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'''Analysis Seminar
 
'''
 
  
The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.
+
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).
  
If you wish to invite a speaker please contact  Brian at street(at)math
+
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.
  
===[[Previous Analysis seminars]]===
+
If you'd like to suggest speakers for the spring  semester please contact David and Andreas.
  
 
= Analysis Seminar Schedule =
 
= Analysis Seminar Schedule =
Line 16: Line 15:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|Sept 11
+
|September 21, VV B139
| Simon Marshall
+
| Dóminique Kemp
 +
| UW-Madison
 +
|[[#Dóminique Kemp  |  Decoupling by way of approximation ]]
 +
|
 +
|-
 +
|September 28, VV B139
 +
| Jack Burkart
 +
| UW-Madison
 +
|[[#Jack Burkart  |  Transcendental Julia Sets with Fractional Packing Dimension ]]
 +
|
 +
|-
 +
|October 5, Online
 +
| Giuseppe Negro
 +
| University of Birmingham
 +
|[[#Giuseppe Negro  |  Stability of sharp Fourier restriction to spheres ]]
 +
|
 +
|-
 +
|October 12, VV B139
 +
|Rajula Srivastava
 +
|UW Madison
 +
|[[#Rajula Srivastava  |  Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups ]]
 +
|
 +
|-
 +
|October 19, Online
 +
|Itamar Oliveira
 +
|Cornell University
 +
|[[#Itamar Oliveira  |  A new approach to the Fourier extension problem for the paraboloid ]]
 +
|
 +
|-
 +
|October 26, VV B139
 +
| Changkeun Oh
 
| UW Madison
 
| UW Madison
|[[#Simon Marshall | Integrals of eigenfunctions on hyperbolic manifolds ]]
+
|[[#Changkeun Oh |   Decoupling inequalities for quadratic forms and beyond ]]
 
|  
 
|  
 
|-
 
|-
|'''Wednesday, Sept 12'''
+
|October 29, Colloquium, Online
| Gunther Uhlmann 
+
| Alexandru Ionescu
| University of Washington
+
| Princeton University
| Distinguished Lecture Series
+
|[[#Alexandru Ionescu  |   Polynomial averages and pointwise ergodic theorems on nilpotent groups]]
| See colloquium website for location
 
 
|-
 
|-
|'''Friday, Sept 14'''
+
|November 2, VV B139
| Gunther Uhlmann 
+
| Liding Yao
| University of Washington
+
| UW Madison
| Distinguished Lecture Series
+
|[[#Liding Yao  |  An In-depth Look of Rychkov's Universal Extension Operators for Lipschitz Domains ]]
| See colloquium website for location
+
|  
 
|-
 
|-
|Sept 18
+
|November 9, VV B139
| Grad Student Seminar
+
| Lingxiao Zhang
 +
| UW Madison
 +
|[[#Lingxiao Zhang  |  Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition ]]
 
|  
 
|  
|
 
|
 
 
|-
 
|-
|Sept 25
+
|November 12, Colloquium, Online
| Grad Student Seminar
+
| Kasso Okoudjou
|
+
| Tufts University
|
+
|[[#Kasso Okoudjou  |  An exploration in analysis on fractals ]]
|
+
|-
 +
|November 16, VV B139
 +
| Rahul Parhi
 +
| UW Madison (EE)
 +
|[[#Rahul Parhi  |    On BV Spaces, Splines, and Neural Networks ]]
 +
| Betsy
 
|-
 
|-
|Oct 9
+
|November 30, VV B139
| Hong Wang
+
| Alexei Poltoratski
| MIT
+
| UW Madison
|[[#Hong Wang |   About Falconer distance problem in the plane ]]
+
|[[#Alexei Poltoratski | Pointwise convergence for the scattering data and non-linear Fourier transform. ]]
| Ruixiang
+
|  
 
|-
 
|-
|Oct 16
+
|December 7, Online
| Polona Durcik
+
| John Green
| Caltech
+
| The University of Edinburgh
|[[#Polona Durcik |   Singular Brascamp-Lieb inequalities and extended boxes in R^n ]]
+
|[[#John Green | Estimates for oscillatory integrals via sublevel set estimates ]]
| Joris
+
|  
 
|-
 
|-
|Oct 23
+
|December 14, VV B139
| Song-Ying Li
+
| Tao Mei
| UC Irvine
+
| Baylor University
|[[#Song-Ying Li Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold ]]
+
|[[#Tao Mei Fourier Multipliers on free groups ]]
| Xianghong
+
| Shaoming
 
|-
 
|-
|Oct 30
+
|Winter break
|Grad student seminar
 
|
 
 
|
 
|
 
|
 
|
 
|-
 
|-
|Nov 6
+
|February 8, VV B139
| Hanlong Fang
+
|Alexander  Nagel
 
| UW Madison
 
| UW Madison
|[[#Hanlong Fang A generalization of the theorem of Weil and Kodaira on prescribing residues ]]
+
|[[#Alex Nagel Global estimates for a class of kernels and multipliers with multiple homogeneities]]
| Brian
+
|  
 
|-
 
|-
||'''Monday, Nov. 12, B139'''
+
|February 15, Online
| Kyle Hambrook
+
| Sebastian Bechtel
| San Jose State University
+
| Institut de Mathématiques de Bordeaux
|[[#Kyle Hambrook |   Fourier Decay and Fourier Restriction for Fractal Measures on Curves ]]
+
|[[#Sebastian Bechtel | Square roots of elliptic systems on open sets]]
| Andreas
+
|  
 
|-
 
|-
|Nov 13
+
|Friday,  February 18,  Colloquium, VVB239
| Laurent Stolovitch
+
| Andreas Seeger
| Université de Nice - Sophia Antipolis
+
| UW Madison
|[[#Laurent Stolovitch Equivalence of Cauchy-Riemann manifolds and multisummability theory ]]
+
|[[#Andreas Seeger | Spherical maximal functions and fractal dimensions of dilation sets]]
|Xianghong
+
|  
 +
|-
 +
|February 22, VV B139
 +
|Tongou Yang
 +
|University of British Comlumbia
 +
|[[#linktoabstract Restricted projections along $C^2$ curves on the sphere ]]
 +
| Shaoming
 
|-
 
|-
|Nov 20
+
|Monday, February 28, 4:30 p.m.,  Online
| Grad Student Seminar
+
| Po Lam Yung
 +
| Australian National University
 +
|[[#Po Lam Yung  |  Revisiting an old argument for Vinogradov's Mean Value Theorem ]]
 
|  
 
|  
|[[#linktoabstract |   ]]
+
|-
 +
|March 8, VV B139
 +
| Brian Street
 +
| UW Madison
 +
|[[#Brian Street | Maximal Subellipticity ]]
 
