Difference between revisions of "Analysis Seminar"

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The 2020-2021 Analysis Seminar will be organized by David Beltran and Andreas Seeger.
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The 2021-2022 Analysis Seminar will be organized by David Beltran and Andreas Seeger.
It will be online for the entire academic year. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).
+
Some of the talks will be in person (room Van Vleck B139) and some will be online. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).
  
Zoom links will be sent to those who have signed up for the Analysis Seminar List.  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 as well as an additional email to David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request.
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Zoom links will be sent to those who have signed up for the Analysis Seminar List.  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 David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request.
  
If you'd like to suggest speakers for the spring semester please contact David and Andreas.
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If you'd like to suggest speakers for the fall semester please contact David and Andreas.
  
 
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= Analysis Seminar Schedule =
 
 
=[[Previous_Analysis_seminars]]=
 
 
 
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars
 
 
 
= Current Analysis Seminar Schedule =
 
 
{| cellpadding="8"
 
{| cellpadding="8"
 
!align="left" | date   
 
!align="left" | date   
Line 21: Line 15:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|September 22
+
|September 21, VV B139
|Alexei Poltoratski
+
| Dóminique Kemp
|UW Madison
+
| UW-Madison
|[[#Alexei Poltoratski Dirac inner functions ]]
+
|[[#Dóminique Kemp Decoupling by way of approximation ]]
 
|  
 
|  
 
|-
 
|-
|September 29
+
|September 28, VV B139
|Joris Roos
+
| Jack Burkart
|University of Massachusetts - Lowell
+
| UW-Madison
|[[#Polona Durcik and Joris Rooslinktoabstract | A triangular Hilbert transform with curvature, I ]]
+
|[[#Jack Burkart |   Transcendental Julia Sets with Fractional Packing Dimension ]]
 
|  
 
|  
 
|-
 
|-
|Wednesday September 30, 4 p.m.
+
|October 5, Online
|Polona Durcik
+
| Giuseppe Negro
|Chapman University
+
| University of Birmingham
|[[#Polona Durcik and Joris Roos | A triangular Hilbert transform with curvature, II ]]
+
|[[#Giuseppe Negro |   Stability of sharp Fourier restriction to spheres ]]
 
|  
 
|  
 
|-
 
|-
|October 6
+
|October 12, VV B139
|Andrew Zimmer
+
|Rajula Srivastava
 
|UW Madison
 
|UW Madison
|[[#Andrew Zimmer Complex analytic problems on domains with good intrinsic geometry ]]
+
|[[#Rajula Srivastava Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups ]]
 
|  
 
|  
 
|-
 
|-
|October 13
+
|October 19, Online
|Hong Wang
+
|Itamar Oliveira
|Princeton/IAS
+
|Cornell University
|[[#Hong Wang Improved decoupling for the parabola ]]
+
|[[#Itamar Oliveira A new approach to the Fourier extension problem for the paraboloid ]]
 
|  
 
|  
 
|-
 
|-
|October 20
+
|October 26, VV B139
|Kevin Luli
+
| Changkeun Oh
|UC Davis
+
| UW Madison
|[[#Kevin Luli Smooth Nonnegative Interpolation ]]
+
|[[#Changkeun Oh Decoupling inequalities for quadratic forms and beyond ]]
 
|  
 
|  
 
|-
 
|-
|October 21, 4.00 p.m.
+
|October 29, TBA
|Niclas Technau
+
| Alexandru Ionescu (Colloquium)
|UW Madison
+
| Princeton University
|[[#Niclas Technau Number theoretic applications of oscillatory integrals ]]
+
|[[#linktoabstract  |  Title ]]
 +
|-
 +
|November 2, VV B139
 +
| Liding Yao
 +
| UW Madison
 +
|[[#linktoabstract  |  Title ]]
 +
|
 +
|-
 +
|November 9, VV B139
 +
| Lingxiao Zhang
 +
| UW Madison
 +
|[[#linktoabstract Title ]]
 
