describe interacting fields of objects with hundreds to thousands of signals and properties. My goals for this seminar are to first present the structure and language of spatial biology data and its current applications, and then to recruit your minds to capture the underlying structures, patterns, and projections that will allow us to translate spatial data into actionable hypotheses to improve the treatment of cancer.

View Abstract

Abstract: Many natural and social phenomena involve individual agents coming together to create group dynamics, whether the agents are drivers in a traffic jam, cells in a developing tissue, or locusts in a swarm. Here I will focus on the example of pattern formation in zebrafish, which are named for their dark and light stripes. Mutant zebrafish, on the other hand, feature different skin patterns, including spots and labyrinth curves. All of these patterns form as the fish grow due to the interactions of tens of thousands of pigment cells. The longterm motivation for my work is to help identify the alterations to cell interactions that lead to mutant patterns. Toward this goal, I will overview our work building agent-based and continuum models to simulate pattern formation and make experimentally testable predictions. Because stochastic, microscopic models are not analytically tractable using traditional techniques, I will also describe how we are applying topological data analysis and approximate Bayesian inference to quantify structure in messy, cell-based patterns and identify rules of behavior from data.

View Abstract

Abstract: A very useful technique called BBM (Brown-Barge-Martin), incorporates inverse limits and natural extensions of the underlying bonding maps to embed attractors (for homeomorphisms) in manifolds. The original idea goes back to the paper of Barge and Martin, where the authors constructed strange attractors from a wide class of inverse limits. One of the crucial steps for this technique to work is the usage of Brown's approximation theorem.

Recently, this technique found several interesting applications and extensions. In this talk we will present a few possible applications of BBM technique in a construction of concrete examples of dynamical systems on surfaces. If time permits, we will relate BBM to Anosov-Katok fast approximation method and construction of smooth diffeomorphisms.

The talk is based on joint work(s) with Jernej Cinc and Michal Kowalewski.

View Abstract

Abstract: Abstract elementary classes (AECs) provide an extension of first order model theory in which we can still attempt a classification theory. The question of when limit models (a kind of surrogate for saturated models for AECs) are isomorphic has connections to important open problems in AECs, such as Shelah's categoricity conjecture. Most work in this area is towards 'positive' results - that is, showing limit models are isomorphic. The question of when limit models are not isomorphic is less explored.

In this talk we give a full characterisation of the spectrum of limit models under general assumptions in a stable AEC - that is, describe completely which models are isomorphic and which are not. Given time we will discuss applications, a more general result in the 'positive' direction, and touch on a recent result which says that all high cofinality limit models are disjoint amalgamation bases. Based partly on joint work with Marcos Mazari-Armida.

View Abstract

Abstract: Electrical Impedance Tomography (EIT) is a PDE-based inverse problem that aims to reconstruct electrical conductivity from boundary measurements of electrical currents and voltages. Mathematically, EIT is modeled by an elliptic equation with a variable conductivity function, and the goal is to infer this function from the Dirichlet-to-Neumann (DtN) map. One fundamental barrier in solving EIT is data scarcity, and in this talk we address this bottleneck by completing full data from partially observed DtN measurements.
Specifically, we train a conditional diffusion model to learn the distribution of DtN data and to infer full measurement vectors given partial observations. Our approach supports flexible source–receiver configurations and can be used as a plug-in preprocessing step with off-the-shelf EIT solvers. Under mild assumptions on the polygonal conductivity class, we derive nonasymptotic end-to-end bounds on the distributional discrepancy between the completed and ground-truth DtN measurements. In numerical experiments, we couple the proposed diffusion-based completion procedure with a deep learning–based inverse solver and compare its performance against the same solver using full measurement data. The results show that diffusion completion enables reconstructions that closely match the full-data baseline while using only 1% of the measurements. In contrast, standard baselines such as matrix completion require 30% of the measurements to achieve similar reconstruction quality.

View Abstract

Abstract: https://blog.umd.edu/brinmrc/files/2026/02/TereSeara-DistinguishedLectureDRAFT02_WEB.pdf

View Abstract

Abstract: In this talk, I will present a recent result in collaboration with I. Baldomá (U.Politecnica de Catalunya) A. Florio (U. Paris Dauphine) and M. Leguil (E. Polytechnique Paris). The object of our study is the existence of Chaos in analytic convex billiards. It is well known that, for a convex billiard table, the corresponding billiard map is a twist map on the cylinder. To obtain the existence of chaos in such a map, we nd hyperbolic periodic orbits whose stable and
unstable manifolds intersect transversally, giving rise to the existence of horseshoes and, therefore, to hyperbolic invariant sets whose dynamics is conjugated to the Bernoulli shift. This is the classical approach to prove chaos; nevertheless, the abundance of transverse homoclinic orbits has been only proved for generic smooth billiard tables. The novelty of our work is that, going back to some ideas of Zehnder of 1972, we prove that the set of
analytic billiards such that "given any rational rotation number, there is a unique periodic orbit in the associated Aubry-Mather set, whose stable and unstable manifolds intersect and whose intersections are all transversal” form a Baire generic set with the usual analytic topology. To our knowledge, this is the rst result proving chaos in a non-perturbative analytic setting.