|  
 
|  
 
|-
 
|-
|Nov 27
+
|March 15: No Seminar
| No Seminar
 
 
|  
 
|  
|[[#linktoabstract  |    ]]
 
 
|  
 
|  
|-
+
|
|Dec 4
 
| No Seminar
 
|[[#linktoabstract  |    ]]
 
 
|  
 
|  
 
|-
 
|-
|Jan 22
+
|March 22
| Brian Cook
+
| Laurent Stolovitch
| Kent
+
| University of Cote d'Azur
|[[#Brian Cook Equidistribution results for integral points on affine homogenous algebraic varieties ]]
+
|[[#linktoabstract Classification of reversible parabolic diffeomorphisms of
| Street
+
$(\mathbb{C}^2,0)$  and of flat CR-singularities of exceptional
 +
hyperbolic type ]]
 +
| Xianghong
 
|-
 
|-
|Jan 29
+
|March 29, VV B139
| No Seminar
+
|Betsy Stovall
 +
|UW Madison
 +
|[[#Betsy Stovall  |  On extremizing sequences for adjoint Fourier restriction to the sphere ]]
 
|  
 
|  
|[[#linktoabstract  |    ]]
 
|
 
|-
 
|Feb 5, '''B239'''
 
| Alexei Poltoratski
 
| Texas A&M
 
|[[#Alexei Poltoratski  |  Completeness of exponentials: Beurling-Malliavin and type problems ]]
 
| Denisov
 
|-
 
|'''Friday, Feb 8'''
 
| Aaron Naber
 
| Northwestern University
 
|[[#linktoabstract  |  A structure theory for spaces with lower Ricci curvature bounds ]]
 
| See colloquium website for location
 
 
|-
 
|-
|Feb 12
+
|April 5, Online
| Shaoming Guo
+
|Malabika Pramanik
| UW Madison
+
|University of British Columbia
|[[#Shaoming Guo | Polynomial Roth theorems in Salem sets   ]]
+
|[[#Malabika Pramanik Dimensionality and Patterns with Curvature]]
 
|  
 
|  
 
|-
 
|-
|'''Wed, Feb 13, B239'''
+
|April 12, VV B139
| Dean Baskin
+
| Hongki Jung
| TAMU
+
| IU Bloomington
|[[# Dean Baskin |   Radiation fields for wave  equations ]]
+
|[[#Hongki Jung | A small cap decoupling for the twisted cubic ]]
| Colloquium
+
| Shaoming
 
|-
 
|-
|'''Friday, Feb 15'''
+
|Friday, April 15, Colloquium, VV B239
| Lillian Pierce
+
| Bernhard Lamel
| Duke
+
| Texas A&M University at Qatar
|[[#Lillian Pierce Short character sums ]]
+
|[[#Bernhard Lamel Convergence and Divergence of Formal Power Series Maps ]]
| Colloquium
+
| Xianghong
 
|-
 
|-
|'''Monday, Feb 18, 3:30 p.m, B239.'''
+
|April 19, Online
| Daniel Tataru
+
| Carmelo Puliatti
| UC Berkeley
+
| Euskal Herriko Unibertsitatea
|[[#Daniel Tataru A Morawetz inequality for water waves ]]
+
|[[#Carmelo Puliatti Gradients of single layer potentials for elliptic operators 
| PDE Seminar
+
with coefficients of Dini mean oscillation-type ]]
 +
| David
 +
|
 +
|
 
|-
 
|-
|Feb 19
+
|April 25-26-27, Distinguished Lecture Series
| Wenjia Jing
+
|Larry Guth
|Tsinghua University
+
|MIT
|Periodic homogenization of Dirichlet problems in perforated domains: a unified proof
+
|[[#Larry Guth | Reflections on decoupling and Vinogradov's mean value problem. ]]
| PDE Seminar
 
 
|-
 
|-
|Feb 26
+
|April 25, 4:00 p.m., Lecture I, VV B239
| No Seminar
 
 
|
 
|
 
|
 
|
 +
|[[#linktoabstract  |  Introduction to decoupling and Vinogradov's mean value problem ]]
 
|-
 
|-
|Mar 5
+
|April 26, 4:00 p.m., Lecture II, Chamberlin 2241
| Loredana Lanzani
+
|
| Syracuse University
 
|[[#Loredana Lanzani  |  On regularity and irregularity of the Cauchy-Szegő projection in several complex variables ]]
 
| Xianghong
 
|-
 
|Mar 12
 
| Trevor Leslie
 
| UW Madison
 
|[[#Trevor Leslie  |  Energy Equality for the Navier-Stokes Equations at the First Possible Blowup Time ]]
 
 
|
 
|
 +
|[[#linktoabstract  |  Features of the proof of decoupling  ]]
 
|-
 
|-
|Mar 19
+
|April 27, 4:00 p.m., Lecture III, VV B239
|Spring Break!
 
|
 
 
|
 
|
 
|
 
|
|-
+
|[[#linktoabstract  |    Open problems ]]
|Mar 26
 
| No seminar
 
|
 
|[[#linktoabstract  |    ]]
 
 
|  
 
|  
 +
|
 
|-
 
|-
|Apr 2
+
|
| Stefan Steinerberger
+
|
| Yale
+
|
|[[#Stefan Steinerberger  |  Wasserstein Distance as a Tool in Analysis ]]
+
|
| Shaoming, Andreas
 
 
|-
 
|-
 
+
|Talks in the Fall semester 2022:
|Apr 9
 
| Franc Forstnerič
 
| Unversity of Ljubljana
 
|[[#Franc Forstnerič  |  Minimal surfaces by way of complex analysis ]]
 
| Xianghong, Andreas
 
 
|-
 
|-
|Apr 16
+
|September 20,  PDE and Analysis Seminar
| Andrew Zimmer
+
|Andrej Zlatoš
| Louisiana State University
+
|UCSD
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Xianghong
+
| Hung Tran
 
|-
 
|-
|Apr 23
+
|Friday, September 23, 4:00 p.m., Colloquium
| Person
+
|Pablo Shmerkin
| Institution
+
|University of British Columbia
 
|[[#linktoabstract  |  Title ]]
 
|[[#linktoabstract  |  Title ]]
| Sponsor
+
|Shaoming and Andreas
|-
 
|Apr 30
 
| Reserved
 
| Institution
 
|[[#linktoabstract  |  Title ]]
 
| Shaoming
 
 
|-
 
|-
 +
|September 24-25, RTG workshop in Harmonic Analysis
 +
|
 +
|
 +
|
 +
|Shaoming and Andreas
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===Simon Marshall===
+
===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 $M$.
  
