Category Archives: Spring 2024

Analysis of micro- and macro-scale models of superfluidity

The HADES seminar on Tuesday, May 7th, will be at 3:30pm in Room 748.

Speaker: Pranava Jayanti

Abstract: We introduce the physics of superfluidity, including two mathematical models. We begin with a micro-scale description of the interacting dynamics between the superfluid and normal fluid phases of Helium-4 at length scales much smaller than the inter-vortex spacing. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of solutions in $\mathbb{T}^n$ (for n=2,3) for a power-type nonlinearity, beginning from small initial data. Depending upon the strength of the nonlinear self-interactions, we obtain solutions that are global or almost-global in time. The main challenge is to control the inter-phase mass transfer in order to ensure the strict positivity of the normal fluid density, while deriving time-independent a priori estimates. We compare two different approaches: purely energy based, versus a combination of energy estimates and maximal regularity. The results are from recent collaborations with Juhi Jang and Igor Kukavica.
We will also briefly discuss some results pertaining to a macro-scale model known as the HVBK equations, some of which is joint work with Konstantina Trivisa.

The C^0 inextendibility of the maximal analytic Schwarzschild spacetime

The HADES seminar on Tuesday, April 30th, will be at 3:30pm in Room 939.

Speaker: Ning Tang

Abstract: The study of low-regularity inextendibility criteria for Lorentzian manifolds is motivated by the strong cosmic censorship conjecture in general relativity. In this expository talk I will present a result by Jan Sbierski on the C^0 inextendibility of the maximal analytic Schwarzschild spacetime. I will start with a proof for the continuous inextendibility of Minkowski spacetime, followed by a comparison between this and the continuous inextendibility of Schwarzschild exterior. Then I will sketch the proof of continuous inextendibility of Schwarzschild interior.

A loosely coupled splitting scheme for a fluid – multilayered poroelastic structure interaction problem

The HADES seminar on Tuesday, April 23rd, will be at 3:30pm in Room 939.

Speaker: Andrew Scharf

Abstract: Multilayered poroelastic structures are found in many biological tissues, such as cartilage and the cornea, and find use in the design of bioartificial organs and other bioengineering applications. Motivated by these applications, we analyze the interaction of a free fluid flow modeled by the time-dependent Stokes equation and a multilayered poroelastic structure consisting of a thick Biot layer and a thin, linear, poroelastic membrane separating it from the Stokes flow. The resulting equations are linearly coupled across the thin structure domain through physical coupling conditions such as the Beavers-Joseph-Saffman condition. I will discuss previous work in which weak solutions were shown to exist by constructing approximate solutions using Rothe’s method. While a number of partitioned numerical schemes have been developed for the interaction of Stokes flow with a thick Biot structure, the existence of an additional thin poroelastic plate in the model presents new challenges related to finite element analysis on multiscale domains. As an important step toward an efficient numerical scheme for this model, we develop a novel, fully discrete partitioned method for the multilayered poroelastic structure problem based on the fixed strain Biot splitting method. This work is carried out jointly with Sunčica Čanić and Jeffrey Kuan at the University of California, Berkeley and Martina Bukač at the University of Notre Dame.

Two dimensional gravity water waves with constant vorticity at low regularity

The HADES seminar on Tuesday, April 16th, will be at 3:30pm in Room 939.

Speaker: Lizhe Wan, University of Wisconsin-Madison

Abstract: In this talk I will discuss the Cauchy problem of two-dimensional gravity water waves with constant vorticity. The water waves system is a nonlinear dispersive system that characterizes the evolution of free boundary fluid flows. I will describe the balanced energy estimates by Ai-Ifrim-Tataru and show that using this method, the water waves system is locally well-posed in $H^{\frac{3}{4}}\times H^{\frac{5}{4}}$. This is a low regularity well-posedness result that effectively lowers $\frac{1}{4}$ Sobolev regularity compared to the previous result.

A probabilistic approach to the fractal uncertainty principle

The HADES seminar on Tuesday, April 9th, will be at 3:30pm in Room 939.

