Difference between revisions of "Analysis Seminar"

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The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.
 
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.
 
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 to accomodate speakers).
+
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
 
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
Line 25: Line 25:
 
|Alexei Poltoratski
 
|Alexei Poltoratski
 
|UW Madison
 
|UW Madison
|[[#linktoabstract Title ]]
+
|[[#Alexei Poltoratski Dirac inner functions ]]
 
|  
 
|  
 
|-
 
|-
 
|September 29
 
|September 29
|Polona Durcik
+
|Joris Roos
| Chapman University
+
|University of Massachusetts - Lowell
|[[#linktoabstract |   Title ]]
+
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]
 
|  
 
|  
 
|-
 
|-
|September 30
+
|Wednesday September 30, 4 p.m.
|Joris Roos
+
|Polona Durcik
|University of Massachusetts - Lowell
+
|Chapman University
|[[#linktoabstract |   Title ]]
+
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]
 
|  
 
|  
 
|-
 
|-
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|Andrew Zimmer
 
|Andrew Zimmer
 
|UW Madison
 
|UW Madison
|[[#linktoabstract Title ]]
+
|[[#Andrew Zimmer Complex analytic problems on domains with good intrinsic geometry ]]
 
|  
 
|  
 
|-
 
|-
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|Hong Wang  
 
|Hong Wang  
 
|Princeton/IAS
 
|Princeton/IAS
|[[#linktoabstract Title ]]
+
|[[#Hong Wang Improved decoupling for the parabola ]]
 
|  
 
|  
 
|-
 
|-
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|Kevin Luli
 
|Kevin Luli
 
|UC Davis
 
|UC Davis
|[[#linktoabstract Title ]]
+
|[[#Kevin Luli Smooth Nonnegative Interpolation ]]
 +
|
 +
|-
 +
|October 21, 4.00 p.m.
 +
|Niclas Technau
 +
|UW Madison
 +
|[[#Niclas Technau  |  Number theoretic applications of oscillatory integrals ]]
 
|  
 
|  
 
|-
 
|-
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|Terence Harris
 
|Terence Harris
 
| Cornell University
 
| Cornell University
|[[#linktoabstract Title ]]
+
|[[#Terence Harris Low dimensional pinned distance sets via spherical averages ]]
 
|  
 
|  
 
|-
 
|-
|November 3
+
|Monday, November 2, 4 p.m.
|No seminar
+
|Yuval Wigderson
|  
+
|Stanford  University
|
+
|[[#Yuval Wigderson  |  New perspectives on the uncertainty principle ]]
 
|  
 
|  
 
|-
 
|-
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|-
 
|-
 
|December 8
 
|December 8
|
+
|Alejandra Gaitán
|  
+
| Purdue University
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
 
|  
 
|  
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|-
 
|-
 
|February 16
 
|February 16
|David Beltran
+
|Krystal Taylor
|UW - Madison
+
|The Ohio State University
 +
|[[#linktoabstract  |  Title ]]
 +
|
 +
|-
 +
|February 23
 +
|Dominique Maldague
 +
|MIT
 +
|[[#linktoabstract  |  Title ]]
 +
|
 +
|-
 +
|March 2
 +
|Diogo Oliveira e Silva
 +
|University of Birmingham
 +
|[[#linktoabstract  |  Title ]]
 +
|
 +
|-
 +
|March 9
 +
|
 +
|
 +
|[[#linktoabstract  |  Title ]]
 +
|
 +
|-
 +
|March 16
 +
|Ziming Shi
 +
|Rutgers University
 +
|[[#linktoabstract  |  Title ]]
 +
|
 +
|-
 +
|March 23
 +
|
 +
|
 +
|[[#linktoabstract  |  Title ]]
 +
|
 +
|-
 +
|March 30
 +
|
 +
|
 +
|[[#linktoabstract  |  Title ]]
 +
|
 +
|-
 +
|April 6
 +
|
 +
|
 +
|[[#linktoabstract  |  Title ]]
 +
|
 +
|-
 +
|April 13
 +
|
 +
|
 +
|[[#linktoabstract  |  Title ]]
 +
|
 +
|-
 +
|April 20
 +
|
 +
|
 +
|[[#linktoabstract  |  Title ]]
 +
|
 +
|-
 +
|April 27
 +
|
 +
|
 +
|[[#linktoabstract  |  Title ]]
 +
|
 +
|-
 +
|May 4
 +
|
 +
|
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===Name===
+
===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
+
Title: Smooth Nonnegative Interpolation
  
Abstract
+
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===
  
===Name===
+
Title: Number theoretic applications of oscillatory integrals
  
Title
+
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
+
===Terence Harris===
  
 +
Title: Low dimensional pinned distance sets via spherical averages
  
===Name===
+
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.
  
Title
+
===Yuval Wigderson===
  
Abstract
+
Title: New perspectives on the uncertainty principle
  
 +
Abstract: The phrase ``uncertainty principle'' refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized''. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.
  
 
===Name===
 
===Name===
Line 145: Line 257:
  
 
Abstract
 
Abstract
 
  
 
===Name===
 
===Name===
Line 152: Line 263:
  
 
Abstract
 
Abstract
 
  
 
=Extras=
 
=Extras=

Latest revision as of 17:46, 23 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).


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 New perspectives on the uncertainty principle
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 Dominique Maldague MIT Title
March 2 Diogo Oliveira e Silva University of Birmingham 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.

Yuval Wigderson

Title: New perspectives on the uncertainty principle

Abstract: The phrase ``uncertainty principle refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.

Name

Title

Abstract

Name

Title

Abstract

Extras

Blank Analysis Seminar Template


Graduate Student Seminar:

https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html