Fall 2023 Analysis Seminar

Organizer: Shaoming Guo

Email: shaomingguo (at) math (dot) wisc (dot) edu

Time: Wed 3:30--4:30

Room: B223

We also have room B211 reserved at 4:25-5:25 for discussions after talks.

All talks will be in-person unless otherwise specified.

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

Week	Date	Speaker	Institution	Title	Host(s)	Notes (e.g. unusual room/day/time)
1	We, Sep. 6
	Fr, Sept. 8	Tushar Das	UW La Crosse	Playing games on fractals: Dynamical and Diophantine	Betsy	Colloquium, 4-5pm in B239
2	Tu, Sept. 12	Rajula Srivastava	Hausdorff Center of Mathematics, Bonn	Counting Rational Points near Flat Hypersurfaces	Andreas	Tuesday 4:00 pm in VV B135
	We, Sept. 13	Niclas Technau	University of Graz	Oscillatory Integrals Count	Andreas
3	We, Sept. 20	Terry Harris	UW Madison	Horizontal Besicovitch sets of measure zero and some related problems	analysis group
4	We, Sept. 27
5	Mo, Oct 2	Edriss Titi	Texas A&M	On the Solvability of the Navier-Stokes and Euler Equations, where do we stand?	Leslie and Sam	Distinguished Lecture 1 (general audience talk), 4-5pm in B239
5	Tu, Oct 3	Edriss Titi	Texas A&M	Oceanic and Atmospheric Dynamics Models - Global Regularity and Finite-time Singularity.	Leslie and Sam	Distinguished Lecture 2 (Analysis seminar), 4-5pm, B135
5	We, Oct 4	Edriss Titi	Texas A&M	Downscaling Data Assimilation Algorithm for Weather and Climate Prediction	Leslie and Sam	Distinguished Lecture 3 (Applied and Computational math seminar), 901
5	We, Oct. 4	Tristan Leger	Princeton	L^p bounds for spectral projectors on hyperbolic surfaces	Simon
6	We, Oct. 11	Bingyang Hu	Auburn	On the curved Trilinear Hilbert transform	Brian
7	We, Oct. 18	Ashley Zhang	Vanderbilt	Convergence of discrete non-linear Fourier transform via spectral problems for canonical systems	Alexei
8	Tu, Oct 25	Gigliola Staffilani	MIT	A Small Window into Wave Turbulence Theory	Mihaela and Leslie	Hilldale Lecture 4-5pm in 1310 Sterling Hall
8	We, Oct. 25	Gigliola Staffilani	MIT	The Schrödinger Equations as an Inspiration of Beautiful Mathematics	Mihaela and Leslie	Special Colloquium 4-5pm in B239
	Fr, Oct 27	Rodrigo Bañuelos	Purdue	Probabilistic tools in discrete harmonic analysis	Betsy	Colloquium, 4-5pm in B239
9	We, Nov. 1	Shaoming Guo	UW Madison	Oscillatory integrals on Riemannian manifolds, and related Kakeya and Nikodym problems
10	We, Nov. 8	Lechao Xiao	Google deepmind	Some connections between Harmonic Analysis and Theory of Deep Learning	Shaoming
11	We, Nov. 15	Neeraja Kulkarni	Caltech	An improved Minkowski dimension estimate for Kakeya sets in higher dimensions using planebrushes.	Jacob
12	We, Nov. 22	No talk	No talk		No talk	Thanksgiving week
13	We, Nov. 29	Changkeun Oh	MIT	Discrete restriction estimates for manifolds avoiding a line.	Shaoming
14	We, Dec. 6
	Fr, Dec 8	Ryan Bushling	the other UW	Some variational problems characterizing families of convex domains	Jake Fiedler	1:20-2:10 in B107 (normal student seminar time/place)
15	We, Dec. 13	Jing-Jing Huang	Nevada	Diophantine approximation on affine subspaces	Shaoming
1	We, Jan. 24, 2024
2	We, Jan. 31	Sunggeum Hong	Chosun University	The Hörmander multiplier theorem for n-linear operators and its applications	Andreas
3	We, Feb. 7	Donald Stull	University of Chicago		Betsy, Shaoming, and Jake F.
4	We, Feb. 14
5	We, Feb. 21	Andrei Martinez-Finkelshtein	Baylor		Sergey
6	We, Feb. 28	Alex Rutar	University of St. Andrews		Andreas
7	We, Mar. 6	Song-Ying Li	UC-Irvine		Xianghong
8	We, Mar. 13
9	We, Mar. 20
	Fr, Mar. 22	Jack Lutz	Iowa State University		Shaoming	department colloquium, 4-5pm
10	We, Mar. 27	Spring recess	spring recess		spring recess
11	We, Apr. 3
12	We, Apr. 10	Victor Bailey	University of Oklahoma		Betsy
13	We, Apr. 17	Jianhui (Franky) Li	Northwestern University		Betsy
14	We, Apr. 24
15	We, May 1