''Integrals of eigenfunctions on hyperbolic manifolds''
+
===Jack Burkart===
  
Let X be a compact hyperbolic manifold, and let Y be a totally geodesic closed submanifold in X.  I will discuss the problem of bounding the integral of a Laplace eigenfunction on X over Y, as the eigenvalue tends to infinity.  I will present an upper bound for these integrals that is sharp on average, and briefly describe ongoing work with Farrell Brumley in which we attempt to produce eigenfunctions with very large periods.
+
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.
  
===Hong Wang===
+
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.
  
''About Falconer distance problem in the plane''
+
===Giuseppe Negro===
  
If E is a compact set of Hausdorff dimension greater than 5/4 on the plane, we prove that there is a point x\in E such that the set of distances between x and E has positive Lebesgue measure. Our result improves upon Wolff's theorem for dim E> 4/3. This is joint work with Larry Guth, Alex Iosevich and Yumeng Ou.
+
Stability of sharp Fourier restriction to spheres
  
===Polona Durcik===
+
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.
  
''Singular Brascamp-Lieb inequalities and extended boxes in R^n''
+
Joint work with E.Carneiro and D.Oliveira e Silva.
  
Brascamp-Lieb inequalities are L^p estimates for certain multilinear forms on functions on Euclidean spaces. In this talk we consider singular Brascamp-Lieb inequalities, which arise when one of the functions is replaced by a Calderon-Zygmund kernel. We focus on a family of multilinear forms in R^n with a certain cubical structure and discuss their connection to some patterns in positive density subsets in R^n. Based on joint works with V. Kovac and C. Thiele.
+
===Rajula Srivastava===
  
 +
Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups
  
===Song-Ying Li===
+
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.
  
''Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold''
+
===Itamar Oliveira===
  
In this talk, I will present my recent works with my collaborators on the lower bound and upper bounds estimates
+
A new approach to the Fourier extension problem for the paraboloid
for the first positive eigenvalues of Kohn Laplacian and sub-Laplacian on a strictly pseudoconvex pseudo-Hermitian CR manifold,
 
which include CR Lichnerowicz-Obata theorem for the lower and upper bounds for the first positive eigenvalue for the
 
Kohn Laplacian on strictly pseudoconvex hypersurfaces.
 
  
 +
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.
  
===Hanlong Fan===
+
===Changkeun Oh===
  
''A generalization of the theorem of Weil and Kodaira on prescribing residues''
+
Decoupling inequalities for quadratic forms and beyond
  
An old theorem of Weil and Kodaira says that: For a K\"ahler manifold X, there exists a closed meromorphic one-form with residue divisor D if and only if D is homologous to zero. In this talk, I will generalize Weil and Kodaira's criterion to non-K\"ahler manifolds.
+
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.
  
===Kyle Hambrook===
+
===Alexandru Ionescu===
  
''Fourier Decay and Fourier Restriction for Fractal Measures on Curves''
+
Polynomial averages and pointwise ergodic theorems on nilpotent groups
  
I will discuss my recent work on some problems concerning
+
I will talk about some recent work on pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two. Our proof is based on almost-orthogonality techniques that go far beyond Fourier transform tools, which are not available in the non-commutative nilpotent setting. In particular we develop what we call a nilpotent circle method}, which allows us to adapt some the ideas of the classical circle method to the setting of nilpotent groups.
Fourier decay and Fourier restriction for fractal measures on curves.
 
  
===Laurent Stolovitch===
+
===Liding Yao===
 +
 
 +
An In-depth Look of Rychkov's Universal Extension Operators for Lipschitz Domains
 +
 
 +
Given a bounded Lipschitz domain $\Omega\subset\mathbb{R}^n$, Rychkov showed that there is a linear extension operator $\mathcal E$ for $\Omega$ which is bounded in Besov and Triebel-Lizorkin spaces. We introduce a class of operators that generalize $\mathcal E$ which are more versatile for applications. We also derive some quantitative blow-up estimates of the extended function and all its derivatives in $\overline{\Omega}^c$ up to boundary. This is a joint work with Ziming Shi.
 +
 
 +
===Lingxiao Zhang===
 +
 
 +
Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition
 +
 
 +
We study operators of the form
 +
$Tf(x)= \psi(x) \int f(\gamma_t(x))K(t)\,dt$
 +
where $\gamma_t(x)$ is a real analytic function of $(t,x)$ mapping from a neighborhood of $(0,0)$ in $\mathbb{R}^N \times \mathbb{R}^n$ into $\mathbb{R}^n$ satisfying $\gamma_0(x)\equiv x$, $\psi(x) \in C_c^\infty(\mathbb{R}^n)$, and $K(t)$ is a `multi-parameter singular kernel' with compact support in $\mathbb{R}^N$; for example when $K(t)$ is a product singular kernel. The celebrated work of Christ, Nagel, Stein, and Wainger studied such operators with smooth $\gamma_t(x)$, in the single-parameter case when $K(t)$ is a Calder\'on-Zygmund kernel. Street and Stein generalized their work to the multi-parameter case, and gave sufficient conditions for the $L^p$-boundedness of such operators. This paper shows that when $\gamma_t(x)$ is real analytic, the sufficient conditions of Street and Stein are also necessary for the $L^p$-boundedness of $T$, for all such kernels $K$.
 +
 
 +
===Kasso Okoudjou===
 +
 
 +
An exploration in analysis on fractals
  
''Equivalence of Cauchy-Riemann manifolds and multisummability theory''
+
Analysis on fractal sets such as the Sierpinski gasket is based on the spectral analysis of a corresponding Laplace operator. In the first part of the talk, I will describe a class of fractals and the analytical tools that they support. In the second part of the talk, I will consider fractal analogs of topics from classical analysis, including the Heisenberg uncertainty principle, the spectral theory of Schrödinger operators, and the theory of orthogonal polynomials.
  
We apply the multisummability theory from Dynamical Systems to CR-geometry. As the main result, we show that two real-analytic hypersurfaces in $\mathbb C^2$ are formally equivalent, if and only if they are $C^\infty$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic Levi-nonflat hypersurfaces in $\mathbb C^2$  are algebraic (and in particular convergent). This is a joint work with I. Kossovskiy and B. Lamel.
+
===Rahul Parhi===
  
 +
On BV Spaces, Splines, and Neural Networks
  
===Brian Cook===
+
Many problems in science and engineering can be phrased as the problem
 +
of reconstructing a function from a finite number of possibly noisy
 +
measurements. The reconstruction problem is inherently ill-posed when
 +
the allowable functions belong to an infinite set. Classical techniques
 +
to solve this problem assume, a priori, that the underlying function has
 +
some kind of regularity, typically Sobolev, Besov, or BV regularity. The
 +
field of applied harmonic analysis is interested in studying efficient
 +
decompositions and representations for functions with certain
 +
regularity. Common representation systems are based on splines and
 +
wavelets. These are well understood mathematically and have been
 +
successfully applied in a variety of signal processing and statistical
 +
tasks. Neural networks are another type of representation system that is
 +
useful in practice, but poorly understood mathematically.
  
