Colloquium Abstracts Spring 2025

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

Chenyang Xu

Princeton University
March 13, 2025

Moduli Theory of Fano Varieties

Moduli spaces, which parametrize classes of geometric objects, are central to algebraic geometry. The construction of moduli spaces for negatively curved curves by Mumford marked a significant milestone in modern moduli theory. For higher-dimensional varieties, the moduli theory of those with negative curvature, developed by Kollár and others, has long been linked to the minimal model program.

However, constructing moduli spaces for positively curved varieties, called Fano varieties, remained a challenge and an open question for higher-dimensional geometers. Recent breakthroughs have revealed that the key lies in the concept of K-stability—a notion introduced in complex geometry by Tian, Donaldson, and others to characterize the existence of Kähler-Einstein metrics.

In this lecture, I will discuss the development of these ideas, highlighting the interplay between algebraic and complex geometry, and the role of K-stabiity in establishing a moduli theory for Fano varieties.

Athanasios E. Tzavaras

King Abdullah University of Science and Technology
March 27, 2025

The Maxwell-Stefan System of Multi-component Diffusion

The Maxwell-Stefan system is a system of nonlinear equations commonly used in the description of diffusion processes in multi-component systems, like gases with many constituents. In this talk I will briefly discuss modeling of multi-component systems, and how the Maxwell-Stefan system emerges through a process of alignment in the high-friction limit of multi-component Euler flows. The main part of the talk will be devoted to : (i) Challenges posed on analysis by the existence and uniqueness theory of the Maxwell-Stefan system; (ii) The connection between minimization of the frictional dissipation and the Maxwell-Stefan system.

Philip Engel

University of Illinois at Chicago
March 28, 2025

Counting Buckyballs

You may have heard of Buckminsterfullerene, a spherical molecule made out of 60 carbon atoms. More generally, we can ask about the following counting problem: How many spherical carbon molecules, also called fullerenes, can you make out of 2n carbon atoms? A related question: How many ways are there to build a sphere out of 2n triangular pieces, so that every corner has at most six triangles next to it? There is a beautiful exact formula which is quite surprising!

Ronan Conlon

University of Texas, Dallas
March 28, 2025

On Finite-time Singularities of the Ricci Flow on Compact Kähler Surfaces

The Ricci flow, introduced by Richard Hamilton in the 1980’s, is a powerful geometric evolution equation that deforms the metric of a Riemannian manifold in a way analogous to heat diffusion. It has proven instrumental in understanding the geometry and topology of manifolds. A rigorous analysis of its behavior at finite time singularities has been key to these applications. In this talk, I will present joint work with Cifarelli-Deruelle and Hallgren-Ma on the behavior of the Ricci flow on a compact Kähler surface at a non-collapsed finite time singularity.