Fall 2024 Analysis Seminar: Difference between revisions

From DEV UW-Math Wiki
Jump to navigation Jump to search
VisualWikitext
No edit summary
No edit summary
 
(61 intermediate revisions by 3 users not shown)
Line 7: Line 7:
* Andreas Seeger: seeger at math dot wisc dot edu
* Andreas Seeger: seeger at math dot wisc dot edu


Time: Wed 3:30--4:30
Time and Room: Wed 3:30--4:30 Van Vleck B119.


Room: B223
In some cases the seminar may be scheduled at different time to accommodate speakers.
 
If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+subscribe (at) g-groups (dot) wisc (dot) edu


We also have room B211 reserved at 4:25-5:25 for discussions after talks.
[[Previous Analysis seminars|Links to previous seminars]]


All talks will be in-person unless otherwise specified.
Link to [[Spring 2025 Analysis Seminar]]


In some cases the seminar may be scheduled at different time to accommodate speakers.


If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+subscribe (at) g-groups (dot) wisc (dot) edu
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 27: Line 27:
!Notes (e.g. unusual room/day/time)
!Notes (e.g. unusual room/day/time)
|-
|-
|9-11
|We, 9-11
|
|Gevorg Mnatsakanyan
|
|UW Madison
|Almost everywhere convergence of the Malmquist Takenaka series
|
|
|
|
|-
|We, 9-18
|Lars Niedorf
|UW Madison
|Restriction type estimates and spectral multipliers on Métivier groups
|
|
|
|
|-
|-
|9-25
|Th, 9-26, 2:25-3:25, VV B215
|
|Niclas Technau
|  
|University of Bonn
|
|Rational points on/near homogeneous hyper-surfaces
|
|Andreas
|
|Note changes of time/date/room, joint with Number Theory Seminar
|
|-
|-
|10-2
|We, 10-2
|
|Sergey Denisov
|
|UW Madison
|
|Applications of inverse spectral theory for canonical systems to NLS.
|
|
|
|
|
|-
|-
|10-9
|We, 10-9
|
|Shukun Wu
|
|Indiana University
|
|On almost everywhere convergence of planar Bochner Riesz means
|
|Shengwen
|
|
|
|-
|-
|10-16
|We, 10-16
|
|Nathan Wagner
|
|Brown University
|
|Dyadic shifts, sparse domination, and commutators in the non-doubling setting
|
|Andreas
|
|
|
|-
|-
|10-23
|We, 10-23
|
|Betsy Stovall
|
|UW Madison
|
|Title: Incidences among flows
|
|
|
|
|
|-
|-
|10-30
|We, 10-30
|Burak Hatinoglu
|Burak Hatinoglu
|Michigan State University
|Michigan State University
|
|Norm estimates of Chebyshev polynomials
|
|Alexei
|
|
|-
|11-6
|
|
|
|
|
|
|-
|11-13
|
|
|
|
|
|
|-
|11-20
|
|
|
|
|
|
|-
|11-27
|No seminar
|
|
|
|
|
|-
|12-4
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|
|-
|-
|
|We, 11-6
|
|Bingyuan Liu
|
|University of Texas Rio Grande Valley
|
|The Diederich--Fornaess index and the dbar-Neumann problem
|
|Xianghong
|
|
|
|-
|-
|
|We, 11-13
|
|Maxim Yattselev
|
|Indiana University (Indianapolis)
|
|On rational approximants of multi-valued functions
|
|Sergey
|
|
|
|-
|-
|
|We, 11-20, 2:25-3:25, Sterling 1333
|
|Li Ji
|
|Macquarie University
|
|Hardy spaces associated to the multiparameter flag structure
|
|Brian
|
|
|
|-
|-
|
|
|
|
|
|
|
|-
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|}
|}




Line 352: Line 112:




===[[Tushar Das]]===
===[[Gevorg Mnatsakanyan]]===
Title: Playing games on fractals: Dynamical & Diophantine
Title: Almost everywhere convergence of the Malmquist Takenaka series


Abstract: We will present sketches of a program, developed in collaboration with Lior Fishman, David Simmons, and Mariusz Urbanski, which extends the parametric geometry of numbers (initiated by Wolfgang Schmidt and Leonhard Summerer) to Diophantine approximation for systems of m linear forms in n variables. Our variational principle (arXiv:1901.06602) provides a unified framework to compute Hausdorff and packing dimensions of a variety of sets of number-theoretic interest,  as well as their dynamical counterparts via the Dani correspondence. Highlights include the introduction of certain combinatorial objects that we call templates, which arise from a dynamical study of Minkowski’s successive minima in the geometry of numbers; as well as a new variant of Schmidt’s game designed to compute the Hausdorff and packing dimensions of any set in a doubling metric space. The talk will be accessible to students and faculty whose interests contain a convex combination of homogeneous dynamics, Diophantine approximation and fractal geometry.
Link to Abstract: [https://people.math.wisc.edu/~seeger/abstracts/gevorg2024.pdf]