''Equidistribution results for integral points on affine homogenous algebraic varieties''
+
In this talk, I will discuss my research which aims to rectify this
 +
issue by understanding the regularity properties of neural networks in a
 +
similar vein to classical methods based on splines and wavelets. In
 +
particular, we will show that neural networks are optimal solutions to
 +
variational problems over BV-type function spaces defined via the Radon
 +
transform. These spaces are non-reflexive Banach spaces, generally
 +
distinct from classical spaces studied in analysis. However, in the
 +
univariate setting, neural networks reduce to splines and these function
 +
spaces reduce to classical univariate BV spaces. If time permits, I will
 +
also discuss approximation properties of these spaces, showing that they
 +
are, in some sense, "small" compared to classical multivariate spaces
 +
such as Sobolev or Besov spaces.
  
Let Q be a homogenous integral polynomial of degree at least two. We consider certain results and questions concerning the distribution of the integral points on the level sets of Q.
+
This is joint work with Robert Nowak.
  
 
===Alexei Poltoratski===
 
===Alexei Poltoratski===
  
''Completeness of exponentials: Beurling-Malliavin and type problems''
+
Title: Pointwise convergence for the scattering data and non-linear Fourier transform.
 +
 
 +
Abstract: This talk is about applications of complex and harmonic analysis in spectral and scattering theory
 +
for differential operators. The scattering transform for the Dirac system of differential equations
 +
can be viewed as the non-linear version of the classical Fourier transform. This connection raises many natural
 +
problems on extensions of classical results of Fourier analysis to non-linear settings. In this talk
 +
I will discuss one of such problems, an extension of Carleson's theorem on pointwise convergence of Fourier series to the non-linear case.
 +
 
 +
===John Green===
 +
 
 +
Estimates for oscillatory integrals via sublevel set estimates.
 +
 
 +
In many situations, oscillatory integral estimates are known to imply sublevel set estimates in a stable manner. Reversing this implication is much more difficult, but understanding when this is true is helpful for understanding scalar oscillatory integral estimates. We shall motivate a line of investigation in which we seek to reverse the implication in the presence of a qualitative structural assumption. After considering some one-dimensional results, we turn to the setting of convex functions in higher dimensions.
 +
 
 +
===Tao Mei===
 +
 
 +
Fourier Multipliers on free groups.
 +
 
 +
In this introductory talk,  I will try to explain what is the noncommutative Lp spaces associated with the free groups, and what are the to be answered questions on  the corresponding Fourier multiplier operators.  At the end, I will explain a recent work on an analogue of Mikhlin’s Lp Fourier multiplier theory on free groups (joint with Eric Ricard and Quanhua Xu).
 +
 
 +
===Alex Nagel===
 +
 
 +
Global estimates for a class of kernels and multipliers with multiple homogeneities
 +
 
 +
In joint work with Fulvio Ricci we obtain global estimates for a class of kernels and multipliers which contain homogeneous Calderon-Zygmund operators for several different homogeneities. This is an extension of earlier work with Ricci, Stein, and Wainger on the local theory.
 +
 
 +
===Sebastian Bechtel===
 +
 
 +
Square roots of elliptic systems on open sets
 +
 
 +
In my talk, we will consider elliptic systems in divergence form with measurable and elliptic complex coefficients on possibly unbounded open sets which are subject to mixed boundary conditions. First, I will present and discuss minimal geometric conditions under which Kato’s square root problem can be solved. In particular, I will present an argument that allows to work on a set that is not supposed to satisfy the interior thickness condition.
 +
Afterwards, we will investigate the question for which integrability parameters p the square root isomorphism $W^{1,2} \to L^2$ extrapolates to an isomorphism $W^{1,p} \to L^p$. We focus on the case $p>2$. I will introduce a critical number that describes the range in which $L$ (compatibly) acts as an isomorphism $W^{1,p} \to W^{-1,p}$. We will then see that this critical number also yields an optimal range in which the square root extrapolates to a $p$-isomorphism, even in the case of mixed boundary conditions.
 +
 
 +
===Tongou Yang===
 +
 
 +
Restricted projections along $C^2$ curves on the sphere
 +
 
 +
Given a $C^2$ closed curve $\gamma(\theta)$ lying on the sphere
 +
$\mathbb S^2$ and a Borel set $A\subseteq \mathbb R^3$. Consider the
 +
projections $P_\theta(A)$ of $A$ into straight lines in the directions
 +
$\gamma(\theta)$. We prove that if $\gamma$ satisfies the torsion
 +
condition: $\det(\gamma,\gamma',\gamma")(\theta)\neq 0$ for any $\theta$,
 +
then for almost every $\theta$, the Hausdorff dimension of $P_\theta(A)$ is
 +
equal to $\min\{1,\dim_H(A)\}$. This solves a conjecture of Fässler and
 +
Orponen. One key feature of our argument is a result of Marcus-Tardos in
 +
topological graph theory. This is a joint work with Malabika Pramanik, Orit Raz and Josh Zahl.
 +
 
 +
===Po Lam Yung===
 +
 
 +
Revisiting an old argument for Vinogradov's Mean Value Theorem
  
This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin problem was solved in the early 1960s and I will present its classical solution along with modern generalizations and applications. I will then discuss history and recent progress in the type problem, which stood open for more than 70 years.
+
We will examine an old argument for the Vinogradov's Mean Value Theorem due to Karatsuba, and interpret it in the language of Fourier decoupling. This is ongoing work in progress with Brian Cook, Kevin Hughes, Zane Kun Li, Akshat Mudgal and Olivier Robert.
  
 +
===Brian Street===
  
===Shaoming Guo===
+
Maximal Subellipticity
  
''Polynomial Roth theorems in Salem sets''
+
The theory of elliptic PDEs stands apart from many other areas of PDEs because sharp results are known for very general linear and fully nonlinear elliptic PDEs.  Many of the classical techniques from harmonic analysis were first developed to prove these sharp results; and the study of elliptic PDEs leans heavily on the Fourier transform and Riemannian geometry. Starting with work of Hörmander, Kohn, Folland, Stein, and Rothschild in the 60s and 70s, a far-reaching generalization of ellipticity was introduced:  now known as maximal subellipticity or maximal hypoellipticity.  In the intervening years, many authors have adapted results from elliptic PDEs to various special cases of maximally subelliptic PDEs. Where elliptic operators are connected to Riemannian geometry, maximally subelliptic operators are connected to sub-Riemannian geometry.  The Fourier transform is no longer a central tool but can be replaced with more modern tools from harmonic analysis.
  
