# Fall 2022 analysis seminar

The 2022-2023 Analysis Seminar will be organized by Shaoming Guo. The regular time and place for the Seminar will be Tuesdays at 4:00 p.m. in Van Vleck B139 (in some cases the seminar may be scheduled at different time to accommodate speakers). If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu. If you are from an institution different than UW-Madison, please send as well as an additional email to Shaoming. If you'd like to suggest speakers for the spring semester please contact Shaoming.

All talks will be in-person unless otherwise specified.

# Analysis Seminar Schedule

 date speaker title host(s) institution 08.23 Gustavo Garrigós University of Murcia Approximation by N-term trigonometric polynomials and greedy algorithms Andreas Seeger 08.30 Simon Myerson Warwick Forms of the Circle Method Shaoming Guo 09.13 (first week of semester) Zane Li UW Madison A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem Analysis group 09.16 (Friday, 1:20-2:10, Room B139) Franky Li UW Madison Affine restriction estimates for surfaces in R^3 via decoupling Analysis group 09.20 (Joint with PDE and Geometric Analysis seminar) Andrej Zlatoš UCSD Homogenization in front propagation models Hung Tran 09.23 Friday, Colloquium Pablo Shmerkin UBC Incidences and line counting: from the discrete to the fractal setting Shaoming Guo and Andreas Seeger 09.23-09.25 RTG workshop in Harmonic Analysis Shaoming Guo and Andreas Seeger 09.27 (online, special time, 3-4pm) Michael Magee Durham The maximal spectral gap of a hyperbolic surface Simon Marshall 10.04 Philip Gressman UPenn Title Shaoming Guo 10.11 Detlef Müller CAU Kiel Maximal averages along hypersurfaces: a geometric conjecture and further progress for 2-surfaces Betsy Stovall and Andreas Seeger 10.14 (1:00 PM Friday. Joint with Geometry & Topology Seminar) Min Ru U of Houston The K-stability and Nevanlinna/Diophantine theory Xianghong Gong 10.18 Madelyne M. Brown UNC Fourier coefficients of restricted eigenfunctions Betsy Stovall 10.24 (Monday, B135) Milivoje Lukic Rice Title Sergey Denisov 11.01 Ziming Shi Rutgers Title Xianghong Gong 11.04 (Friday, 1:20-2:10, in room tbd) Sarah Tammen MIT Betsy Stovall 11.08 Robert Fraser Wichita State University Title Andreas Seeger 11.15 Brian Cook Virginia Tech Title Brian Street 11.22 Thanksgiving 11.29 Jaume de Dios Pont UCLA Title Betsy Stovall 12.06 Shengwen Gan MIT Title Shaoming Guo and Andreas Seeger 12.13 Óscar Domínguez Universidad Complutense Madrid and University of Lyons Title Andreas Seeger and Brian Street

# Abstracts

### Gustavo Garrigós

Title: Approximation by N-term trigonometric polynomials and greedy algorithms

### Simon Myerson

Title: Forms of the circle method

Abstract: The circle method is an analytic proof strategy, typically used in number theory when one wants to estimate the number of integer lattice points in some interesting set. Traditionally the first step is to evaluate the innocent integral $\int_0^1 e^{2 \pi i t n} dt$ to give 1 if $n = 0$ and 0 if $n$ is any other integer. Since Heath-Brown’s delta-method in the 90s this simplest step has been embellished with carefully constructed partitions of unity. In this informal discussion I will interpret these as different versions of the circle method and suggest how to understand their relative advantages.

### Andrej Zlatos

Title: Homogenization in front propagation models

Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit. I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients. These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.

### Zane Li

Title: A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem

Abstract: There are two proofs of Vinogradov's Mean Value Theorem (VMVT), the harmonic analysis decoupling proof by Bourgain, Demeter, and Guth from 2015 and the number theoretic efficient congruencing proof by Wooley from 2017. While there has been recent work illustrating the relation between these two methods, VMVT has been around since 1935. It is then natural to ask: What does old partial progress on VMVT look like in harmonic analysis language? How similar or different does it look from current decoupling proofs? We talk about a refinement of a 1973 argument of Karatsuba that showed partial progress towards VMVT and interpret this in decoupling language. This yields an argument that only uses rather simple geometry of the moment curve. This is joint work with Brian Cook, Kevin Hughes, Olivier Robert, Akshat Mudgal, and Po-Lam Yung.

### Jianhui Li

Title: Affine restriction estimates for surfaces in \mathbb{R}^3 via decoupling

Abstract: We will discuss some L^2 restriction estimates for smooth compact surfaces in \mathbb{R}^3 with weights that respect affine transformations. The key ingredient is a decoupling inequality. The results are also uniform for polynomial surfaces of bounded degrees and coefficients. Part of the work is joint with Tongou Yang.

### Pablo Shmerkin

Title: Incidences and line counting: from the discrete to the fractal setting

Abstract: How many lines are spanned by a set of planar points?. If the points are collinear, then the answer is clearly "one". If they are not collinear, however, several different answers exist when sets are finite and "how many" is measured by cardinality. I will discuss a bit of the history of this problem and present a recent extension to the continuum setting, obtained in collaboration with T. Orponen and H. Wang. No specialized background will be assumed.

### Michael Magee

Title: The maximal spectral gap of a hyperbolic surface

Abstract: A hyperbolic surface is a surface with metric of constant curvature -1. The spectral gap between the first two eigenvalues of the Laplacian on a closed hyperbolic surface contains a good deal of information about the surface, including its connectivity, dynamical properties of its geodesic flow, and error terms in geodesic counting problems. For arithmetic hyperbolic surfaces the spectral gap is also the subject of one of the biggest open problems in automorphic forms: Selberg's eigenvalue conjecture.

A conjecture of Buser from the 1980s stated that there exists a sequence of closed hyperbolic surfaces with genera tending to infinity and spectral gap tending to 1/4. (The value 1/4 here is the asymptotically optimal one.) We proved that such a sequence does exist. I'll discuss the very interesting background of this problem in detail as well as some ideas of the proof.

This is joint work with Will Hide.

### Detlef Müller

Tirlw: Maximal averages along hypersurfaces: a geometric conjecture and further progress for 2-surfaces.

Title: Fourier coefficients of restricted eigenfunctions

Abstract: We will discuss the growth of Laplace eigenfunctions on a compact manifold when restricted to a submanifold. We analyze the behavior of the restricted eigenfunctions by studying their Fourier coefficients with respect to an arbitrary orthonormal basis for the submanifold. We give an explicit bound on these coefficients depending on how the defect measures for the two collections of functions, the eigenfunctions and the basis, relate.

### Min Ru

Title: The K-stability and Nevanlinna/Diophantine theory

Abstract: In the recent paper with P. Vojta, we introduced the so-called beta-constant, and used it to extend the Cartan's Second Main Theorem in Nevanlinna theory and Schmidt's subspace theorem in Diophantine approximation. It turns out the beta-constant is also used in the algebro-geometric stability criterion in the Fano's case. In this talk, I'll describe and explore the somewhat mysterious connection. The talk is based on the recent joint paper with Yan He entitled "The stability threshold and Diophantine approximation", Proc. AMS, 2022.

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