# Colloquium 2022

The colloquium typically meets Mondays at 4:00 PM in Room 617 on the sixth floor of Wachman Hall.

The colloquium is preceded by tea starting at 3:30 in the Faculty Lounge, adjacent to Room 617. Click on title for abstract.

• Monday January 24, 2022 at 16:00, Wachman 617
Morrey's problem and quasiconformal mappings

Andre Guerra, Institute for Advanced Study

Quasiconvexity is a fundamental notion in the vectorial Calculus of Variations and is essentially equivalent to the applicability of the Direct Method. A fundamental problem, considered by Morrey in the 50s and 60s, is whether quasiconvexity is equivalent to ellipticity (in the sense of Legendre-Hadamard). In 1992 Vladimir Sverak showed that in 3 or higher dimensions they are not equivalent, but the two-dimensional case remains open. In this case one can expect a "complex analysis miracle", and we will discuss deep connections of Morrey's problem to old questions in Quasiconformal Analysis.

• Monday September 19, 2022 at 16:00, Wachman 617
Integral points on elliptic curves

Wei Ho, IAS, Princeton, Michigan

Elliptic curves are fundamental and well-studied objects in arithmetic geometry. However, much is still not known about many basic properties, such as the number of rational points on a "random" elliptic curve. We will discuss some conjectures and theorems about this "arithmetic statistics" problem, and then show how they can be applied to answer a related question about the number of integral points on elliptic curves over Q. In particular, we show that the second moment (and the average) for the number of integral points on elliptic curves over any number field is bounded (joint work with Levent Alpöge).

• Monday September 26, 2022 at 16:00, Wachman 617
Maps between configuration spaces and moduli spaces

Lei Chen, University of Maryland

In this talk, I will survey what is known or conjectured about maps between configuration spaces and moduli spaces of surfaces. Among them, we consider two categories: continuous maps and also holomorphic maps. We will also talk about some results and conjectures about maps between finite covers of those spaces. Unlike the case of the whole spaces, much less is known about their finite covers.

• Monday October 17, 2022 at 16:00, Wachman 617
Some congruences and consequences in number theory and beyond

Ellen Eischen, University of Oregon

In the middle of the nineteenth century, Kummer observed striking congruences between certain values of the Riemann zeta function, which have important consequences in number theory.  In spite of its potential, this topic lay mostly dormant for nearly a century until breakthroughs by Iwasawa in the middle of the twentieth century. Since then, advances in arithmetic geometry and number theory (in particular, for modular forms, certain analytic functions that play a central role in number theory) have revealed similarly consequential congruences in the context of other arithmetic data.  This remains an active area of research. In this talk, I will survey old and new tools for studying such congruences. I will conclude with some unexpected challenges that arise when one tries to take what would seem like immediate next steps beyond the current state of the art.

• Monday October 24, 2022 at 16:00, Wachman 617
Stochastic Nash evolution

Govind Menon, Brown University

We introduce a probabilistic formulation for the Nash embedding theorems. Our approach inverts the usual relation between mathematics and physics. We use rigorous mathematical results, including Nash’s work, results of De Lellis and Szekelyhidi, and work of the speaker, Rezakhanlou and co-workers, to guide the design of algorithms and evolution equations.

We use relaxation as in Nash’s work, but replace his iteration (in low codimension) or continuous flow (in high codimension) with a stochastic flow. The main issue in the derivation of our flow is a principled resolution of a semidefinite program. The same fundamental structure applies to several hard constraint systems and nonlinear PDE.

• Monday November 14, 2022 at 16:00, Wachman 617
Formation and development of singularities for the compressible Euler equations

We consider the compressible Euler equations of fluid dynamics, in multiple space dimensions. In this talk, we discuss the formation and development of stable singularities, from smooth initial conditions. The questions we address are: given smooth initial conditions, precisely how does the first singularity arise? is the mechanism stable? how can one geometrically characterize the preshock (the boundary of the space-time set on which the solution remains smooth)? precisely how does the entropy-producing shock wave instantaneously develop from the preshock? does uniqueness hold once the shock has formed? do other singularities instantaneously arise after the preshock? In this level of detail, the problem was previously open even in one space dimension. We discuss a sequence of joint works with Steve Shkoller, Tristan Buckmaster, and Theodore Drivas, in which we have developed a multidimensional theory to answer the above questions.

• Tuesday November 29, 2022 at 16:00, Wachman 617
Classification of divergent trajectories

Nattalie Tamam, University of Michigan

The study of group actions gained significant interest in the past several decades, as group actions are a powerful tool when approaching problems from number theory and geometry. We will focus on the dynamical equivalent to vectors with 'infinitely good' diophantine approximation. From this dynamical point of view, Weiss conjectured a complete classification of the relevant trajectories. We will discuss the steps and different tools used in proving this conjecture. This is in part a joint work with Omri Solan, and in part a joint work with Lingmin Liao, Ronggang Shi and Omri Solan.

• Friday December 2, 2022 at 16:00, Wachman 617
Wave propagation on rotating cosmic string spacetimes

Katrina Morgan, Northwestern University

Energy methods have historically been a useful tool for studying waves on different background geometries. Under the right conditions, solutions to the wave equation satisfy energy estimates which state that the energy of the solution $u$ at time $t$ is controlled by the energy of the initial data. However, such techniques are not always available, such as in the case of rotating cosmic string spacetimes. These geometries are solutions to the Einstein field equations which exhibit a singularity along a timelike curve corresponding to a one-dimensional source of angular momentum. They have a notable unusual feature: they admit closed timelike curves near the so-called string" and are thus not globally hyperbolic. In joint work with Jared Wunsch, we show that \textit{forward in time} solutions to the wave equation (in an appropriate microlocal sense) do exist on rotating cosmic string spacetimes, despite the causality issues present in the geometry. Our techniques involve proving a statement on propagation of singularities which provides a microlocal version of an energy estimate that allows us to establish existence of solutions.

• Tuesday December 6, 2022 at 16:00, Wachman 617
A PDE perspective on the hydrodynamics of flexible filaments

Laurel Ohm, Princeton University

Many fundamental biophysical processes, from cell division to cellular motility, involve dynamics of thin structures immersed in a very viscous fluid. Various popular models have been developed to describe this interaction mathematically, but much of our understanding of these models is only at the level of numerics and formal asymptotics. Here we seek to develop the PDE theory of filament hydrodynamics.

First, we propose a PDE framework for analyzing the error introduced by slender body theory (SBT), a common approximation used to facilitate computational simulations of immersed filaments in 3D. Given data prescribed only along a 1D curve, we develop a novel type of boundary value problem and obtain an error estimate for SBT in terms of the fiber radius. This places slender body theory on firm theoretical footing.

Second, we consider a classical elastohydrodynamic model for the motion of an immersed inextensible filament. We highlight how the analysis can help to better understand undulatory swimming at low Reynolds number. This includes the development of a novel numerical method to simulate inextensible swimmers.

• Thursday December 8, 2022 at 16:00, Wachman 617
Designing low rank methods via rational functions

Heather Wilber, UT Austin

Low rank approximation methods are a central pillar of modern scientific computing. They are the powerhouse behind many fast and superfast methods relied upon on for computing solutions to various partial differential equations, linear systems, and matrix equations.  In this talk, we focus on the role that rational approximation methods can play in the design of such algorithms. We illustrate how rational approximation tools can help us design highly effective low rank methods in the context of two very different (but surprisingly related!) kinds of problems: (1) the development of a direct solver for linear systems involving non-uniform discrete Fourier transform matrices, and (2) the development of solvers for the spectral fractional Poisson equation on geometrically complicated domains.