Let P(t) be a polynomial of one real variable. I will report a result on searching for patterns of the form (x, x+t, x+P(t)) within Salem sets, whose Hausdorff dimension is sufficiently close to one. Joint work with Fraser and Pramanik.
+
===Laurent Stolovitch===
  
 +
Classification of reversible parabolic diffeomorphisms of $(\mathbb{C}^2,0)$  and of flat CR-singularities of exceptional hyperbolic type
  
 +
The aim of this joint work with Martin Klimes is twofold:
  
 +
First we study holomorphic germs of parabolic diffeomorphisms of $(\mathbb{C}^2,0)$ that are reversed by a holomorphic reflection and posses an analytic first integral with non-degenerate critical point at the origin. We find a canonical formal normal form and provide a complete analytic classification (in formal generic cases) in terms of a collection of functional invariants.
  
===Dean Baskin===
+
Related to it, we solve the problem of both formal and analytic classification of germs of real analytic surfaces in $\mathbb{C}^2$ with non-degenerate CR singularities of exceptional hyperbolic type, under the assumption that the surface is holomorphically flat, i.e. that it can be locally holomorphically embedded in a real hypersurface of $\mathbb{C}^2$.
  
''Radiation fields for wave equations''
+
===Betsy Stovall===
  
Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.
+
On extremizing sequences for adjoint Fourier restriction to the sphere
  
===Lillian Pierce===
+
In this talk we will provide a soft answer to the question, "What properties must a function $f$ obeying $\|Ef\|_q \geq C \|f\|_p$ have?," where $E$ denotes the spherical extension operator.  We will use our answer (called a linear profile decomposition) to establish new results about the existence of extremizers (functions obeying $\|Ef\|_q = \|E\|\|f\|_p$) for $E$.  This is joint work with Taryn C. Flock.
  
''Short character sums''
+
===Malabika Pramanik===
  
A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.
+
https://people.math.wisc.edu/~seeger/seminar/Malabika-Analysis-Seminar-2022-Title-Abstract.pdf
  
===Loredana Lanzani===
+
===Hongki Jung===
  
''On regularity and irregularity of the Cauchy-Szegő projection in several complex variables''
+
A small cap decoupling for the twisted cubic
  
This talk is a survey of my latest, and now final, collaboration with Eli Stein.
+
===Bernhard Lamel===
  
It is known that for bounded domains $D$ in $\mathbb C^n$ that are of class $C^2$ and are strongly pseudo-convex, the Cauchy-Szegő projection is bounded in $L^p(\text{b}D, d\Sigma)$ for $1<p<\infty$. (Here $d\Sigma$ is induced Lebesgue measure.)  We show, using appropriate worm domains, that this fails for any $p\neq 2$, when we assume that the domain in question is only weakly pseudo-convex. Our starting point are the ideas of Kiselman-Barrett introduced more than 30 years ago in the analysis of the Bergman projection. However the study of the Cauchy-Szegő projection raises a number of new issues and obstacles that need to be overcome. We will also compare these results to the analogous problem for the Cauchy-Leray integral, where however the relevant counter-example is of much simpler nature.
+
Convergence and Divergence of Formal Power Series Maps
  
===Trevor Leslie===
+
Consider two real-analytic hypersurfaces (i.e. defined by convergent power series) in complex spaces. A formal holomorphic map is said to take one into the other if the composition of the power series defining the target with the map (which is just another formal power series) is a (formal) multiple of the defining power series of the source. In this talk, we are going to be interested in conditions for formal holomorphic maps to necessarily be convergent. Now, a formal holomorphic map taking the real line to itself is just a formal power series with real coefficients; this example also gives rise to real hypersurfaces in higher dimensional complex spaces having divergent formal self-maps. On the other hand, a formal map taking the unit sphere in higher dimensional complex space to itself is necessarily a rational map with poles outside of the sphere, in particular, the formal power series defining it converges. The convergence theory for formal self-maps of real hypersurfaces has been developed in the late 1990s and early 2000s. For formal embeddings, “ideal" conditions had been long conjectured. I’m going to give an introduction to this problem and talk about some joint work from 2018 with Nordine Mir giving a basically complete answer to the question when a formal map taking a real-analytic hypersurface in complex space into another one is necessarily convergent.
  
''Energy Equality for the Navier-Stokes Equations at the First Possible Blowup Time''
+
===Carmelo Puliatti===
  
In this talk, we discuss the problem of energy equality for strong solutions of the Navier-Stokes Equations (NSE) at the first time where such solutions may lose regularity.  Our approach is motivated by a famous theorem of Caffarelli, Kohn, and Nirenberg, which states that the set of singular points associated to a suitable weak solution of the NSE has parabolic Hausdorff dimension of at most 1.  In particular, we furnish sufficient conditions for energy equality which depend on the dimension of the singularity set in addition to time and space integrability assumptions; in doing so we improve upon the classical results when attention is restricted to the first blowup time.  When our method is inconclusive, we are able to quantify the possible failure of energy equality in terms of the lower local dimension and the ''concentration dimension'' of a certain measure associated to the solution.  The work described is joint with Roman Shvydkoy (UIC).
+
Gradients of single layer potentials for elliptic operators 
 +
with coefficients of Dini mean oscillation-type
  
===Stefan Steinerberger===
+
We consider a uniformly elliptic operator $L_A$ in divergence form 
 +
associated with a matrix A with real, bounded, and possibly 
 +
non-symmetric coefficients. If a proper $L^1$-mean oscillation of the 
 +
coefficients of A satisfies suitable Dini-type assumptions, we prove 
 +
the following: if \mu is a compactly supported Radon measure in 
 +
$\mathbb{R}^{n+1}, n >= 2$,  the $L^2(\mu)$-operator norm of the gradient of the 
 +
single layer potential $T_\mu$ associated with $L_A$ is comparable to the 
 +
$L^2$-norm of the n-dimensional Riesz transform $R_\mu$, modulo an 
 +
additive constant.
 +
This makes possible to obtain direct generalizations of some deep 
 +
geometric results, initially proved for the Riesz transform, which 
 +
were recently extended to $T_\mu$ under a H\"older continuity assumption 
 +
on the coefficients of the matrix $A$.
  
''Wasserstein Distance as a Tool in Analysis''
+
This is a joint work with Alejandro Molero, Mihalis Mourgoglou, and 
 +
Xavier Tolsa.
  