Speaker: Xiaolong Han

Abstract: The Fourier uncertainty principle describes a fundamental phenomenon that a function and its Fourier transform cannot simultaneously localize. Dyatlov and his collaborators recently introduced a concept of Fractal Uncertainty Principle (FUP). It is a mathematical formulation concerning the limit of localization of a function and its Fourier transform on sets with certain fractal structure.

The FUP has quickly become an emerging topic in Fourier analysis and also has important applications to other fields such as quantum chaos. In this talk, we report on an ongoing project concerning the FUP when the fractal sets are constructed via certain random procedures. Examples include random Cantor sets in the discrete or continuous setting. We present the FUP with a much more favorable estimate than the ones in the deterministic cases. We also propose questions and applications of the FUP by this probabilistic approach. The talk is based on joint works with Suresh Eswarathasan and Pouria Salekani.

The Sign of Scalar Curvature on Kähler Blowups

The HADES seminar on Tuesday, April 2nd, will be at 3:30pm in Room 939.

Speaker: Garrett Brown

Abstract: Blowing up is a construction in complex geometry that can be thought of as the analog to connected sum in smooth topology. In this talk we will show that the property of having a positive (or negative) scalar curvature Kähler metric is preserved under blowing up points on a compact complex manifold of any dimension. This is done by solving a certain prescribed scalar curvature equation. The most crucial step is establishing uniform estimates for the linearized scalar curvature operators of a family of metrics on the blowup, for which the underlying geometry plays an interesting role. In the case of positive scalar curvature in two complex dimensions, this answers a question of Hitchin and Lebrun in the affirmative and completes the classification of positive scalar curvature Kähler surfaces.

Modified scattering for the three dimensional Maxwell-Dirac system

The HADES seminar on Tuesday, March 19th, will be at 3:30pm in Room 939.

Speaker: Mihaela Ifrim

Abstract: In this work we prove global well-posedness for the massive Maxwell-Dirac equation in the Lorentz gauge in $\mathbb{R}^{1+3}$, for small and localized initial data, as well as modified scattering for the solutions.  In doing so, we heuristically exploit the close connection between massive  Maxwell-Dirac and the  wave-Klein-Gordon equations, while  developing a novel approach which applies directly at the level of the Dirac equations.  This is joint work with Sebastian Herr and Martin Spitz.

Singularity formation in 3d incompressible fluids: the role of angular regularity

The HADES seminar on Tuesday, March 12th, will be at 2:00pm in Room 748. (NOTE THE UNUSUAL SPACE AND TIME)

Speaker: Federico Pasqualotto

Abstract: In this talk, I will review recent results concerning the singularity formation problem for 3d incompressible fluids. In particular, I will focus on the role of angular regularity and explain why higher angular regularity makes blow-up constructions harder. I will finally outline recent work in collaboration with Tarek Elgindi for the 3d Euler equations on R^3, in which we construct the first singularity scenario entirely smooth in the angular variable.

Semiclassical analysis of twisted TMDs: exponentially flat and trivial bands

The HADES seminar on Tuesday, March 5th, will be at 3:30pm in Room 939.

Speaker: Mengxuan Yang

Abstract: Recent experiments discovered fractional Chern insulator states at zero magnetic field in twisted bilayer $\text{MoTe}_2$ and $\text{WSe}_2$, which are different types of twisted transition metal dichalcogenides (TMDs). In this talk, using Floquet theory, harmonic oscillators and construction of quasi-modes, I will present some simple mathematical results of band properties of TMDs when the twisting angles are small, including: absence of flat bands, trivial topology of lower bands and exponential flatness of lower bands. This is a joint work with Simon Becker.

Resonances on hyperbolic surfaces… and Berkovich space

The HADES seminar on Tuesday, February 20th, will be at 3:30pm in Room 939.

Speaker: Zhongkai Tao

Abstract: Hyperbolic surfaces are surfaces with constant negative curvature -1. They appear in many places: number theory, PDE, geometry, and topology… and they have many special properties. Despite a lot of studies and efforts put into this subject, the spectral theory of hyperbolic surfaces remains mysterious, especially in the infinite volume case. I will introduce some basic notions of the spectral theory on hyperbolic surfaces, and advertise some open problems. Then I will talk about recent developments on degeneration of hyperbolic surfaces, which uses some new tools from non-Archimedean geometry, and how this would potentially help us understand hyperbolic surfaces.