Difference between revisions of "Analysis Seminar"

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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  Betsy at stovall(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 fall semester please contact David and Andreas.
  
= 2017-2018 Analysis Seminar Schedule =
+
= Analysis Seminar Schedule =
 
{| cellpadding="8"
 
{| cellpadding="8"
 
!align="left" | date   
 
!align="left" | date   
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!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|September 8 in B239 (Colloquium)
+
|September 21, VV B139
| Tess Anderson
+
| Dóminique Kemp
| UW Madison
+
| UW-Madison
|[[#linktoabstract A Spherical Maximal Function along the Primes]]
+
|[[#Dóminique Kemp Decoupling by way of approximation ]]
|Tonghai
+
|  
 
|-
 
|-
|September 19
+
|September 28, VV B139
| Brian Street
+
| Jack Burkart
| UW Madison
+
| UW-Madison
|[[#Brian Street Convenient Coordinates ]]
+
|[[#Jack Burkart Transcendental Julia Sets with Fractional Packing Dimension ]]
| Betsy
+
|  
 
|-
 
|-
|September 26
+
|October 5, Online
| Hiroyoshi Mitake
+
| Giuseppe Negro
| Hiroshima University
+
| University of Birmingham
|[[#Hiroyoshi Mitake Derivation of multi-layered interface system and its application ]]
+
|[[#Giuseppe Negro Stability of sharp Fourier restriction to spheres ]]
| Hung
+
|  
 
|-
 
|-
|October 3
+
|October 12, VV B139
| Joris Roos
+
|Rajula Srivastava
| UW Madison
+
|UW Madison
|[[#Joris Roos A polynomial Roth theorem on the real line ]]
+
|[[#Rajula Srivastava Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups ]]
| Betsy
+
|  
 
|-
 
|-
|October 10
+
|October 19, Online
| Michael Greenblatt
+
|Itamar Oliveira
| UI Chicago
+
|Cornell University
|[[#Michael Greenblatt Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]
+
|[[#Itamar Oliveira A new approach to the Fourier extension problem for the paraboloid ]]
| Andreas
+
|  
 
|-
 
|-
|October 17
+
|October 26, VV B139
| David Beltran
+
| Changkeun Oh
| Basque Center of Applied Mathematics
+
| UW Madison
|[[#David Beltran Fefferman-Stein inequalities ]]
+
|[[#linktoabstract Title ]]
| Andreas
+
|  
 
|-
 
|-
|Wednesday, October 18, 4:00 p.m.  in B131
+
|October 29, TBA
|Jonathan Hickman
+
| Alexandru Ionescu (Colloquium)
|University of Chicago
+
| Princeton University
|[[#Jonathan Hickman | Factorising X^n  ]]
+
|[[#linktoabstract |   Title ]]
|Andreas
 
 
|-
 
|-
|October 24
+
|November 2, VV B139
| Xiaochun Li
+
| Liding Yao
| UIUC
+
| UW Madison
|[[#Xiaochun Li  |  Recent progress on the pointwise convergence problems of Schroedinger equations ]]
+
|[[#linktoabstract Title ]]
| Betsy
+
|  
|-
 
|Thursday, October 26, 4:30 p.m. in B139
 
| Fedor Nazarov
 
| Kent State University
 
|[[#Fedor Nazarov  |  The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp  ]]
 
| Sergey, Andreas
 
|-
 
|Friday, October 27, 4:00 p.m.  in B239
 
| Stefanie Petermichl
 
| University of Toulouse
 
|[[#Stefanie Petermichl  | Higher order Journé commutators  ]]
 
| Betsy, Andreas
 
|-
 
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)
 
| Shaoming Guo
 
| Indiana University
 
|[[#Shaoming Guo Parsell-Vinogradov systems in higher dimensions ]]
 
| Andreas
 
 
|-
 
|-
|November 14
+
|November 9, VV B139
| Naser Talebizadeh Sardari
+
| Lingxiao Zhang
 
| UW Madison
 
| UW Madison
|[[#Naser Talebizadeh Sardari Quadratic forms and the semiclassical eigenfunction hypothesis ]]
+
|[[#linktoabstract Title ]]
| Betsy
+
|  
 
|-
 
|-
|November 28
+
|November 12, TBA
| Xianghong Chen
+
| Kasso Okoudjou (Colloquium)
| UW Milwaukee
+
| Tufts University
|[[#Xianghong Chen Some transfer operators on the circle with trigonometric weights ]]
+
|[[#linktoabstract Title ]]
| Betsy
 