===[[Rajula Srivastava]]===
===[[Lars Niedorf]]===
Title: Counting Rational Points near Flat Hypersurfaces
Title: Restriction type estimates and spectral multipliers on Métivier groups


Abstract: How many rational points with denominator of a given size lie
Abstract: We present a restriction type estimate for sub-Laplacians on arbitrary two-step stratified Lie groups. Although weaker than previously known estimates for the subclass of Heisenberg type groups, these estimates turn out to be sufficient to prove an Lp-spectral multiplier theorem with sharp regularity condition s > d|1/p-1/2| for sub-Laplacians on Métivier groups.
within a given distance from a compact hypersurface? In this talk, we
shall describe how the geometry of the surface plays a key role in
determining this count, and present a heuristic for the same. In a
recent breakthrough, J.J. Huang proved that this guess is indeed true
for hypersurfaces with non-vanishing Gaussian curvature.  What about
hypersurfaces with curvature only vanishing up to a finite order, at a
single point? We shall offer a new heuristic in this regime which also
incorporates the contribution arising from "local flatness". Further, we
will describe how ideas from Harmonic Analysis can be used to establish
the indicated estimates for hypersurfaces of this type immersed by
homogeneous functions. In particular, we shall use a powerful
bootstrapping argument relying on Poisson summation, duality between
flat and "rough" hypersurfaces, and the method of stationary phase. A
crucial role is played by a dyadic scaling argument exploiting the
homogeneous nature of the hypersurface. Based on joint work with N. Technau.


===[[Niclas Technau]]===
===[[Niclas Technau]]===
Title: Oscillatory Integrals Count
Title: Rational points on/near homogeneous hyper-surfaces
 
Abstract: This talk is about phrasing (number theoretic) counting problems in terms oscillatory integrals.
Abstract: How many rational points are on/near a compact hyper-surface? This question is related to Serre's Dimension Growth Conjecture.
We shall provide a simple introduction to the topic, mention open questions, and
We survey the state of the art, and explain a standard random model. Furthermore, we report on recent joint work with Rajula Srivastava (Uni/MPIM Bonn).
report on joint work with Sam Chow, as well as on joint work with Chris Lutsko.
Our arguments are rooted in Fourier analysis and, in particular, clarify the role of curvature in the random model.
 
 
===[[Terry Harris]]===


===[[Sergey Denisov]]===
Title: Applications of inverse spectral theory for canonical systems to NLS


Title: Horizontal Besicovitch sets of measure zero and some related problems
Abstract: For nice initial data, NLS can be integrated using the inverse scattering theory for the Dirac equation on the line. We will discuss the connection of the Dirac equation to canonical systems and use the recent characterization of the Szegő class of measures on the real line to obtain a new semi-conserved quantity for the NLS. The bounds for the negative Sobolev norms will be presented as an application for the L2 NLS solutions (based on joint work with Roman Bessonov).


Abstract:   It is shown that horizontal Besicovitch sets of measure zero exist in R^3. The proof is constructive and uses point-line duality analogously to Kahane’s construction of measure zero Besicovitch sets in the plane. Some consequences and related examples are shown for the SL_2 Kakeya maximal function.
===[[Shukun Wu]]===
Title: On almost everywhere convergence of planar Bochner Riesz means


===[[Tristan Leger]]===
Abstract: We prove that the planar Bochner Riesz mean converges almost everywhere for any L^p function in the optimal range, for 5/3<p<2. Our approach is based on a weighted L^2 estimate, which may be of independent interest. For example, up to an epsilon loss, we can reprove Cordoba's L^4 orthogonality by solely considering L^2 space and using L^2 orthogonality. This is a joint work with Xiaochun Li.


