# Analysis Seminar

The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger. It will be online at least for the Fall semester, with details to be announced in September. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar earlier, or on different days, to accomodate speakers).

Zoom links will be sent to those who have signed up for the Analysis Seminar List. For instructions how to sign up for seminar lists, see https://www.math.wisc.edu/node/230

If you'd like to suggest speakers for the spring semester please contact David and Andreas (dbeltran at math, seeger at math).

# Current Analysis Seminar Schedule

 date speaker title host(s) institution September 22 Alexei Poltoratski UW Madison Dirac inner functions September 29 Joris Roos University of Massachusetts - Lowell A triangular Hilbert transform with curvature, I Wednesday September 30, 4 p.m. Polona Durcik Chapman University A triangular Hilbert transform with curvature, II October 6 Andrew Zimmer UW Madison Complex analytic problems on domains with good intrinsic geometry October 13 Hong Wang Princeton/IAS Improved decoupling for the parabola October 20 Kevin Luli UC Davis Smooth Nonnegative Interpolation October 21, 4.00 p.m. Niclas Technau UW Madison Number theoretic applications of oscillatory integrals October 27 Terence Harris Cornell University Low dimensional pinned distance sets via spherical averages Monday, November 2, 4 p.m. Yuval Wigderson Stanford University Title November 10 Óscar Domínguez Universidad Complutense de Madrid Title November 17 Tamas Titkos BBS U of Applied Sciences and Renyi Institute Title November 24 Shukun Wu University of Illinois (Urbana-Champaign) Title December 1 Jonathan Hickman The University of Edinburgh Title December 8 Alejandra Gaitán Purdue University Title February 2 Jongchon Kim UBC Title February 9 Bingyang Hu Purdue University Title February 16 Krystal Taylor The Ohio State University Title February 23 Title March 2 Title March 9 Title March 16 Ziming Shi Rutgers University Title March 23 Title March 30 Title April 6 Title April 13 Title April 20 Title April 27 Title May 4 Title

# Abstracts

### Alexei Poltoratski

Title: Dirac inner functions

Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations. We will discuss connections between problems in complex function theory, spectral and scattering problems for differential operators and the non-linear Fourier transform.

### Polona Durcik and Joris Roos

Title: A triangular Hilbert transform with curvature, I & II.

Abstract: The triangular Hilbert is a two-dimensional bilinear singular originating in time-frequency analysis. No Lp bounds are currently known for this operator. In these two talks we discuss a recent joint work with Michael Christ on a variant of the triangular Hilbert transform involving curvature. This object is closely related to the bilinear Hilbert transform with curvature and a maximally modulated singular integral of Stein-Wainger type. As an application we also discuss a quantitative nonlinear Roth type theorem on patterns in the Euclidean plane. The second talk will focus on the proof of a key ingredient, a certain regularity estimate for a local operator.

### Andrew Zimmer

Title: Complex analytic problems on domains with good intrinsic geometry

Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).

### Hong Wang

Title: Improved decoupling for the parabola

Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.

### Kevin Luli

Title: Smooth Nonnegative Interpolation

Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets.

### Niclas Technau

Title: Number theoretic applications of oscillatory integrals

Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.

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