# Difference between revisions of "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 Sublevel Set Estimates in Higher Dimensions: Symmetry and Uniformity 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 (1:20 PM Friday, 901 Van Vleck. 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 An approach to universality using Weyl m-functions Sergey Denisov 11.01 Ziming Shi Rutgers Sobolev Differentiability Properties of the Modulus of Real Analytic Functions Xianghong Gong 11.04 (Friday, 1:20-2:10, in room tbd) Sarah Tammen MIT Incidence Estimates for Slabs Betsy Stovall 11.08 Robert Fraser Wichita State University Explicit Salem Sets in $\mathbb{R}^n$ Andreas Seeger 11.15 Brian Cook Virginia Tech Spherical multiple recurrence Brian Street 11.22 Thanksgiving 11.29 Jaume de Dios Pont UCLA Uniform boundedness in operators parametrized by polynomial curves Betsy Stovall 12.02 Donggeun Ryou Rochester A variant of the lambda p set in Orlicz spaces Shaoming Guo 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.

### Philip Gressman

Title: Sublevel Set Estimates in Higher Dimensions: Symmetry and Uniformity

Abstract: Using even simple derivative bounds, it is possible to understand the behavior of smooth functions of a single real variable in very precise ways. In contrast, when one moves to dimensions 2 and higher, current best approaches fail to yield the same kind of sharp, uniform inequalities that are relatively easy to obtain in 1D. I will discuss a number of related problems which attempt to illuminate some of the reasons for this discrepancy and to formulate new ways of working in higher dimensions to recover some of the robustness that is available in 1D. Of particular interest will be sublevel sets and uniform estimates for integrals of the sort found in the theory of critical integrability exponents. One main result will show how machinery developed for the study of affine Hausdorff measure can be used to build nonlinear differential operators whose nonvanishing implies uniform sublevel set estimates and bounds for related integrals.

### Detlef Müller

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

### 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.

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.

### Milivoje Lukic

Title: An approach to universality using Weyl m-functions

Abstract:

I will describe an approach to universality limits for orthogonal polynomials on the real line which is completely local and uses only the boundary behavior of the Weyl $m$-function at the point. We show that bulk universality of the Christoffel--Darboux kernel holds for any point where the imaginary part of the $m$-function has a positive finite nontangential limit. This approach is based on studying a matrix version of the Christoffel--Darboux kernel and the realization that bulk universality for this kernel at a point is equivalent to the fact that the corresponding $m$-function has normal limits at the same point. Our approach automatically applies to other self-adjoint systems with $2\times 2$ transfer matrices such as continuum Schr\"odinger and Dirac operators. We also obtain analogous results for orthogonal polynomials on the unit circle. This is joint work with Benjamin Eichinger and Brian Simanek.

### Ziming Shi

Title: Sobolev Differentiability Properties of the Logarithmic Modulus of Real Analytic Functions

Abstract: Let $f$ be the germ of a real analytic function at the origin in $\mathbb{R}^n$ for $n \geq 2$, and suppose the codimension of the zero set of $f$ at $\mathbf{0}$ is at least $2$. We show that $\log |f|$ is $W^{1,1}_{\operatorname{loc}}$ near $\mathbf{0}$. In particular, this implies the differential inequality $|\nabla f |\leq V |f|$ holds with $V \in L^1_{\operatorname{loc}}$. As an application, we derive an inequality relating the {\L}ojasiewicz exponent and the singularity exponent for such functions. This is joint work with Ruixiang Zhang.

### Sarah Tammen

Title: Incidence Estimates for Slabs

Abstract: We discuss incidence estimates for neighborhoods of hyperplanes in R^n, after the work of Guth, Solomon, and Wang, who proved an analogue of the Szemerédi-Trotter theorem to estimate incidences of tubes.  We use induction on scales and the high-low method of Vinh, along with new geometric insights.

### Robert Fraser

Title: Explicit Salem Sets in $\displaystyle{ \mathbb{R}^n }$

### Brian Cook

Title: Spherical multiple recurrence

Abstract: We will look at the question of pointwise convergence for a family of ergodic averages over the lattice points on spheres in dimension five and higher, and discuss aspects of extending these types of results to multilinear averages for more general algebraic surfaces.

### Jaume de Dios Pont

Title: Uniform boundedness in operators parametrized by polynomial curves

Abstract: Multiple results in harmonic analysis involving integrals of functions over curves (such as restriction theorems, convolution estimates, maximal function estimates or decoupling estimates) depend strongly on the non-vanishing of the torsion of the associated curve. Over the past years there has been considerable interest in extending these results to a degenerate case where the torsion vanishes at a finite number of points by using the affine arc-length as an alternative integration measure. As a model case, multiple results have been proven in which the coordinate functions of the curve are polynomials. In this case one expects the bounds of the operators to depend only on the degree of the polynomial. In this talk I will introduce and motivate the concept of affine arclength measure, provide new decomposition theorems for polynomial curves over characteristic zero local fields, and provide some applications to uniformity results in harmonic analysis.

### Donggeun Ryou

Title: A variant of the lambda p set in Orlicz spaces

Abstract: When $p>2$, let $S$ be a set of integers and consider trigonometric polynomials whose Fourier coefficients are supported on S. For various sets S, the range of $p$ has been studied where $L^p$ norms of trigonometric polynomials are bounded by their $L^2$ norms. However, in the opposite direction, we can fix p and think of a set S which satisfies the inequality $\|f\|_p \leq C \|f\|_2$ for some constant C. This set S is called a $\Lambda(p)$-set. In this talk, we will introduce $\Lambda(\Phi)$-sets which are defined in terms of Orlicz norms. And we will discuss some results about $\Lambda(\Phi)$-sets which extends known results about $\Lambda(p)$-sets.