Title: L^p bounds for spectral projectors on hyperbolic surfaces
===[[Nathan Wagner]]===
Title: Dyadic shifts, sparse domination, and commutators in the non-doubling setting


Abstract: In this talk I will present L^p boundedness results for spectral projectors on hyperbolic surfaces. I will focus on the case where the spectral window has small width. Indeed the negative curvature assumption leads to improvements over the universal bounds of C.Sogge, thus illustrating how these objects are sensitive to the global geometry of the underlying manifold. I will also explain how this relates to the general theme of delocalization of eigenfunctions of the Laplacian on hyperbolic surfaces.
Abstract: In this talk, we will discuss a dyadic variant of the Hilbert transform, which is a useful model of its continuous counterpart and the prototypical example of a so-called "Haar shift". After discussing some background and motivation in the Lebesgue measure case, we will turn to the situation where the L2 Haar functions are defined with respect to a locally finite Borel measure μ, which may not satisfy the dyadic doubling condition. In this more general setting, Lopez-Sanchez, Martell, and Parcet identified a weak regularity condition on the measure  μ which characterizes weak-type and Lp estimates for this dyadic Hilbert transform. I then will discuss joint work with Jose Conde Alonso and Jill Pipher, where we obtain a domination of the dyadic Hilbert transform (and more generally, Haar shifts) by a modified sparse form.  As an application, we characterize the class of weights where the dyadic Hilbert transform and related operators are bounded. A surprising novelty is that the usual (dyadic) Muckenhoupt A2 condition is necessary, but no longer sufficient in the non-doubling setting, and our modified weight condition reflects the "complexity" of the underlying Haar shift. Finally, we will examine a different dyadic Haar shift model of the Hilbert transform and its relationship to BMO (bounded mean oscillation) functions via commutators in the non-doubling setting (joint with Tainara Borges, Jose Conde Alonso, and Jill Pipher).
This is based on joint work with Jean-Philippe Anker and Pierre Germain.


 
===[[Betsy Stovall|Betsy Stovall]]===
===[[Bingyang Hu]]===
Title:  Incidences among flows
 
Title: On the curved Trilinear Hilbert transform
   
Abstract: The goal of this talk is to discuss the Lp boundedness of the trilinear Hilbert transform along the moment curve. More precisely, we show that the operator
$$
H_C(f_1, f_2, f_3)(x):=p.v. \int_{\mathbb R} f_1(x-t)f_2(x+t^2)f_3(x+t^3) \frac{dt}{t}, \quad x \in \mathbb R
$$
is bounded from $L^{p_1}(\mathbb R) \times L^{p_2}(\mathbb R) \times L^{p_3}(\mathbb R}$ into $L^r(\mathbb R)$ within the Banach H\"older range $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=\frac{1}{r}$ with $1<p_1, p_3<\infty$, $1<p_2 \le \infty$ and $1 \le r <\infty$.
The main difficulty in approaching this problem(compared to the classical approach to the bilinear Hilbert transform) is the lack of absolute summability after we apply the time-frequency discretization(which is known as the LGC-methodology introduced by V. Lie in 2019). To overcome such a difficulty, we develop a new, versatile approch -- referred to as Rank II LGC (which is also motived by the study of the non-resonant bilinear Hilbert-Carleson operator by C. Benea, F. Bernicot, V. Lie, and V. Vitturi in 2022), whose control is achieved via the following interdependent elements:
1). a sparse-uniform deomposition of the input functions adapted to an appropriate time-frequency foliation of the phase-space;
2). a structural analysis of suitable maximal "joint Fourier coefficients";
3). a level set analysis with respect to the time-frequency correlation set.
   
   
This is a joint work with my postdoc advisor Victor Lie from Purdue.
Abstract:  
 
In this talk, we will discuss a family of combinatorial problems that may be viewed as discretized and (possibly) perturbed versions of various continuum incidence problems in harmonic analysis. Namely, given a finite collection L of integral curves of some “small” family of “nice” vector fields, how many incidences can be formed under various definitions of “small,” “nice,” and “incidence”?  Well-studied special cases include the Joints Problem in R^n, the Szeméredi—Trotter Theorem for point-line intersections in the plane, and work of Magyar—Stein—Wainger, Pierce, and others, in which the discretization is to the integers. In this talk, we will discuss results and examples both old and new; the new results are joint with Huang and Tammen.  
 
===[[Rodrigo Bañuelos]]===
 
Title: Probabilistic tools in discrete harmonic analysis
 
Abstract: The discrete Hilbert transform was introduced by David Hilbert at the beginning of the 20th century as an example of a singular quadratic form. Its boundedness on the space of square summable sequences appeared in H. Weyl’s doctoral dissertation (under Hilbert) in 1908. In 1925, M. Riesz proved that the continuous version of this operator is bounded on L^p(R), 1 < p < \infty, and that the same holds for the discrete version on the integers. Shortly thereafter (1926), E. C. Titchmarsh gave a different proof and from it concluded that the operators have the same p-norm. Unfortunately, Titchmarsh’s argument for equality was incorrect. The question of equality of the norms had been a “simple tantalizing" problem ever since.