Wasserstein Distance is a way of measuring the distance between two probability distributions (minimizing it is a main problem in Optimal Transport). We will give a gentle Introduction into what it means and then use it to prove (1) a completely elementary but possibly new and quite curious inequality for real-valued functions and (2) a statement along the following lines: linear combinations of eigenfunctions of elliptic operators corresponding to high frequencies oscillate a lot and vanish on a large set of co-dimension 1 (this is already interesting for trigonometric polynomials on the 2-torus, sums of finitely many sines and cosines, whose sum has to vanish on long lines) and (3) some statements in Basic Analytic Number Theory that drop out for free as a byproduct.
+
===Larry Guth===
  
===Franc Forstnerič===
+
Series title: Reflections on decoupling and Vinogradov's mean value problem.
  
``Minimal surfaces by way of complex analysis''
+
Series abstract: Decoupling is a recent development in Fourier analysis that has solved several longstanding problems.  The goal of the lectures is to describe this development to a general mathematical audience. We will focus on one particular application of decoupling: Vinogradov's mean value problem from analytic number theory.  This problem is about the number of solutions of a certain system of diophantine equations.  It was raised in the 1930s and resolved in the last decade. We will give some context about this problem, but the main goal of the lectures is to explore the ideas that go into the proof.  The method of decoupling came as a big surprise to me, and I think to other people working in the field. The main idea in the proof of decoupling is to combine estimates from many different scales.  We will describe this process and reflect on why it is helpful.
  
After a brief historical introduction, I will present some recent developments in the theory of minimal surfaces in Euclidean spaces which have been obtained by complex analytic methods. The emphasis will be on results pertaining to the global theory of minimal surfaces including Runge and Mergelyan approximation, the conformal Calabi-Yau problem, properly immersed and embedded minimal surfaces, and a new result on the Gauss map of minimal surfaces.
+
 
 +
Lecture 1: Introduction to decoupling and Vinogradov's mean value problem.
 +
Abstract: In this lecture, we introduce Vinogradov's problem and give an overview of the proof.
 +
 
 +
Lecture 2: Features of the proof of decoupling.
 +
Abstract: In this lecture, we look more closely at some features of the proof of decoupling.  The first feature we examine is the exact form of writing the inequality, which is especially suited for doing induction and connecting information from different scales. The second feature we examine is called the wave packet decomposition.  This structure has roots in quantum physics and in information theory.
 +
 
 +
Lecture 3: Open problems.
 +
Abstract: In this lecture, we discuss some open problems in number theory that look superficially similar to Vinogradov mean value conjecture, such as Hardy and Littlewood's Hypothesis K*.  In this lecture, we probe the limitations of decoupling by exploring why the techniques from the first two lectures don't work on these open problems.  Hopefully this will give a sense of some of the issues and difficulties involved in these problems.
 +
 
 +
=[[Previous_Analysis_seminars]]=
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 +
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars
  
 
=Extras=
 
=Extras=
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[[Blank Analysis Seminar Template]]
 
[[Blank Analysis Seminar Template]]
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Graduate Student Seminar:
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https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html

Latest revision as of 00:07, 28 April 2022

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 spring 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, Colloquium, Online Alexandru Ionescu Princeton University Polynomial averages and pointwise ergodic theorems on nilpotent groups
November 2, VV B139 Liding Yao UW Madison An In-depth Look of Rychkov's Universal Extension Operators for Lipschitz Domains
November 9, VV B139 Lingxiao Zhang UW Madison Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition
November 12, Colloquium, Online Kasso Okoudjou Tufts University An exploration in analysis on fractals
November 16, VV B139 Rahul Parhi UW Madison (EE) On BV Spaces, Splines, and Neural Networks Betsy
November 30, VV B139 Alexei Poltoratski UW Madison Pointwise convergence for the scattering data and non-linear Fourier transform.
December 7, Online John Green The University of Edinburgh Estimates for oscillatory integrals via sublevel set estimates
December 14, VV B139 Tao Mei Baylor University Fourier Multipliers on free groups Shaoming
Winter break
February 8, VV B139 Alexander Nagel UW Madison Global estimates for a class of kernels and multipliers with multiple homogeneities
February 15, Online Sebastian Bechtel Institut de Mathématiques de Bordeaux Square roots of elliptic systems on open sets
Friday, February 18, Colloquium, VVB239 Andreas Seeger UW Madison Spherical maximal functions and fractal dimensions of dilation sets
February 22, VV B139 Tongou Yang University of British Comlumbia Restricted projections along $C^2$ curves on the sphere Shaoming
Monday, February 28, 4:30 p.m., Online Po Lam Yung Australian National University Revisiting an old argument for Vinogradov's Mean Value Theorem
March 8, VV B139 Brian Street UW Madison Maximal Subellipticity
March 15: No Seminar
March 22 Laurent Stolovitch University of Cote d'Azur Classification of reversible parabolic diffeomorphisms of

$(\mathbb{C}^2,0)$ and of flat CR-singularities of exceptional hyperbolic type

Xianghong
March 29, VV B139 Betsy Stovall UW Madison On extremizing sequences for adjoint Fourier restriction to the sphere
April 5, Online Malabika Pramanik University of British Columbia Dimensionality and Patterns with Curvature
April 12, VV B139 Hongki Jung IU Bloomington A small cap decoupling for the twisted cubic Shaoming
Friday, April 15, Colloquium, VV B239 Bernhard Lamel Texas A&M University at Qatar Convergence and Divergence of Formal Power Series Maps Xianghong
April 19, Online Carmelo Puliatti Euskal Herriko Unibertsitatea Gradients of single layer potentials for elliptic operators

with coefficients of Dini mean oscillation-type

David
April 25-26-27, Distinguished Lecture Series Larry Guth MIT Reflections on decoupling and Vinogradov's mean value problem.
April 25, 4:00 p.m., Lecture I, VV B239 Introduction to decoupling and Vinogradov's mean value problem
April 26, 4:00 p.m., Lecture II, Chamberlin 2241 Features of the proof of decoupling
April 27, 4:00 p.m., Lecture III, VV B239 Open problems
Talks in the Fall semester 2022:
September 20, PDE and Analysis Seminar Andrej Zlatoš UCSD Title Hung Tran
Friday, September 23, 4:00 p.m., Colloquium Pablo Shmerkin University of British Columbia Title Shaoming and Andreas
September 24-25, RTG workshop in Harmonic Analysis Shaoming and Andreas

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.

Alexandru Ionescu

Polynomial averages and pointwise ergodic theorems on nilpotent groups

I will talk about some recent work on pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two. Our proof is based on almost-orthogonality techniques that go far beyond Fourier transform tools, which are not available in the non-commutative nilpotent setting. In particular we develop what we call a nilpotent circle method}, which allows us to adapt some the ideas of the classical circle method to the setting of nilpotent groups.