Abstracts

Tushar Das

Title: Playing games on fractals: Dynamical & Diophantine

Abstract: We will present sketches of a program, developed in collaboration with Lior Fishman, David Simmons, and Mariusz Urbanski, which extends the parametric geometry of numbers (initiated by Wolfgang Schmidt and Leonhard Summerer) to Diophantine approximation for systems of m linear forms in n variables. Our variational principle (arXiv:1901.06602) provides a unified framework to compute Hausdorff and packing dimensions of a variety of sets of number-theoretic interest,  as well as their dynamical counterparts via the Dani correspondence. Highlights include the introduction of certain combinatorial objects that we call templates, which arise from a dynamical study of Minkowski’s successive minima in the geometry of numbers; as well as a new variant of Schmidt’s game designed to compute the Hausdorff and packing dimensions of any set in a doubling metric space. The talk will be accessible to students and faculty whose interests contain a convex combination of homogeneous dynamics, Diophantine approximation and fractal geometry.

Rajula Srivastava

Title: Counting Rational Points near Flat Hypersurfaces

Abstract: How many rational points with denominator of a given size lie within a given distance from a compact hypersurface? In this talk, we shall describe how the geometry of the surface plays a key role in determining this count, and present a heuristic for the same. In a recent breakthrough, J.J. Huang proved that this guess is indeed true for hypersurfaces with non-vanishing Gaussian curvature. What about hypersurfaces with curvature only vanishing up to a finite order, at a single point? We shall offer a new heuristic in this regime which also incorporates the contribution arising from "local flatness". Further, we will describe how ideas from Harmonic Analysis can be used to establish the indicated estimates for hypersurfaces of this type immersed by homogeneous functions. In particular, we shall use a powerful bootstrapping argument relying on Poisson summation, duality between flat and "rough" hypersurfaces, and the method of stationary phase. A crucial role is played by a dyadic scaling argument exploiting the homogeneous nature of the hypersurface. Based on joint work with N. Technau.

Niclas Technau

Title: Oscillatory Integrals Count

Abstract: This talk is about phrasing (number theoretic) counting problems in terms oscillatory integrals. We shall provide a simple introduction to the topic, mention open questions, and report on joint work with Sam Chow, as well as on joint work with Chris Lutsko.

Terry Harris

Title: Horizontal Besicovitch sets of measure zero and some related problems

Abstract: It is shown that horizontal Besicovitch sets of measure zero exist in R^3. The proof is constructive and uses point-line duality analogously to Kahane’s construction of measure zero Besicovitch sets in the plane. Some consequences and related examples are shown for the SL_2 Kakeya maximal function.

Tristan Leger

Title: L^p bounds for spectral projectors on hyperbolic surfaces

Abstract: In this talk I will present L^p boundedness results for spectral projectors on hyperbolic surfaces. I will focus on the case where the spectral window has small width. Indeed the negative curvature assumption leads to improvements over the universal bounds of C.Sogge, thus illustrating how these objects are sensitive to the global geometry of the underlying manifold. I will also explain how this relates to the general theme of delocalization of eigenfunctions of the Laplacian on hyperbolic surfaces. This is based on joint work with Jean-Philippe Anker and Pierre Germain.

Bingyang Hu

Title: On the curved Trilinear Hilbert transform

Abstract: The goal of this talk is to discuss the Lp boundedness of the trilinear Hilbert transform along the moment curve. More precisely, we show that the operator $$ H_C(f_1, f_2, f_3)(x):=p.v. \int_{\mathbb R} f_1(x-t)f_2(x+t^2)f_3(x+t^3) \frac{dt}{t}, \quad x \in \mathbb R $$ is bounded from $L^{p_1}(\mathbb R) \times L^{p_2}(\mathbb R) \times L^{p_3}(\mathbb R}$ into $L^r(\mathbb R)$ within the Banach H\"older range $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=\frac{1}{r}$ with $1<p_1, p_3<\infty$, $1<p_2 \le \infty$ and $1 \le r <\infty$.

The main difficulty in approaching this problem(compared to the classical approach to the bilinear Hilbert transform) is the lack of absolute summability after we apply the time-frequency discretization(which is known as the LGC-methodology introduced by V. Lie in 2019). To overcome such a difficulty, we develop a new, versatile approch -- referred to as Rank II LGC (which is also motived by the study of the non-resonant bilinear Hilbert-Carleson operator by C. Benea, F. Bernicot, V. Lie, and V. Vitturi in 2022), whose control is achieved via the following interdependent elements:

1). a sparse-uniform deomposition of the input functions adapted to an appropriate time-frequency foliation of the phase-space;

2). a structural analysis of suitable maximal "joint Fourier coefficients";

3). a level set analysis with respect to the time-frequency correlation set.

This is a joint work with my postdoc advisor Victor Lie from Purdue.

Rodrigo Bañuelos

Title: Probabilistic tools in discrete harmonic analysis

Abstract: The discrete Hilbert transform was introduced by David Hilbert at the beginning of the 20th century as an example of a singular quadratic form. Its boundedness on the space of square summable sequences appeared in H. Weyl’s doctoral dissertation (under Hilbert) in 1908. In 1925, M. Riesz proved that the continuous version of this operator is bounded on L^p(R), 1 < p < \infty, and that the same holds for the discrete version on the integers. Shortly thereafter (1926), E. C. Titchmarsh gave a different proof and from it concluded that the operators have the same p-norm. Unfortunately, Titchmarsh’s argument for equality was incorrect. The question of equality of the norms had been a “simple tantalizing" problem ever since.