|  
 
|  
 
|-
 
|-
|October 27
+
|November 12, TBA
|Terence Harris
+
| Kasso Okoudjou (Colloquium)
| Cornell University
+
| Tufts University
|[[#Terence Harris Low dimensional pinned distance sets via spherical averages ]]
+
|[[#linktoabstract Title ]]
 +
|-
 +
|November 16, VV B139
 +
| Rahul Parhi
 +
| UW Madison (EE)
 +
|[[#linktoabstract  |  Title ]]
 
|  
 
|  
 
|-
 
|-
|Monday, November 2, 4 p.m.
+
|November 30, VV B139
|Yuval Wigderson
+
| Alexei Poltoratski
|Stanford  University
+
| UW Madison
|[[#Yuval Wigderson New perspectives on the uncertainty principle ]]
+
|[[#linktoabstract Title ]]
 
|  
 
|  
 
|-
 
|-
|November 10, 10 a.m.
+
|December 7
|Óscar Domínguez
+
| Person
| Universidad Complutense de Madrid
+
| Institution
|[[#Oscar Dominguez | New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions ]]
+
|[[#linktoabstract |   Title ]]
 
|  
 
|  
 
|-
 
|-
|November 17
+
|December 14
|Tamas Titkos
+
| reserved
|BBS U of Applied Sciences and Renyi Institute
+
| Institution
|[[#Tamas Titkos | Isometries of Wasserstein spaces ]]
+
|[[#linktoabstract  |   Title ]]
 
|  
 
|  
 
|-
 
|-
|November 24
+
|February 1
|Shukun Wu
+
| Person
|University of Illinois (Urbana-Champaign)
+
| Institution
||[[#Shukun Wu | On the Bochner-Riesz operator and the maximal Bochner-Riesz operator ]]  
+
|[[#linktoabstract |   Title ]]
 
|  
 
|  
 
|-
 
|-
|December 1
+
|February 8
| Jonathan Hickman
+
| Person
| The University of Edinburgh
+
| Institution
|[[#Jonathan Hickman | Sobolev improving for averages over space curves ]]
+
|[[#linktoabstract |   Title ]]
 
|  
 
|  
 
|-
 
|-
|February 2, 7:00 p.m.
+
|February 15
|Hanlong Fang
+
| Person
|UW Madison
+
| Institution
|[[#Hanlong Fang | Canonical blow-ups of Grassmann manifolds ]]
+
|[[#linktoabstract  |   Title ]]
 
|  
 
|  
 
|-
 
|-
|February 9
+
|February 22
|Bingyang Hu
+
| Person
|Purdue University
+
| Institution
|[[#Bingyang Hu | Some structure theorems on general doubling measures ]]
+
|[[#linktoabstract |   Title ]]
 
|  
 
|  
 
|-
 
|-
|February 16
+
|March 1
|Krystal Taylor
+
| Person
|The Ohio State University
+
| Institution
|[[#Krystal Taylor Quantifications of the Besicovitch Projection theorem in a nonlinear setting  ]]
+
|[[#linktoabstract Title ]]
|
+
|  
 
|-
 
|-
|February 23
+
|March 8
|Dominique Maldague
+
| Brian Street
|MIT
+
| UW Madison
|[[#Dominique Maldague | A new proof of decoupling for the parabola ]]
+
|[[#linktoabstract |   Title ]]
|
+
|  
 
|-
 
|-
|March 2
+
|March 15: No Seminar
|Diogo Oliveira e Silva
+
| Person
|University of Birmingham
+
| Institution
|[[#Diogo Oliveira e Silva Global maximizers for spherical restriction ]]
+
|[[#linktoabstract Title ]]
|
+
|  
 
|-
 
|-
|March 9
+
|March 23
|Oleg Safronov
+
| Person
|University of North Carolina Charlotte
+
| Institution
|[[#Oleg Safronov | Relations between discrete and continuous spectra of differential operators ]]
+
|[[#linktoabstract |   Title ]]
|
+
|  
 
|-
 
|-
|March 16
+
|March 30
|Ziming Shi
+
| Person
|Rutgers University
+
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
|
+
|  
 
|-
 
|-
|March 23
+
|April 5
|Xiumin Du
+
| Person
|Northwestern University
+
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
|
+
|  
 
