Kalender

In this talk, we discuss how temporal causal structures can be modelled using a path-dependent stochastic differential equation (SDE). We then consider a rich class of path-dependent SDEs, called signature SDEs, which can model general path-dependent phenomena. We provide conditions that ensure the existence and uniqueness of solutions to a general signature SDE. Path signatures are iterated integrals of a given path with the property that any sufficiently nice function of the path can be approximated by a linear functional of its signatures. This is why we model the drift and diffusion of our signature SDE as linear functions of path signatures, and then introduce the Expected Signature Matching Method (ESMM) for linear signature SDEs, which enables inference of the signature-dependent drift and diffusion coefficients from observed trajectories. Furthermore, we show that the ESMM is consistent: given sufficiently many samples and Picard iterations used by the method, the parameters estimated by the ESMM approach the true parameter with arbitrary precision. We discuss the asymptotic distribution of the estimator obtained from the ESMM, and finally, demonstrate on a variety of empirical simulations that our ESMM accurately infers the drift and diffusion parameters from observed trajectories. While parameter estimation is often restricted by the need for a suitable parametric model, this study makes progress toward a completely general framework for SDE parameter estimation, using signature terms to model arbitrary path-independent and path-dependent processes.This talk is based on joint work with Vincent Guan, Elina Robeva, and Darrick Lee.

The problem of finding homogeneous Einstein metrics on a compact homogeneous space reduces to solving a system of Laurent polynomial equations. This is an example of a vertically parametrized system; such systems arise in algebraic statistics and chemical reaction networks. We prove that the number of isolated solutions of this system is bounded above by the central Delannoy numbers and we describe the discriminant locus where the number of isolated solutions drops in terms of the principal A-determinant.

We develop AI models to better understand proteins and the information they encode. The first model, Contrastive Learning Sequence–Structure (CLSS), aims to map the protein universe by characterizing relationships between amino acid sequences and structures. CLSS is a self-supervised contrastive learning model trained on large and diverse sets of protein domains to co-embed sequence and structure into a shared high-dimensional space, where distance reflects sequence–structure similarity. This representation naturally captures both evolutionary relationships and structural variation. We find that CLSS refines expert knowledge of the global organization of protein space, highlights transitional forms that resist hierarchical classification, and reveals linkages between domains from seemingly separate lineages, thereby improving our understanding of evolutionary design. The second model focuses on codon selection. Codon usage is shaped by selective pressures that optimize multiple, overlapping signals that remain only partially understood. We trained AI models to predict gene codons from amino acid sequences in four organisms (S. cerevisiae, S. pombe, E. coli, and B. subtilis). The AI models significantly outperformed frequency-based baselines, indicating that dependencies between codons within genes can be learned. Performance gains were greater for highly expressed genes and in bacteria compared to eukaryotes, consistent with stronger selective pressure under larger effective population sizes. In S. cerevisiae and bacteria, accuracy also increased with protein length, suggesting that the models captured signals related to co-translational folding. Incorporating information from homologous proteins provided only minor additional benefit, potentially reflecting complex codon-usage patterns in rapidly evolving genes. Together, these studies provide practical tools and demonstrate how AI can be used to study how evolution has shaped the protein universe and its encodings.

Using periodic surfaces as a scaffold is a convenient route to making periodic entanglements, which are interesting in the context of physics, biomaterials and chemical frameworks. I will present a systematic way of enumerating and characterising new tangled periodic structures, using low-dimensional topology and combinatorics. As a second part, the morphometric approach to solvation free energy is a geometry-based theory that incorporates a weighted combination of geometric measures over the solvent accessible surface for solute configurations in a solvent. I will demonstrate that employing this geometric technique in simulating the self assembly of sphere clusters, viruses and short flexible tubes results in an assortment of interesting geometric structures. This gives insight into the role of shape in the physical process of self assembly.

In the analysis of three-dimensional biological microstructures such as organoids, microscopy frequently yields two-dimensional optical sections without access to their orientation. Motivated by the question of whether such random planar sections determine the underlying three-dimensional structure, we investigate a discrete analogue in which the ambient structure is the vertex set of a Platonic solid and the observed data are congruence classes of planar intersections. For the regular dodecahedron with vertex set V, we define the planar statistic of a subset X⊆V of vertices as the distribution of isometry types of inclusions Π∩X⊆Π∩V⊆V, and ask whether this statistic determines X up to isometry. We show that this is not the case: there exist two non-isometric 7-element subsets with identical planar statistics. As a consequence, there exist two polytopes in R3, whose distribution of isometry classes of two-dimensional intersections is identical, while the polytopes are not themselves isometric. This result is an analogue of classical non-uniqueness phenomena in geometric tomography.

Flow-firing, introduced by Felzenszwalb-Klivans, is a 2D analogue of chip-firing: an integer flow on the edges of a cell complex evolves by repeatedly applying local rerouting moves around faces. In their original work, the only proven confluent family on the grid stabilized (independent of firing choices) into an Aztec diamond, a centered diamond-shaped patch of unit squares. In this talk I will explain how far this phenomenon extends. For a natural family of conservative “pulse” initial conditions, we prove a three-regime theorem: there is a small-support regime with unique stabilization to the Aztec diamond, an intermediate regime where stabilization occurs but the terminal state is not unique (though the Aztec diamond can still occur), and a large-support regime where confluence fails, including a range where the Aztec-diamond outcome is impossible.