 
|-
 
|-
|Monday, December 4, 4:00, B139
+
|November 16, VV B139
| Bartosz Langowski and Tomasz Szarek
+
| Rahul Parhi
| Institute of Mathematics, Polish Academy of Sciences
+
| UW Madison (EE)
|[[#Bartosz Langowski and Tomasz Szarek Discrete Harmonic Analysis in the Non-Commutative Setting ]]
+
|[[#linktoabstract Title ]]
| Betsy
+
|  
 
|-
 
|-
|Wednesday, December 13, 4:00, B239 (Colloquium)
+
|November 30, VV B139
|Bobby Wilson
+
| Alexei Poltoratski
|MIT
 
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]
 
| Andreas
 
|-
 
| Monday, February 5, 3:00-3:50, B341  (PDE-GA seminar)
 
| Andreas Seeger
 
| UW
 
|[[#Andreas Seeger |  Singular integrals and a problem on mixing flows]]
 
|
 
|-
 
|February 6
 
| Dong Dong
 
| UIUC
 
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]
 
|Betsy
 
|-
 
|February 13
 
| Sergey  Denisov
 
 
| UW Madison
 
| UW Madison
| [[#Sergey Denisov | Spectral Szegő theorem on the real line]]
+
|[[#linktoabstract  |   Title ]]
 
|  
 
|  
 
|-
 
|-
|February 20
+
|December 7
| Ruixiang Zhang
+
| Person
| IAS (Princeton)
+
| Institution
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]
+
|[[#linktoabstract  |   Title ]]
| Betsy, Jordan, Andreas
+
|  
 
|-
 
|-
|February 27
+
|December 14
|Detlef Müller
+
| Person
|University of Kiel
+
| Institution
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]
+
|[[#linktoabstract  |   Title ]]
|Betsy, Andreas
+
|  
|-
 
|Wednesday, March 7, 4:00 p.m.
 
| Winfried Sickel
 
|Friedrich-Schiller-Universität Jena
 
| [[#Winfried Sickel | On the regularity of compositions of functions]]
 
|Andreas
 
|-
 
|March 20
 
| Betsy Stovall
 
| UW
 
| [[#linkofabstract | Two endpoint bounds via inverse problems]]
 
|
 
|-
 
|April 10
 
| Martina Neuman
 
| UC Berkeley
 
| [[#Martina Neuman | Gowers-Host-Kra norms and Gowers structure on Euclidean spaces]]
 
| Betsy
 
|-
 
|Friday, April 13, 4:00 p.m. (Colloquium, 911 VV)
 
|Jill Pipher
 
|Brown
 
| [[#Jill Pipher | Mathematical ideas in cryptography]]
 
|WIMAW
 
 
|-
 
|-
|April 17
+
|Date
 +
| Person
 +
| Institution
 +
|[[#linktoabstract  |  Title ]]
 
|  
 
|  
|
 
| [[#linkofabstract | Title]]
 
|
 
|-
 
|April 24
 
| Lenka Slavíková
 
| University of Missouri
 
| [[#Lenka Slavíková | <math>L^2 \times L^2 \to L^1</math> boundedness criteria]]
 
|Betsy, Andreas
 
|-
 
|May 1 '''at 3:30pm'''
 
| Xianghong Gong
 
| UW
 
| [[#Xianghong Gong | Smooth equivalence of deformations of domains in complex euclidean spaces]]
 
|
 
|-
 
| '''May 2 in B239 at 4pm'''
 
| Keith Rush
 
| senior data scientist with the Milwaukee Brewers
 
| [[#Keith Rush | Guerilla warfare: ruling the data jungle]]
 
|-
 
| '''May 7''' in '''B223'''
 
| Ebru Toprak
 
| UIUC
 
| [[#Ebru Toprak |Dispersive estimates for massive Dirac equations]]
 
|Betsy
 
|-
 
| '''May 15'''
 
| Gennady Uraltsev
 
| Cornell
 
| [[#linkofabstract | TBA]]
 
| Andreas, Betsy
 
|-
 
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]
 
|
 
|
 
|
 
|Betsy, Andreas
 
 
|-
 
|-
 +
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===Brian Street===
+
===Dóminique Kemp===
  
Title:  Convenient Coordinates
+
Decoupling by way of approximation
  
Abstract:  We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields.  We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic).  By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger.  When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".
+
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 $M$.
  
