Colloquium Abstracts Fall 2026

Abstracts will be posted here for the colloquium talks when they are available.

Debaditya Raychaudhury

University of New Mexico
September 10, 2026

Geometry and Topology of Singularities via Hodge Theory

Algebraic geometry is the study of algebraic varieties that are defined as the solution sets of systems of polynomial equations. The topology of algebraic varieties plays a central role in algebraic geometry, particularly in the classification of singularities. Among the most studied singularity classes are the so-called Du Bois and rational singularities, and in recent years their higher analogues have emerged as natural extensions and have become active areas of research. In this talk, I will explain how these notions give rise to refined measures of singularities that connect algebro-geometric and topological viewpoints through Hodge-theoretic techniques. I will also discuss their applications to the study of singularities of secant varieties, a classical and important class of varieties with a history spanning more than a century and broad impact across mathematics.

Nate Lesnevich

Oklahoma State University
September 17, 2026

Splines, Graphs, and Symmetry

A spline is a function built by gluing polynomials together so that they agree where they meet. This gluing condition can be recorded combinatorially on a labeled graph, turning an analytic problem into an algebraic one. When the graph comes from the symmetries of a reflection group (such as the symmetric group) the resulting collection of splines is simultaneously a ring, a module over the polynomials, and a graded representation of the group. In certain cases, it is also the equivariant cohomology ring of an algebraic variety, and so deep geometric questions can translate into questions about splines. This talk introduces splines on symmetry groups from first principles and shares what is known about their structure more generally.

Leonardo Abbrescia

Georgia Tech University
September 24, 2026

Classical Determinism in Compressible Fluid Mechanics

Perhaps the most fundamental questions one can ask for a given PDE are: do classical (i.e., the equations are satisfied pointwise) solutions exist? If they exist, are they unique? Are the solutions global and, if not, *why*? In this talk we will investigate these questions through the lens of the compressible Euler equations (CEE). These are the fundamental equations of fluid mechanics and remain a fertile source of outstanding mathematical challenges, even though they are among the earliest PDE systems written down (they were formulated by Euler in 1757). We will present comprehensive answers to the questions posed above for a new rich class of smooth initial data that form singularities in their dynamic evolution. That is, the solutions launched by the data are unique but the set in spacetime which describes the extent on which they remain classical solutions has a boundary. For a related set of initial data, we dynamic solution is truly global and lives across all of space and time (to the future and past!). This is joint work with Jared Speck and Dongxiao Yu.