Liding Yao

An In-depth Look of Rychkov's Universal Extension Operators for Lipschitz Domains

Given a bounded Lipschitz domain $\Omega\subset\mathbb{R}^n$, Rychkov showed that there is a linear extension operator $\mathcal E$ for $\Omega$ which is bounded in Besov and Triebel-Lizorkin spaces. We introduce a class of operators that generalize $\mathcal E$ which are more versatile for applications. We also derive some quantitative blow-up estimates of the extended function and all its derivatives in $\overline{\Omega}^c$ up to boundary. This is a joint work with Ziming Shi.

Lingxiao Zhang

Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition

We study operators of the form $Tf(x)= \psi(x) \int f(\gamma_t(x))K(t)\,dt$ where $\gamma_t(x)$ is a real analytic function of $(t,x)$ mapping from a neighborhood of $(0,0)$ in $\mathbb{R}^N \times \mathbb{R}^n$ into $\mathbb{R}^n$ satisfying $\gamma_0(x)\equiv x$, $\psi(x) \in C_c^\infty(\mathbb{R}^n)$, and $K(t)$ is a `multi-parameter singular kernel' with compact support in $\mathbb{R}^N$; for example when $K(t)$ is a product singular kernel. The celebrated work of Christ, Nagel, Stein, and Wainger studied such operators with smooth $\gamma_t(x)$, in the single-parameter case when $K(t)$ is a Calder\'on-Zygmund kernel. Street and Stein generalized their work to the multi-parameter case, and gave sufficient conditions for the $L^p$-boundedness of such operators. This paper shows that when $\gamma_t(x)$ is real analytic, the sufficient conditions of Street and Stein are also necessary for the $L^p$-boundedness of $T$, for all such kernels $K$.

Kasso Okoudjou

An exploration in analysis on fractals

Analysis on fractal sets such as the Sierpinski gasket is based on the spectral analysis of a corresponding Laplace operator. In the first part of the talk, I will describe a class of fractals and the analytical tools that they support. In the second part of the talk, I will consider fractal analogs of topics from classical analysis, including the Heisenberg uncertainty principle, the spectral theory of Schrödinger operators, and the theory of orthogonal polynomials.

Rahul Parhi

On BV Spaces, Splines, and Neural Networks

Many problems in science and engineering can be phrased as the problem of reconstructing a function from a finite number of possibly noisy measurements. The reconstruction problem is inherently ill-posed when the allowable functions belong to an infinite set. Classical techniques to solve this problem assume, a priori, that the underlying function has some kind of regularity, typically Sobolev, Besov, or BV regularity. The field of applied harmonic analysis is interested in studying efficient decompositions and representations for functions with certain regularity. Common representation systems are based on splines and wavelets. These are well understood mathematically and have been successfully applied in a variety of signal processing and statistical tasks. Neural networks are another type of representation system that is useful in practice, but poorly understood mathematically.

In this talk, I will discuss my research which aims to rectify this issue by understanding the regularity properties of neural networks in a similar vein to classical methods based on splines and wavelets. In particular, we will show that neural networks are optimal solutions to variational problems over BV-type function spaces defined via the Radon transform. These spaces are non-reflexive Banach spaces, generally distinct from classical spaces studied in analysis. However, in the univariate setting, neural networks reduce to splines and these function spaces reduce to classical univariate BV spaces. If time permits, I will also discuss approximation properties of these spaces, showing that they are, in some sense, "small" compared to classical multivariate spaces such as Sobolev or Besov spaces.

This is joint work with Robert Nowak.

Alexei Poltoratski

Title: Pointwise convergence for the scattering data and non-linear Fourier transform.

Abstract: This talk is about applications of complex and harmonic analysis in spectral and scattering theory for differential operators. The scattering transform for the Dirac system of differential equations can be viewed as the non-linear version of the classical Fourier transform. This connection raises many natural problems on extensions of classical results of Fourier analysis to non-linear settings. In this talk I will discuss one of such problems, an extension of Carleson's theorem on pointwise convergence of Fourier series to the non-linear case.

John Green

Estimates for oscillatory integrals via sublevel set estimates.

In many situations, oscillatory integral estimates are known to imply sublevel set estimates in a stable manner. Reversing this implication is much more difficult, but understanding when this is true is helpful for understanding scalar oscillatory integral estimates. We shall motivate a line of investigation in which we seek to reverse the implication in the presence of a qualitative structural assumption. After considering some one-dimensional results, we turn to the setting of convex functions in higher dimensions.

Tao Mei

Fourier Multipliers on free groups.

In this introductory talk, I will try to explain what is the noncommutative Lp spaces associated with the free groups, and what are the to be answered questions on the corresponding Fourier multiplier operators. At the end, I will explain a recent work on an analogue of Mikhlin’s Lp Fourier multiplier theory on free groups (joint with Eric Ricard and Quanhua Xu).

Alex Nagel

Global estimates for a class of kernels and multipliers with multiple homogeneities

In joint work with Fulvio Ricci we obtain global estimates for a class of kernels and multipliers which contain homogeneous Calderon-Zygmund operators for several different homogeneities. This is an extension of earlier work with Ricci, Stein, and Wainger on the local theory.

Sebastian Bechtel

Square roots of elliptic systems on open sets

In my talk, we will consider elliptic systems in divergence form with measurable and elliptic complex coefficients on possibly unbounded open sets which are subject to mixed boundary conditions. First, I will present and discuss minimal geometric conditions under which Kato’s square root problem can be solved. In particular, I will present an argument that allows to work on a set that is not supposed to satisfy the interior thickness condition. Afterwards, we will investigate the question for which integrability parameters p the square root isomorphism $W^{1,2} \to L^2$ extrapolates to an isomorphism $W^{1,p} \to L^p$. We focus on the case $p>2$. I will introduce a critical number that describes the range in which $L$ (compatibly) acts as an isomorphism $W^{1,p} \to W^{-1,p}$. We will then see that this critical number also yields an optimal range in which the square root extrapolates to a $p$-isomorphism, even in the case of mixed boundary conditions.

Tongou Yang

Restricted projections along $C^2$ curves on the sphere

Given a $C^2$ closed curve $\gamma(\theta)$ lying on the sphere $\mathbb S^2$ and a Borel set $A\subseteq \mathbb R^3$. Consider the projections $P_\theta(A)$ of $A$ into straight lines in the directions $\gamma(\theta)$. We prove that if $\gamma$ satisfies the torsion condition: $\det(\gamma,\gamma',\gamma")(\theta)\neq 0$ for any $\theta$, then for almost every $\theta$, the Hausdorff dimension of $P_\theta(A)$ is equal to $\min\{1,\dim_H(A)\}$. This solves a conjecture of Fässler and Orponen. One key feature of our argument is a result of Marcus-Tardos in topological graph theory. This is a joint work with Malabika Pramanik, Orit Raz and Josh Zahl.

