Georgia Institute of Technology College of Sciences

Given a collection of multivariate polynomials $F, F_1, …F_n$, to determine whether $F$ can be written as $F= \sum G_iF_i$ is an important problem, both in theory and practice. In this talk, I will explain how some problems of this nature, even in 2 or 3 variables, can become interesting and difficult. An unexpected connection to a conjecture on sumsets will be explained. As the time of writing, the conjecture is unsolved by both humans and AIs.  

Given a closed subvariety Z in a smooth complex variety X, the local cohomology sheaves with support in Z are holonomic D-modules, and thus have finite filtration with simple composition factors. We determine the D-module structure on local cohomology in the case when X is a Grassmannian and Z is a Schubert variety, including a combinatorial formula describing the composition factors and the weight filtration in the sense of mixed Hodge modules. Upon restriction to the opposite big cell, these calculations recover several previously known results concerning local cohomology with support in determinantal varieties.

A Brownian motion tree (BMT) model is a Gaussian model whose associated set of covariance matrices is linearly constrained according to common ancestry in a phylogenetic tree. This talk will discuss the complexity of inferring the maximum likelihood (ML) estimator for a BMT model by computing its ML-degree.  The talk will highlight an explicit formula for the ML-degree of the BMT model on a star tree. We will also show that the ML-degree of a BMT model is independent of the choice of the root. This talk is based on work (doi.org/10.1016/j.jsc.2025.102482) with J. I. Coon, S. Cox, & A. Maraj.

The theory of $p$-adic differential equations first rose to prominence after Dwork used them to prove the rationality of zeta functions of a positive characteristic variety in 1960. Since then, there has been growing interest in the category of convergent $F$-isocrystals and the subcategory of overconvergent $F$-isocrystals due to this subcategory having good cohomology theory with finiteness properties. Recent work by Grubb, Kedlaya and Upton examines when a convergent $F$-isocrystal is overconvergent by restricting to smooth curves on the scheme under a mild tameness assumption (measured by the Swan conductor). In my talk, I will introduce the above categories and talk about work in progress about bounding the Swan conductor of an overconvergent $F$-isocrystal in terms of data associated with the corresponding convergent $F$-isocrystal.