|-
 
|-
|March 30, 10:00  a.m.
+
|April 12
|Etienne Le Masson
+
| Person
|Cergy Paris University
+
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
|
+
|  
 
|-
 
|-
|April 6
+
|April 19
|Theresa Anderson
+
| Person
|Purdue University
+
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
|
+
|  
 
|-
 
|-
|April 13
+
|April 22, Colloquium
|Nathan Wagner
+
|Detlef Müller
|Washington University St. Louis
+
|University of Kiel
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
|
+
|  
 
|-
 
|-
|April 20
+
|April 29
|Jongchon Kim
+
| Person
| University of British Columbia
+
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
|
+
|  
 
|-
 
|-
|April 27
+
|May 3
|Yumeng Ou
+
| Person
|University of Pennsylvania
+
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
|
+
|  
 
|-
 
|-
|May 4
+
|Date
|
+
| Person
|
+
| Institution
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
 +
|
 +
|-
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===Alexei Poltoratski===
+
===Dóminique Kemp===
  
Title: Dirac inner functions
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Decoupling by way of approximation
  
Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations.
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Since Bourgain and Demeter's seminal 2017 decoupling result for nondegenerate hypersurfaces, several attempts have been made to extend the theory to degenerate hypersurfaces $M$. In this talk, we will discuss using surfaces derived from the local Taylor expansions of $M$ in order to obtain "approximate" decoupling results. By themselves, these approximate decouplings do not avail much. However, upon considerate iteration, for a specifically chosen $M$, they culminate in a decoupling partition of $M$ into caps small enough either as originally desired or otherwise genuinely nondegenerate at the local scale. A key feature that will be discussed is the notion of approximating a non-convex hypersurface $M$ by convex hypersurfaces at various scales. In this manner, contrary to initial intuition, non-trivial $\ell^2$ decoupling results will be obtained for $M$.
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===
+
===Jack Burkart===
  
Title: A triangular Hilbert transform with curvature, I & II.
+
Transcendental Julia Sets with Fractional Packing Dimension
  
Abstract: The triangular Hilbert is a two-dimensional bilinear singular
+
If f is an entire function, the Julia set of f is the set of all points such that f and its iterates locally do not form a normal family; nearby points have very different orbits under iteration by f. A topic of interest in complex dynamics is studying the fractal geometry of the Julia set.
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===
+
In this talk, we will discuss my thesis result where I construct non-polynomial (transcendental) entire functions whose Julia set has packing dimension strictly between (1,2). We will introduce various notions of dimension and basic objects in complex dynamics, and discuss a history of dimension results in complex dynamics. We will discuss some key aspects of the proof, which include a use of Whitney decompositions of domains as a tool to calculate the packing dimension, and some open questions I am thinking about.
  
Title:  Complex analytic problems on domains with good intrinsic geometry
+
===Giuseppe Negro===
  
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).
+
Stability of sharp Fourier restriction to spheres
  
===Hong Wang===
+
In dimension $d\in\{3, 4, 5, 6, 7\}$, we establish that the constant functions maximize the weighted $L^2(S^{d-1}) - L^4(R^d)$ Fourier extension estimate on the sphere, provided that the weight function is sufficiently regular and small, in a proper and effective sense which we will make precise. One of the main tools is an integration by parts identity, which generalizes the so-called "magic identity" of Foschi for the unweighted inequality with $d=3$, which is exactly the classical Stein-Tomas estimate.
  
Title: Improved decoupling for the parabola
+
Joint work with E.Carneiro and D.Oliveira e Silva.
  
Abstract: In 2014, Bourgain and Demeter proved the  $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. 
+
===Rajula Srivastava===
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===
+
Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups
  
Title: Smooth Nonnegative Interpolation
+
We discuss $L^p\to L^q$ estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups, sharp up to endpoints. The proof shall be reduced to estimates for standard oscillatory integrals of Carleson-Sj\"olin-H\"ormander type, relying on the maximal possible number of nonvanishing curvatures for a cone in the fibers of the associated canonical relation. We shall also discuss a new counterexample which shows the sharpness of one of the edges in the region of boundedness. Based on joint work with Joris Roos and Andreas Seeger.
  