In this general colloquium talk the speaker will discuss a probabilistic construction, based on Doob’s “h-Brownian motion," that leads to sharp inequalities for a collection of discrete operators on the d-dimensional lattice Z^d, d ≥ 1. The case d = 1 verifies equality of the norms for the discrete and continuous Hilbert transforms. The case d > 1 leads to similar questions and conjectures for other Calderón-Zygmund singular integrals in higher dimensions.

Shaoming Guo

Title: Oscillatory integrals on Riemannian manifolds, and related Kakeya and Nikodym problems

Abstract: The talk is about oscillatory integral operators on manifolds. Manifolds of constant sectional curvatures are particularly interesting, and we will see that very good estimates on these manifolds can be expected. We will also discuss Kakeya and Nikodym problems on general manifolds, in particular, manifolds satisfying Sogge’s chaotic curvatures.

Lechao Xiao

Title: Some connections between Harmonic Analysis and Theory of Deep Learning

Abstract: The past decade has witnessed a remarkable surge in breakthroughs in artificial intelligence (AI), with the potential to profoundly impact various aspects of our lives. However, the fundamental mathematical principles underlying the success of deep learning, the core technology behind these breakthroughs, is still far from well-understood. In this presentation, I will share some interesting connections between the theory of deep learning and harmonic analysis. The first half provides a gentle introduction to machine learning and deep learning. The second half focuses on two technical topics:

An uncertainty principle between space and frequency and its significance in overcoming the curse of dimensionality.

The multi-scale Marchenko-Pastur law and its interplay with the multiple-descent learning curve phenomenon.

Neeraja Kulkarni

Title: An improved Minkowski dimension estimate for Kakeya sets in higher dimensions using planebrushes.

Abstract: A Kakeya set is defined as a compact subset of $\mathbb{R}^n$ which contains a line segment of length 1 in every direction. The Kakeya conjecture says that every Kakeya set has Minkowski and Hausdorff dimensions equal to $n$. In this talk, I will discuss an improved Minkowski dimension estimate for Kakeya sets in dimensions $n\geq 5$. The improved estimate comes from using a geometric argument called a ``$k$-planebrush, which is a higher dimensional analogue of Wolff's ``hairbrush argument from 1995. The $k$-planebrush argument is used in conjunction with a previously known "k-linear" result on Kakeya sets proved by Hickman-Rogers-Zhang (and concurrently by Zahl) in 2019 along with an x-ray transform estimate which is a corollary of Hickman-Rogers-Zhang (and Zahl). The x-ray transform estimate is used to deduce that the Kakeya set has a structural property called ``stickiness, which was first introduced in a paper by Katz-Laba-Tao in 1999. Sticky Kakeya sets exhibit a self-similar structure which is exploited by the $k$-planebrush argument.

Changkeun

Title: Discrete restriction estimates for manifolds avoiding a line.

Abstract: We identify a new way to divide the d-neighborhood of surfaces in R^3. We decompose the d-neighborhood of surfaces into a finitely-overlapping collection of rectangular boxes S. We obtain an (l^2,L^p) decoupling estimate using this decomposition, for the sharp range of exponents. The decoupling theorem we prove is new for the hyperbolic paraboloid, and recovers the Tomas-Stein restriction inequality. Our decoupling inequality leads to new exponential sum estimates where the frequencies lie on surfaces which do not contain a line. In this talk, I'll focus on explaning backgrounds and theorems rather than giving proofs. This is joint work with Larry Guth and Dominique Maldague.

Ryan Bushlin

Friday, December 8, 1:20-2:10 in B107

Some variational problems characterizing families of convex domains

We prove a singular integral identity for the surface measure of (n - 1)-rectifiable sets in Euclidean n-space satisfying the orientation cancellation condition. In particular, sets of finite perimeter enjoy this property, and from this observation follows a geometric inequality in which equality is attained precisely by the convex sets. More generally, the integral identity has anisotropic analogues whose corresponding inequalities characterize some geometrically simple subfamilies of the family of convex sets.

Jing-Jing Huang

Diophantine approximation on affine subspaces

We establish a clear-cut criterion for an affine subspace of R^n to be extremal (i.e. the Dirichlet exponent 1/n is best possible almost everywhere), confirming a conjecture of Kleinbock. We also extend the classical theorem of Khintchine on metric diophantine approximation to affine subspaces. Moreover, an exact formula for the Hausdorff dimension of the set of very well approximable vectors on the affine subspace is obtained. The above results are proved as consequences of our novel estimates for the number of rational points lying close to the affine subspace. The proof of this counting estimate is Fourier analytic in nature and in particular utilizes the large sieve inequality.

Links to previous seminars

Link to the analysis seminar in 2024 Spring