===Hiroyoshi Mitake===
+
===Jack Burkart===
  
Title:  Derivation of multi-layered interface system and its application
+
Transcendental Julia Sets with Fractional Packing Dimension
  
Abstract:  In this talk, I will propose a multi-layered interface system which can
+
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.
be formally derived by the singular limit of the weakly coupled system of  
 
the Allen-Cahn equation.  By using the level set approach, this system can be
 
written as a quasi-monotone degenerate parabolic system.  
 
We give results of the well-posedness of viscosity solutions, and study the  
 
singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.
 
  
===Joris Roos===
+
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: A polynomial Roth theorem on the real line
+
===Giuseppe Negro===
  
Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.
+
Stability of sharp Fourier restriction to spheres
  
===Michael Greenblatt===
+
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:  Maximal averages and Radon transforms for two-dimensional hypersurfaces
+
Joint work with E.Carneiro and D.Oliveira e Silva.
  
Abstract:  A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.
+
===Rajula Srivastava===
  
===David Beltran===
+
Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups
  
Title:  Fefferman Stein Inequalities
+
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:  Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.
+
===Itamar Oliveira===
  
===Jonathan Hickman===
+
A new approach to the Fourier extension problem for the paraboloid
  
Title: Factorising X^n.
+
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.
  
Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.
+
=[[Previous_Analysis_seminars]]=
  
===Xiaochun Li ===
+
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars
  
Title:  Recent progress on the pointwise convergence problems of Schrodinger equations
+
=Extras=
  
Abstract:  Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3.  This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and  the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.
+
[[Blank Analysis Seminar Template]]
 
 
===Fedor Nazarov=== 
 
  
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden
 
conjecture is sharp.
 
  
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for
+
Graduate Student Seminar:
the norm of the Hilbert transform on the line as an operator from $L^1(w)$
 
to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work
 
with Andrei Lerner and Sheldy Ombrosi.
 
  
===Stefanie Petermichl===
+
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html
Title: Higher order Journé commutators
 
 
 
Abstract: We consider questions that stem from operator theory via Hankel and
 
Toeplitz forms and target (weak) factorisation of Hardy spaces. In
 
more basic terms, let us consider a function on the unit circle in its
 
Fourier representation. Let P_+ denote the projection onto
 
non-negative and P_- onto negative frequencies. Let b denote
 
multiplication by the symbol function b. It is a classical theorem by
 
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and
 
only if b is in an appropriate space of functions of bounded mean
 
oscillation. The necessity makes use of a classical factorisation
 
theorem of complex function theory on the disk. This type of question
 
can be reformulated in terms of commutators [b,H]=bH-Hb with the
 
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such
 
as in the real variable setting, in the multi-parameter setting or
 
other, these classifications can be very difficult.
 
 
 
Such lines were begun by Coifman, Rochberg, Weiss (real variables) and
 
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of
 
spaces of bounded mean oscillation via L^p boundedness of commutators.
 
We present here an endpoint to this theory, bringing all such
 
characterisation results under one roof.
 
 
 
The tools used go deep into modern advances in dyadic harmonic
 
analysis, while preserving the Ansatz from classical operator theory.
 
 
 
===Shaoming Guo ===
 
Title: Parsell-Vinogradov systems in higher dimensions
 
 
 
Abstract:
 
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.
 
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.
 
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.
 
 
 
===Naser Talebizadeh Sardari===
 
 
 
Title: Quadratic forms and the semiclassical eigenfunction hypothesis
 
 
 
Abstract:  Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>,  and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math>  in terms of <math>n</math>, <math>k</math>, and <math>D</math>. As a corollary, we prove  a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given  eigenspace of the Laplacian on the flat torus <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.
 
 
 
===Xianghong Chen===
 
 
 
Title:  Some transfer operators on the circle with trigonometric weights
 
 
 
Abstract:  A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer.
 
 
 
===Bobby Wilson===
 
 
 
Title: Projections in Banach Spaces and Harmonic Analysis
 
 
 
Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.
 
 
 
===Andreas Seeger===
 
 
 
Title: Singular integrals and a problem on mixing flows
 
 
 
Abstract: The talk will be about  results related to Bressan's mixing problem. We present  an inequality for the change of a  Bianchini semi-norm of characteristic functions under the  flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator  for which one proves bounds  on Hardy spaces. This is joint work with Mahir Hadžić,  Charles Smart and    Brian Street.
 