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.
+
===Itamar Oliveira===
  
===Niclas Technau===
+
A new approach to the Fourier extension problem for the paraboloid
  
Title: Number theoretic applications of oscillatory integrals
+
An equivalent formulation of the Fourier Extension (F.E.) conjecture for a compact piece of the paraboloid states that the F.E. operator maps $ L^{2+\frac{2}{d}}([0,1]^{d}) $ to $L^{2+\frac{2}{d}+\varepsilon}(\mathbb{R}^{d+1}) $ for every $\varepsilon>0 $. It has been fully solved only for $ d=1 $ and there are many partial results in higher dimensions regarding the range of $ (p,q) $ for which $L^{p}([0,1]^{d}) $ is mapped to $ L^{q}(\mathbb{R}^{d+1}) $. One can reduce matters to proving that a model operator satisfies the same mapping properties, and we will show that the conjecture holds in higher dimensions for tensor functions, meaning for all $ g $  of the form $ g(x_{1},\ldots,x_{d})=g_{1}(x_{1})\cdot\ldots\cdot g_{d}(x_{d}) $. We will present this theorem as a proof of concept of a more general framework and set of techniques that can also address multilinear versions of this problem and get similar results. This is joint work with Camil Muscalu.
  
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.
+
===Changkeun Oh===
  
===Terence Harris===
+
Decoupling inequalities for quadratic forms and beyond
  
Title: Low dimensional pinned distance sets via spherical averages
+
In this talk, I will present some recent progress on decoupling inequalities for some translation- and dilation-invariant systems (TDI systems in short). In particular, I will emphasize decoupling inequalities for quadratic forms. If time permits, I will also discuss some interesting phenomenon related to Brascamp-Lieb inequalities that appears in the study of a cubic TDI system. Joint work with Shaoming Guo, Pavel Zorin-Kranich, and Ruixiang Zhang.
  
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.
+
=[[Previous_Analysis_seminars]]=
  
===Yuval Wigderson===
+
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars
  
Title: New perspectives on the uncertainty principle
+
=Extras=
 
 
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.
 
  
===Oscar Dominguez===
 
 
Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions
 
 
Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.
 
 
===Tamas Titkos===
 
 
Title: Isometries of Wasserstein spaces
 
 
Abstract: Due to its nice theoretical properties and an astonishing number of
 
applications via optimal transport problems, probably the most
 
intensively studied metric nowadays is the p-Wasserstein metric. Given
 
a complete and separable metric space $X$ and a real number $p\geq1$,
 
one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection
 
of Borel probability measures with finite $p$-th moment, endowed with a
 
distance which is calculated by means of transport plans \cite{5}.
 
 
The main aim of our research project is to reveal the structure of the
 
isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although
 
$\mathrm{Isom}(X)$ embeds naturally into
 
$\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding
 
turned out to be surjective in many cases (see e.g. [1]), these two
 
groups are not isomorphic in general. Kloeckner in [2] described
 
the isometry group of the quadratic Wasserstein space
 
$\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$
 
is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$
 
is extremely rich. Namely, it contains a large subgroup of wild behaving
 
isometries that distort the shape of measures. Following this line of
 
investigation, in \cite{3} we described
 
$\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and
 
$\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.
 
 
In this talk I will survey first some of the earlier results in the
 
subject, and then I will present the key results of [3]. If time
 
permits, I will also report on our most recent manuscript [4] in
 
which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading)
 
and D\'aniel Virosztek (IST Austria).
 
 
[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein
 
spaces: isometric rigidity in negative curvature}, International
 
Mathematics Research Notices, 2016 (5), 1368--1386.
 
 
[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean
 
spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di
 
Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.
 
 
[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of
 
Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373
 
(2020), 5855--5883.
 
 
[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of
 
Wasserstein spaces: The Hilbertian case}, submitted manuscript.
 
 
[5] C. Villani, \emph{Optimal Transport: Old and New,}
 
(Grundlehren der mathematischen Wissenschaften)
 
Springer, 2009.
 
 
===Shukun Wu===
 
 
Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator
 
 
Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem.
 
 
 
===Jonathan Hickman===
 
 
Title: Sobolev improving for averages over space curves
 
 
Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity.  Joint with D. Beltran, S. Guo and A. Seeger.
 
