Colloquium Abstracts Spring 2024

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

Yi Wang

Institute of Applied Mathematics, AMSS
Chinese Academy of Sciences
February 15, 2024

Time-asymptotic Stability of Generic Riemann Solutions to the Compressible Navier-Stokes Equations

Since 1980s, the time-asymptotic stability of single wave pattern (viscous shock wave, rarefaction wave and viscous contact wave) to 1D compressible Navier-Stokes equations have been well-established and each wave pattern has quite different stability frameworks. Due to the incompatibility of those different stability frameworks, it is open until our works to prove the time-asymptotic stability of generic Riemann solutions, in particular, consisting of different multiple wave patterns, to compressible Navier-Stokes equations. The talk is concerned with our recent progress on the time-asymptotic stability of generic Riemann solutions to the 1D compressible isentropic/full Navier-Stokes equations. Last but not least, I will talk about our recent developments on the time-asymptotic stability of planar viscous shock wave to the multi-dimensional compressible Navier-Stokes and the extensions to the non-convex conservation laws.

Valery Alexeev

University of Georgia
February 22, 2024

Compact Moduli of K3 Surfaces and Tropical Spheres with 24 Singular Points

I will talk about geometric compactifications of moduli spaces of K3 surfaces, similar in spirit to the Deligne-Mumford moduli spaces of stable curves. Constructions borrow ideas from the tropical and integral-affine geometry and mirror symmetry. The main result is that in many common situations there exists a geometric compactification which is toroidal, and many of these compactifications can be described explicitly using tropical spheres with 24 singular points. Much of this talk is based on the joint work with Philip Engel.

Ronghua Pan

Georgia Institute of Technology
March 7, 2024

Rayleigh-Taylor Instability and Beyond

It is known in physics that steady state of fluids under the influence of uniform gravity is stable if and only if the convection is absent. In the context of incompressible fluids, convection happens when heavier fluids is on top of lighter fluids, known as Rayleigh-Taylor instability. However, in real world, heat transfer plays an important role in convection of fluids, such as the weather changes, and or cooking a meal. In this context, the compressibility of the fluids becomes important. Indeed, using the more realistic model of compressible flow with heat transfer, the behavior of solutions is much closer to the real world and more complicated. We will discuss these topics in this lecture, including some on-going research projects. The lecture will be accessible to audiences with basic knowledge on multivariable calculus, and little of differential equations
that partially answers it. This is joint work with Pedro Solórzano and Fred Wilhelm.

Catherine Searle

Wichita State University
March 28, 2024

When is an Alexandrov Space Smoothable?

Alexandrov spaces are finite-dimensional length spaces with a lower curvature bound in the triangle comparison sense. They are a natural generalization of Riemannian manifolds with a lower sectional curvature bound. In this talk, I will discuss the problem of when anAlexandrov space is smoothable. We will review the history of this question and discuss a new result.

Dori Bejleri

University of Maryland
April 9, 2024

Moduli of Boundary Polarized Calabi-Yau Pairs

The theories of KSBA stability and K-stability furnish compact moduli spaces of general type pairs and Fano pairs respectively. However, much less is known about the moduli theory of Calabi-Yau pairs. In this talk I will present an approach to constructing a moduli space of Calabi-Yau pairs which should interpolate between KSBA and K-stable moduli via wall-crossing. I will explain how this approach can be used to construct projective moduli spaces of plane curve pairs. This is based on joint work with K. Ascher, H. Blum, K. DeVleming, G. Inchiostro, Y. Liu, X. Wang.

Kenneth Ascher

University of California, Irvine
April 11, 2024 (at 3:00 pm)

Moduli of K3 Surfaces, Wall-crossing, and the Hassett-Keel Program

Explicit descriptions of low degree K3 surfaces lead to natural compactifications coming from geometric invariant theory (GIT) and Hodge theory. The relationship between these two compactifications was studied by Shah and Looijenga, and revisited in the work of Laza and O’Grady. This latter work also provided a conjectural description for the case of degree four K3 surfaces. I will survey these results, and discuss a verification of this conjectural picture using tools from K-moduli. This is based on joint work with Kristin DeVleming and Yuchen Liu.

Ning (Patricia) Ning

Texas A&M University
April 11, 2024

Variable Target Scalable Particle Filter

We aim to address the challenge of tracking a variable number of interacting targets as they enter and exit a scene, while maintaining precise trajectory records for each target throughout their presence. Tracking multiple high-dimensional targets that interact with each other over a long time period poses significant challenges in terms of accuracy, efficiency, and computational feasibility. Hence, it is crucial to develop methods that effectively handle these complexities, especially in scenarios involving continuous tracking of numerous spatiotemporal interacting targets. To tackle this issue, we propose the Variable Target Scalable Particle Filter (VTSPF) within an online machine learning framework for the spatiotemporal Markov random field of varying dimensions under partial observation. The VTSPF efficiently tracks multiple moving targets exhibiting complex interactions over large graphs and taking values in general Polish spaces. Importantly, we rigorously demonstrate scalability in both spatial and temporal dimension.

Alexander Tovbia

University of Central Florida
April 18, 2024

Spectral Theory of Soliton Gases for Integrable Equations: Some Basic Facts and Recent Developments

We present a brief overview of the theory of soliton gases for integrable systems, such as NLS and KdV, using  algebro-geometric approach. We also mention some related open problems and their connection with potential theory.

Qile Chen

Boston College
April 25, 2024

Quantum Lefschetz via Logarithmic Gauged Linear Sigma Models

Gromov-Witten invariants are topological invariants which virtually count Riemann surfaces in a given complex  projective manifold. These invariants which play a key role in mirror symmetry inspired by theoretical physics, are in general very difficult to calculate, especially for genus > 1. In this talk, I will report on an on-going project with Felix Janda and Yongbin Ruan on Logarithmic Gauged Linear Sigma Models. A goal of this project are explicit formulas computing higher genus Gromov-Witten invariants of a complete intersection using Gromov-Witten type invariants of its ambient manifold, usually referred to as Quantum Lefschetz formulas. As examples, we will explain how this approach leads to effective higher genus calculation for complete intersections in both Calabi-Yau and Fano cases. In particular, in many interesting cases of Calabi-Yau three-fold complete intersections, the number of free parameters from the BCOV  theory by physicists arises naturally from our geometric approach.