Author Archives: ioltman

A probabilistic approach to the fractal uncertainty principle

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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

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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.

Semiclassical analysis of twisted TMDs: exponentially flat and trivial bands

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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

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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.

Asymptotic behavior of global solutions to a scalar quasilinear wave equation satisfying the weak null condition

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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  ($|x|\ll t$).

Wave dynamics and semiclassical analysis: from graphs to manifolds

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The HADES seminar on Tuesday, December 5th, will be at 3:30pm in Room 748 (not in 740 this week!).

Speaker: Akshat Kumar

Abstract: Graph Laplacians and Markov processes are intimately connected and ubiquitous in the study of graph structures. They have led to significant advances in a class of geometric inverse problems known as “manifold learning”, wherein one wishes to learn the geometry of a Riemannian submanifold from finite Euclidean point samples. The data gives rise to the geometry-encoding neighbourhood graphs. Present-day techniques are dominated primarily by the low spectral resolution of the graph Laplacians, while finer aspects of the underlying geometry, such as the geodesic flow, are observed only in the high spectral regime.

We establish a data-driven uncertainty principle that dictates the scaling of the wavelength $h$, with respect to the density of samples, at which graph Laplacians for neighbourhood graphs are approximately $h$-pseudodifferential operators. This sets the stage for a semiclassical approach to the high-frequency analysis of wave dynamics on weighted graphs. We thus establish a discrete version of Egorov’s theorem and achieve convergence rates for the recovery of geodesics on the underlying manifolds through quantum dynamics on the approximating graphs. I will show examples on samples of model manifolds and briefly discuss some applications to real-world datasets.

Optimal enhanced dissipation for geodesic flows

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The HADES seminar on Tuesday, October 31st will be at 3:30pm in Room 740.

Speaker: Maciej Zworski

Abstract: We consider geodesic flows on negatively curved compact manifolds or more generally contact Anosov flows (all these concepts will be pedagogically explained). The object is to show that if $ X $ is the generator of the flow and $ \Delta $, a (negative) Laplacian, then solutions to the convection diffusion equation, $ \partial_t u = X u +  \nu \Delta $, $ \nu \geq 0 $,  satisfy \[    \| u ( t )   –  \underline u \|_{L^2 ( M) } \leq C \nu^{-K} e^{ – \beta t } \| u( 0 )  \|_{L^2 ( M) }, \] where $ \underline u $ is the (conserved) average of $ u (0) $ with respect to the contact volume form and $K $ is a fixed constant. This provides many examples of very precise {\em optimal enhanced dissipation} in the sense of recent works of Bedrossian–Blumenthal–Punshon-Smith and Elgindi–Liss–Mattingly. The proof is based on results by Dyatlov and the speaker on stochastic stability of Pollicott–Ruelle resonances,  another concept which will be introduced and explained. The talk is based on joint work with Zhongkai Tao.

Sharp Furstenberg Sets Estimate in the Plane

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The HADES seminar on Tuesday, October 24th will be at 3:30pm in Room 740.

Speaker: Kevin Ren

Abstract: Fix a real number 0 < s <= 1. A set E in the plane is a s-Furstenberg set if there exists a line in every direction that intersects E in a set with Hausdorff dimension s. For example, a planar Kakeya set is a special case of a 1-Furstenberg set, and indeed we know that 1-Furstenberg sets have Hausdorff dimension 2. However, obtaining a sharp lower bound for the Hausdorff dimension of s-Furstenberg sets for any 0 < s < 1 has been a challenging open problem for half a century. In this talk, I will illustrate the rich connections between the Furstenberg sets conjecture and other important topics in geometric measure theory and harmonic analysis, and show how exploring these connections can fully resolve the Furstenberg conjecture. Joint works with Yuqiu Fu and Hong Wang.

Existence of more self-similar implosion profiles for the Euler-Poisson system

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The HADES seminar on Tuesday, October 10th will be at 3:30pm in Room 740.

Speaker: Ely Sandine

Abstract: I will discuss implosion for the equations describing a gas which is compressible, isothermal and self-gravitating. Under the hypotheses of radial symmetry and self-similarity, the equations reduce to a system of ODEs which has been extensively studied by the astrophysics community using numerical methods. One such solution, discovered by Larson and Penston in 1969, was recently rigorously proved to exist by Guo, Hadžić and Jang. In this talk, I will discuss rigorous existence of a subset of the discrete family of solutions found numerically by Hunter in 1977.