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 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.

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.

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.

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.

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.

The HADES seminar on Tuesday, February 13th, will be at 3:30pm in Room 939.

Speaker: Dongxiao Yu

Abstract: I will discuss the long time dynamics of a scalar quasilinear wave equation in three space dimensions. This equation satisfies the weak null condition and has global existence for sufficiently small $C_c^\infty$ initial data. In the talk, I will first present an asymptotic completeness result which describes the asymptotic behavior of global solutions to the scalar quasilinear wave equation near the light cone ($|x|\approx t$). Then, I will discuss a work in progress on the asymptotic behavior inside the light cone  ($|x|\ll t$).

The HADES seminar on Tuesday, February 6th, will be at 3:30pm in Room 939.

Speaker: Izak Oltman

Abstract: I will apply exact WKB theory of analytic finite difference equations to estimate the spectrum of the Scottish flag matrix.

This is based on joint work with Frédéric Klopp and Shengtong Zhang.

The HADES seminar on Tuesday, January 30th, will be at 3:30pm in Room 939 (not in 740 this semester!).

Speaker: Joonhyun La

Abstract: In this talk, we will prove local well-posedness of kinetic wave equation arising from MMT equation, which is introduced by Majda, Mclaughlin, and Tabak and is one of the standard toy models to study wave turbulence. Surprisingly, our result reveals a regularization effect of the collision operator, which resembles the situation of non-cutoff Boltzmann. This talk is based on a joint work with Pierre Germain (Imperial College London) and Katherine Zhiyuan Zhang (Northeastern).