Po Lam Yung

Revisiting an old argument for Vinogradov's Mean Value Theorem

We will examine an old argument for the Vinogradov's Mean Value Theorem due to Karatsuba, and interpret it in the language of Fourier decoupling. This is ongoing work in progress with Brian Cook, Kevin Hughes, Zane Kun Li, Akshat Mudgal and Olivier Robert.

Brian Street

Maximal Subellipticity

The theory of elliptic PDEs stands apart from many other areas of PDEs because sharp results are known for very general linear and fully nonlinear elliptic PDEs. Many of the classical techniques from harmonic analysis were first developed to prove these sharp results; and the study of elliptic PDEs leans heavily on the Fourier transform and Riemannian geometry. Starting with work of Hörmander, Kohn, Folland, Stein, and Rothschild in the 60s and 70s, a far-reaching generalization of ellipticity was introduced: now known as maximal subellipticity or maximal hypoellipticity. In the intervening years, many authors have adapted results from elliptic PDEs to various special cases of maximally subelliptic PDEs. Where elliptic operators are connected to Riemannian geometry, maximally subelliptic operators are connected to sub-Riemannian geometry. The Fourier transform is no longer a central tool but can be replaced with more modern tools from harmonic analysis.

Laurent Stolovitch

Classification of reversible parabolic diffeomorphisms of $(\mathbb{C}^2,0)$ and of flat CR-singularities of exceptional hyperbolic type

The aim of this joint work with Martin Klimes is twofold:

First we study holomorphic germs of parabolic diffeomorphisms of $(\mathbb{C}^2,0)$ that are reversed by a holomorphic reflection and posses an analytic first integral with non-degenerate critical point at the origin. We find a canonical formal normal form and provide a complete analytic classification (in formal generic cases) in terms of a collection of functional invariants.

Related to it, we solve the problem of both formal and analytic classification of germs of real analytic surfaces in $\mathbb{C}^2$ with non-degenerate CR singularities of exceptional hyperbolic type, under the assumption that the surface is holomorphically flat, i.e. that it can be locally holomorphically embedded in a real hypersurface of $\mathbb{C}^2$.

Betsy Stovall

On extremizing sequences for adjoint Fourier restriction to the sphere

In this talk we will provide a soft answer to the question, "What properties must a function $f$ obeying $\|Ef\|_q \geq C \|f\|_p$ have?," where $E$ denotes the spherical extension operator. We will use our answer (called a linear profile decomposition) to establish new results about the existence of extremizers (functions obeying $\|Ef\|_q = \|E\|\|f\|_p$) for $E$. This is joint work with Taryn C. Flock.

Malabika Pramanik

https://people.math.wisc.edu/~seeger/seminar/Malabika-Analysis-Seminar-2022-Title-Abstract.pdf

Hongki Jung

A small cap decoupling for the twisted cubic

Bernhard Lamel

Convergence and Divergence of Formal Power Series Maps

Consider two real-analytic hypersurfaces (i.e. defined by convergent power series) in complex spaces. A formal holomorphic map is said to take one into the other if the composition of the power series defining the target with the map (which is just another formal power series) is a (formal) multiple of the defining power series of the source. In this talk, we are going to be interested in conditions for formal holomorphic maps to necessarily be convergent. Now, a formal holomorphic map taking the real line to itself is just a formal power series with real coefficients; this example also gives rise to real hypersurfaces in higher dimensional complex spaces having divergent formal self-maps. On the other hand, a formal map taking the unit sphere in higher dimensional complex space to itself is necessarily a rational map with poles outside of the sphere, in particular, the formal power series defining it converges. The convergence theory for formal self-maps of real hypersurfaces has been developed in the late 1990s and early 2000s. For formal embeddings, “ideal" conditions had been long conjectured. I’m going to give an introduction to this problem and talk about some joint work from 2018 with Nordine Mir giving a basically complete answer to the question when a formal map taking a real-analytic hypersurface in complex space into another one is necessarily convergent.

Carmelo Puliatti

Gradients of single layer potentials for elliptic operators with coefficients of Dini mean oscillation-type

We consider a uniformly elliptic operator $L_A$ in divergence form associated with a matrix A with real, bounded, and possibly non-symmetric coefficients. If a proper $L^1$-mean oscillation of the coefficients of A satisfies suitable Dini-type assumptions, we prove the following: if \mu is a compactly supported Radon measure in $\mathbb{R}^{n+1}, n >= 2$, the $L^2(\mu)$-operator norm of the gradient of the single layer potential $T_\mu$ associated with $L_A$ is comparable to the $L^2$-norm of the n-dimensional Riesz transform $R_\mu$, modulo an additive constant. This makes possible to obtain direct generalizations of some deep geometric results, initially proved for the Riesz transform, which were recently extended to $T_\mu$ under a H\"older continuity assumption on the coefficients of the matrix $A$.

This is a joint work with Alejandro Molero, Mihalis Mourgoglou, and Xavier Tolsa.

Larry Guth

Series title: Reflections on decoupling and Vinogradov's mean value problem.

Series abstract: Decoupling is a recent development in Fourier analysis that has solved several longstanding problems. The goal of the lectures is to describe this development to a general mathematical audience. We will focus on one particular application of decoupling: Vinogradov's mean value problem from analytic number theory. This problem is about the number of solutions of a certain system of diophantine equations. It was raised in the 1930s and resolved in the last decade. We will give some context about this problem, but the main goal of the lectures is to explore the ideas that go into the proof. The method of decoupling came as a big surprise to me, and I think to other people working in the field. The main idea in the proof of decoupling is to combine estimates from many different scales. We will describe this process and reflect on why it is helpful.


Lecture 1: Introduction to decoupling and Vinogradov's mean value problem. Abstract: In this lecture, we introduce Vinogradov's problem and give an overview of the proof.

Lecture 2: Features of the proof of decoupling. Abstract: In this lecture, we look more closely at some features of the proof of decoupling. The first feature we examine is the exact form of writing the inequality, which is especially suited for doing induction and connecting information from different scales. The second feature we examine is called the wave packet decomposition. This structure has roots in quantum physics and in information theory.

Lecture 3: Open problems. Abstract: In this lecture, we discuss some open problems in number theory that look superficially similar to Vinogradov mean value conjecture, such as Hardy and Littlewood's Hypothesis K*. In this lecture, we probe the limitations of decoupling by exploring why the techniques from the first two lectures don't work on these open problems. Hopefully this will give a sense of some of the issues and difficulties involved in these problems.

Previous_Analysis_seminars

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

Extras

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Graduate Student Seminar:

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