# 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.

## Xuan Wu

University of Illinois Urbana-Champaign

October 19*The Scaling Limit of the KPZ Equation*

The KPZ equation was introduced to describe random growing interfaces by Kardar, Parisi, and Zhang in 1986. Since its introduction, the KPZ equation and its large-time asymptotics have been a major research subject in mathematics and physics. The convergence of its fundamental solutions has been a long-standing open problem. In this talk, I will review results in this direction and present my recent work that resolves this problem.

## Volker Mehrmann

Mathematics Distinguished Lecture

Technical University of Berlin

October 26*Energy Based Mathematical Modeling, Simulation, and Control of Real World Systems*

Most real world dynamical systems consist of subsystems from different physical domains, modelled by partial-differential equations, ordinary differential equations, and algebraic equations, combined with input and output connections. To deal with such complex system, in recent years the class of dissipative port-Hamiltonian (pH) descriptor systems has emerged as a very successful mathematical modeling methodology. The main reasons are that the network based interconnection of pH systems is again pH, Galerkin projection in PDE discretization and model reduction preserve the pH structure and the physical properties are encoded in the geometric properties of the flow as well as the algebraic properties of the equations.

Furthermore, dissipative pH systems form a very robust representation under structured perturbations and directly indicate Lyapunov functions for stability analysis.

Another advantage of energy based modeling via pH systems is that each separate model of a physical system can be a whole model catalog from which models can be chosen in an adaptive way within simulation and optimization methods.

We discuss the model class of constrained pH systems and show how many classical real world mathematical models can be formulated in this class. We illustrate the results with some real world examples from gas transport and district heating systems and point out emerging mathematical challenges

## Junliang Shen

Yale University

November 2*Geometry of the P=W Conjecture and Beyond*

Given a compact Riemann surface, nonabelian Hodge theory relates topological and algebro-geometric objects associated to it. Specifically, complex representations of the fundamental group are in correspondence with algebraic vector bundles, equipped with an extra structure called a Higgs field. This gives a transcendental matching between two very different moduli spaces for C: the character variety (parametrizing representations of the fundamental group) and the so-called Hitchin moduli space (parametrizing vector bundles with Higgs field). In 2010, de Cataldo, Hausel, and Migliorini proposed the P=W conjecture, which gives a precise link between the topology of the Hitchin space and the Hodge theory of the character variety, imposing surprising constraints on each side. I will introduce the conjecture, review its recent proofs, and discuss how the geometry hidden behind the P=W phenomenon is connected to other branches of mathematics

## 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.