 
===Hanlong Fang===
 
 
Title: Canonical blow-ups of Grassmann manifolds
 
 
Abstract:  We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification.
 
 
===Bingyang Hu===
 
 
Title: Some structure theorems on general doubling measures.
 
 
Abstract: In this talk, we will first  several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely,  we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Muckenhoupt Ap weights and reverse Holder weights. This is a joint work with Tess Anderson.
 
 
===Krystal Taylor===
 
 
Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting
 
 
Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections.
 
It turns out that many nonlinear projection-type operators also have special geometry that allows us to build similar relationships between a set and its "projections", just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage points.  In particular, we will see that the multi-scale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projection-type operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set.
 
 
===Dominique Maldague===
 
 
Title: A new proof of decoupling for the parabola
 
 
Abstract: Decoupling has to do with measuring the size of functions with specialized Fourier support (in our case, in a neighborhood of the truncated parabola). Bourgain and Demeter resolved the l^2 decoupling conjecture in 2014, using ingredients like the multilinear Kakeya inequality, L^2 orthogonality, and induction-on-scales. I will present the ideas that go into a new proof of decoupling and make some comparison between the two approaches. This is related to recent joint work with Larry Guth and Hong Wang, as well as forthcoming joint work with Yuqiu Fu and Larry Guth.
 
 
===Diogo Oliveira e Silva===
 
 
Title: Global maximizers for spherical restriction
 
 
Abstract: We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler--Lagrange equation being smooth, a fact of independent interest which we discuss. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character $e^{i\xi\cdot\omega}$, for some $\xi$, thereby extending previous work of Christ & Shao (2012) to arbitrary dimensions $d\geq 2$ and general even exponents. This talk is based on results obtained with René Quilodrán.
 
 
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=Extras=
 
 
[[Blank Analysis Seminar Template]]
 
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Latest revision as of 13:06, 18 October 2021

The 2021-2022 Analysis Seminar will be organized by David Beltran and Andreas Seeger. Some of the talks will be in person (room Van Vleck B139) and some will be online. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).

Zoom links will be sent to those who have signed up for the Analysis Seminar List. 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 David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request.

If you'd like to suggest speakers for the fall semester please contact David and Andreas.

Analysis Seminar Schedule

date speaker institution title host(s)
September 21, VV B139 Dóminique Kemp UW-Madison Decoupling by way of approximation
September 28, VV B139 Jack Burkart UW-Madison Transcendental Julia Sets with Fractional Packing Dimension
October 5, Online Giuseppe Negro University of Birmingham Stability of sharp Fourier restriction to spheres
October 12, VV B139 Rajula Srivastava UW Madison Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups
October 19, Online Itamar Oliveira Cornell University A new approach to the Fourier extension problem for the paraboloid
October 26, VV B139 Changkeun Oh UW Madison Decoupling inequalities for quadratic forms and beyond
October 29, TBA Alexandru Ionescu (Colloquium) Princeton University Title
November 2, VV B139 Liding Yao UW Madison Title
November 9, VV B139 Lingxiao Zhang UW Madison Title
November 12, TBA Kasso Okoudjou (Colloquium) Tufts University Title
November 16, VV B139 Rahul Parhi UW Madison (EE) Title
November 30, VV B139 Alexei Poltoratski UW Madison Title
December 7 Person Institution Title
December 14 reserved Institution Title
February 1 Person Institution Title
February 8 Person Institution Title
February 15 Person Institution Title
February 22 Person Institution Title
March 1 Person Institution Title
March 8 Brian Street UW Madison Title
March 15: No Seminar Person Institution Title
March 23 Person Institution Title
March 30 Person Institution Title
April 5 Person Institution Title
April 12 Person Institution Title
April 19 Person Institution Title
April 22, Colloquium Detlef Müller University of Kiel Title
April 29 Person Institution Title
May 3 Person Institution Title
Date Person Institution Title

Abstracts

Dóminique Kemp

Decoupling by way of approximation

Since Bourgain and Demeter's seminal 2017 decoupling result for nondegenerate hypersurfaces, several attempts have been made to extend the theory to degenerate hypersurfaces $M$. In this talk, we will discuss using surfaces derived from the local Taylor expansions of $M$ in order to obtain "approximate" decoupling results. By themselves, these approximate decouplings do not avail much. However, upon considerate iteration, for a specifically chosen $M$, they culminate in a decoupling partition of $M$ into caps small enough either as originally desired or otherwise genuinely nondegenerate at the local scale. A key feature that will be discussed is the notion of approximating a non-convex hypersurface $M$ by convex hypersurfaces at various scales. In this manner, contrary to initial intuition, non-trivial $\ell^2$ decoupling results will be obtained for $M$.