 
 
===Dong Dong===
 
 
 
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory
 
 
 
Abstract:  This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.
 
 
 
===Sergey Denisov===
 
 
 
Title:  Spectral Szegő  theorem on the real line
 
 
 
Abstract:  For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.
 
 
 
===Ruixiang Zhang===
 
 
 
Title:  The (Euclidean) Fractal Uncertainty Principle
 
 
 
Abstract:  On the real line, a  version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work  the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).
 
 
 
===Detlef Müller===
 
 
 
Title: On Fourier restriction for a non-quadratic hyperbolic surface
 
 
 
Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about  hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically  in the presence of a perturbation term,  and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.
 
 
 
===Winfried Sickel===
 
 
 
Title: On the regularity of compositions of functions
 
 
 
Abstract: Let <math>E</math> denote a Banach space of locally integrable functions on <math>\mathbb{R}</math>. To each continuous function <math>f:\mathbb{R} \to \mathbb{R}</math>
 
we associate the composition operator
 
<math>T_f(g):= f\circ g</math>, <math>g\in E</math>.
 
The properties of <math>T_f</math> strongly depend on the chosen function space <math>E</math>.
 
In my talk I will concentrate on Sobolev spaces <math>W^m_p</math> and  Slobodeckij spaces <math>W^s_p</math>.
 
The main aim will consist in giving a survey on necessary and sufficient conditions on <math>f</math>
 
such that the composition operator maps such a space <math>E</math> into itself.
 
 
 
===Martina Neuman===
 
 
 
Title:  Gowers-Host-Kra norms and Gowers structure on Euclidean spaces
 
 
 
Abstract:  The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.
 
 
 
===Jill Pipher===
 
 
 
Title:  Mathematical ideas in cryptography
 
 
 
Abstract:  This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research,
 
including homomorphic encryption.
 
 
 
===Lenka Slavíková===
 
 
 
Title:  <math>L^2 \times L^2 \to L^1</math> boundedness criteria
 
 
 
Abstract:  It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function <math>m</math> is bounded from <math>L^2</math> to itself if and only if <math>m</math> belongs to the space <math>L^\infty</math>. In this talk we will investigate the <math>L^2 \times L^2 \to L^1</math> boundedness of bilinear multiplier operators which is as central in the bilinear theory as the <math>L^2</math> boundedness is in the linear multiplier theory. We will present a sharp <math>L^2 \times L^2 \to L^1</math> boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the <math>L^q</math> integrability of this function; precisely we will show that boundedness holds if and only if <math>q<4</math>. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.
 
 
 
===Xianghong Gong===
 
 
 
Title:  Smooth equivalence of deformations of domains in complex euclidean spaces
 
 
 
Abstract:  We prove that two smooth families of 2-connected domains in the complex plane are smoothly equivalent if they are equivalent under a possibly discontinuous family of biholomorphisms. We construct two smooth families of smoothly bounded domains in C^n for n>=1 that are equivalent under discontinuous families of biholomorphisms but not under any continuous family of biholomorphisms. Finally, we give sufficient conditions for the smooth equivalence of two smooth families of domains. This is joint work with Hervé  Gaussier.
 
 
 
===Ebru Toprak===
 
 
 
Title:  Dispersive estimates for massive Dirac equations
 
 
 
Abstract:  In this talk, I will cover some existing L^1 \rightarrow L^\infty dispersive estimates for the linear Schr\"odinger equation with potential and present a related study on the two and three dimensional massive Dirac equation. In two dimension, we show that the t^{-1} decay rate holds if the threshold energies are regular or if there are s-wave resonances at the threshold. We further show that, if the threshold energies are regular then a faster decay rate of t^{-1}(\log t)^{-2} is attained for large t, at the cost of logarithmic spatial weights, which is not the case for the free Dirac equation. In three dimension, we show that the solution operator is composed of a finite rank operator that decays at the rate t^{-1/2} plus a term that decays at the rate t^{-3/2}. This is a joint work with M.Burak Erdo\u{g}an and William Green.
 
 
 
=Extras=
 
[[Blank Analysis Seminar Template]]
 

Latest revision as of 18:09, 15 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 Title
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 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.

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