Difference between revisions of "Analysis Seminar"
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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. | 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. | ||
− | === | + | ===Terence Harris=== |
− | Title | + | Title: Low dimensional pinned distance sets via spherical averages |
− | Abstract | + | Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set. |
===Name=== | ===Name=== |
Revision as of 18:32, 18 October 2020
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).
Contents
Previous_Analysis_seminars
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars
Current Analysis Seminar Schedule
date | speaker | institution | title | host(s) |
---|---|---|---|---|
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.
Terence Harris
Title: Low dimensional pinned distance sets via spherical averages
Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.
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Blank Analysis Seminar Template
Graduate Student Seminar:
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html