Jack Burkart

Transcendental Julia Sets with Fractional Packing Dimension

If f is an entire function, the Julia set of f is the set of all points such that f and its iterates locally do not form a normal family; nearby points have very different orbits under iteration by f. A topic of interest in complex dynamics is studying the fractal geometry of the Julia set.

In this talk, we will discuss my thesis result where I construct non-polynomial (transcendental) entire functions whose Julia set has packing dimension strictly between (1,2). We will introduce various notions of dimension and basic objects in complex dynamics, and discuss a history of dimension results in complex dynamics. We will discuss some key aspects of the proof, which include a use of Whitney decompositions of domains as a tool to calculate the packing dimension, and some open questions I am thinking about.

Giuseppe Negro

Stability of sharp Fourier restriction to spheres

In dimension $d\in\{3, 4, 5, 6, 7\}$, we establish that the constant functions maximize the weighted $L^2(S^{d-1}) - L^4(R^d)$ Fourier extension estimate on the sphere, provided that the weight function is sufficiently regular and small, in a proper and effective sense which we will make precise. One of the main tools is an integration by parts identity, which generalizes the so-called "magic identity" of Foschi for the unweighted inequality with $d=3$, which is exactly the classical Stein-Tomas estimate.

Joint work with E.Carneiro and D.Oliveira e Silva.

Rajula Srivastava

Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups

We discuss $L^p\to L^q$ estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups, sharp up to endpoints. The proof shall be reduced to estimates for standard oscillatory integrals of Carleson-Sj\"olin-H\"ormander type, relying on the maximal possible number of nonvanishing curvatures for a cone in the fibers of the associated canonical relation. We shall also discuss a new counterexample which shows the sharpness of one of the edges in the region of boundedness. Based on joint work with Joris Roos and Andreas Seeger.

Itamar Oliveira

A new approach to the Fourier extension problem for the paraboloid

An equivalent formulation of the Fourier Extension (F.E.) conjecture for a compact piece of the paraboloid states that the F.E. operator maps $ L^{2+\frac{2}{d}}([0,1]^{d}) $ to $L^{2+\frac{2}{d}+\varepsilon}(\mathbb{R}^{d+1}) $ for every $\varepsilon>0 $. It has been fully solved only for $ d=1 $ and there are many partial results in higher dimensions regarding the range of $ (p,q) $ for which $L^{p}([0,1]^{d}) $ is mapped to $ L^{q}(\mathbb{R}^{d+1}) $. One can reduce matters to proving that a model operator satisfies the same mapping properties, and we will show that the conjecture holds in higher dimensions for tensor functions, meaning for all $ g $ of the form $ g(x_{1},\ldots,x_{d})=g_{1}(x_{1})\cdot\ldots\cdot g_{d}(x_{d}) $. We will present this theorem as a proof of concept of a more general framework and set of techniques that can also address multilinear versions of this problem and get similar results. This is joint work with Camil Muscalu.

Changkeun Oh

Decoupling inequalities for quadratic forms and beyond

In this talk, I will present some recent progress on decoupling inequalities for some translation- and dilation-invariant systems (TDI systems in short). In particular, I will emphasize decoupling inequalities for quadratic forms. If time permits, I will also discuss some interesting phenomenon related to Brascamp-Lieb inequalities that appears in the study of a cubic TDI system. Joint work with Shaoming Guo, Pavel Zorin-Kranich, and Ruixiang Zhang.

Previous_Analysis_seminars

https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars

Extras

Blank Analysis Seminar Template


Graduate Student Seminar:

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