Colloquium Abstracts Fall 2023

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

Gábor Székelyhidi

Northwestern University
September 14, 2023

Singularities of the Lagrangian Mean Curvature Flow

The Lagrangian mean curvature flow is conjectured by Thomas-Yau and Joyce to, roughly speaking, decompose a Lagrangian in a Calabi-Yau manifold into a union of special Lagrangians. I will give an introduction to this conjecture, and discuss some recent progress, in joint work with Jason Lotay and Felix Schulze, analyzing the behavior of the flow near certain singularities. 

Ruobing Zhang

Princeton University
September 21, 2023

Moduli Space of Einstein Metrics: Topology, Analysis, and Metric Geometry

The geometry of Einstein manifolds has been a central topic in differential geometry. This talk concerns the structure of moduli space of Einstein metrics and its compactification, with a focus on how Einstein metrics can degenerate. We will introduce recent major progress and propose several open questions in the field.

Ziquan Zhuang

John Hopkins University
October 5

Stable Degeneration of Singularities

A theorem of Donaldson and Sun says that the metric tangent cone of a smoothable Kähler–Einstein Fano variety underlies some algebraic structure, and they conjecture that the metric tangent cone only depends on the algebraic structure of the singularity. Later Li and Xu extend this speculation and conjecture that every Kawamata log terminal singularity (a singularity class in birational geometry) has a canonical “stable” degeneration induced by the valuation that minimizes the normalized volume. I will talk about some recent works around the solution of these conjectures. Based on joint work with Chenyang Xu.

Robin Young

University of Massachusetts
October 6

The Nonlinear Theory of Sound

We prove the existence of nonlinear sound waves, which are smooth, time periodic, oscillatory solutions to the compressible Euler equations, in one space dimension.  In the mid-19th century, Riemann proved that compressions always form shocks in the simpler isentropic system, which is inconsistent with sound wave solutions of the (linear) wave equation.  We prove that for generic entropy profiles, the fully nonlinear compressible Euler equations support perturbations of the linearized solutions for every frequency.  This shows that Riemann's result is a highly degenerate special case and brings the mathematics of the compressible Euler equations back into line with two centuries of verified Acoustics technology. This is joint work with Blake Temple.

Mohammad Tehrani

University of Iowa
October 12

On Compactifications of SL(2,C) Character Varieties

Consider an algebraic reductive Lie group G and a finitely generated group P. Let X_P(G) denote the moduli space of G-representations of P up to conjugation with elements of G. A special case of interest is X_{g,n}(SL(2,C)), where G = SL(2;C) and P is the fundamental group of an n-punctured genus g surface. This case has been extensively studied and lies at the intersection of many interesting subjects.

It is well-known that X_{g,n}(SL(2;C)) is a quasi-projective affine variety of complex dimension 3(2g + n-2), admitting a natural fibration over C^n with HyperKahler fibers. Various compactications of X_{g,n}(SL(2,C)) and other related moduli spaces can be found in the literature, including those related to Teichmuller space, Hitchin moduli spaces, and Fock-Goncharov A and X moduli spaces. 

This talk delves into complex projective compactifications driven by Mirror Symmetry and P=W conjecture. We introduce a class of projective compactifications determined by ideal triangulations of the surface and provide explicit results on the boundary divisors. Notably, we confirm a well-known conjecture asserting that the boundary complex of any positive dimensional relative character variety is a sphere. I will discuss a few examples. This work is part of an ongoing collaboration with Charlie Frohman.

KC Kong

Department of Physics & Astronomy
University of Kansas
November 30, 2023

Particle Physics, Combinatorial Optimization and Quantum Algorithms

Our knowledge of the fundamental particles and their interactions is summarized by the standard model of particle physics. Mathematically, the theory describes these forces and particles as the dynamics of elegant geometric objects. Now advancing our understanding in this field has required experiments that operate at higher energies and intensities, which produce extremely large and information-rich data samples. The use of machine-learning techniques and quantum algorithms is revolutionizing how we interpret these data samples, greatly increasing the discovery potential of present and future experiments. In this talk, I will provide a brief overview of the standard model of elementary particle physics, and introduce simple examples where quantum algorithms could be useful.