In this general colloquium talk the speaker will discuss a probabilistic construction, based on Doob’s “h-Brownian motion," that leads to sharp inequalities for a collection of discrete operators on the d-dimensional lattice Z^d, d ≥ 1. The case d = 1 verifies equality of the norms for the discrete and continuous Hilbert transforms. The case d > 1 leads to similar questions and conjectures for other Calderón-Zygmund singular integrals in higher dimensions.
===[[Burak Hatinoglu]]===
Title: Norm estimates of Chebyshev polynomials


Abstract: The Chebyshev polynomial of a compact set in the complex plane is the unique monic polynomial of a given degree minimizing the sup-norm among the monic polynomials of the same degree. As a classical object in the approximation theory, its history and connections with potential theory go back to the first half of the 20th century, Fekete, Szego and Faber. However, in the last 20 years many remarkable improvements were made in the study of Chebyshev polynomials. In this talk, after reviewing basics and classical theorems, I will consider recent results on norm estimates of Chebyshev polynomials, discuss a Cantor-type set construction approach, and an application to the spectral theory of periodic operators.


===[[Shaoming Guo]]===
===[[Bingyuan Liu|Bingyuan Liu]] ===
Title: The Diederich--Fornaess idex and the dbar-Neumann problem


Title: Oscillatory integrals on Riemannian manifolds, and related Kakeya and Nikodym problems
Abstract: Introduced in 1977, the Diederich–Fornaess index was developed to help construct bounded plurisubharmonic functions on bounded pseudoconvex domains. For the past three decades, it is believed that the Diederich–Fornaess index is linked with the global regularity of the dbar-Neumann operator. A longstanding open question has been whether a Diederich–Fornaess index of 1 implies this global regularity. In this talk, we will overview the background and present and sketch a proof of a recent theorem, jointly proved by Emil Straube and me. This theorem answers this open question affirmatively for (0, n-1) forms.


===[[Maxim Yattselev]]===
Title: On Rational Approximants of Multi-Valued Functions


Abstract: The talk is about oscillatory integral operators on manifolds. Manifolds of constant sectional curvatures are particularly interesting, and we will see that very good estimates on these manifolds can be expected. We will also discuss Kakeya and Nikodym problems on general manifolds, in particular, manifolds satisfying Sogge’s chaotic curvatures.
Abstract: Let D  be a bounded Jordan domain and  A be its complement on the Riemann sphere. The asymptotic behavior in D of the best rational approximants in the uniform norm on A of functions holomorphic on A that admit a multi-valued continuation to quasi every point of D with finitely many possible branches will be discussed.


===[[Ji Li]]===
Title: Hardy spaces associated to multiparameter flag structures


 
The theory of multi-parameter flag singular integral originates from the study of the ∂ ̄-problem on the Heisenberg group by D. Phong and E.M. Stein. In our recent work, we established a complete flag Hardy space theory on the Heisenberg group, including characterisations via Littlewood–Paley area function, square function, non-tangential and radial maximal functions, atoms, and the flag Riesz transforms. It provided a unified approach for proving the L^p boundedness of different types of singular integrals, and led to the endpoint L\log L\to L^{1,\infty}  estimates. The representations of flag BMO functions are also provided.
 
 
 
 
 
===[[Lechao Xiao]]===
 
Title: Some connections between Harmonic Analysis and Theory of Deep Learning
 
 
 
 
Abstract: The past decade has witnessed a remarkable surge in breakthroughs in artificial intelligence (AI), with the potential to profoundly impact various aspects of our lives. However, the fundamental mathematical principles underlying the success of deep learning, the core technology behind these breakthroughs, is still far from well-understood. In this presentation, I will share some interesting connections between the theory of deep learning and harmonic analysis.
The first half provides a gentle introduction to machine learning and deep learning. The second half focuses on two technical topics:
 
An uncertainty principle between space and frequency and its significance in overcoming the curse of dimensionality.
 
The multi-scale Marchenko-Pastur law and its interplay with the multiple-descent learning curve phenomenon.
 
 
 
 
===[[Neeraja Kulkarni]]===
 
Title: An improved Minkowski dimension estimate for Kakeya sets in higher dimensions using planebrushes.
 
 
Abstract: A Kakeya set is defined as a compact subset of $\mathbb{R}^n$ which contains a line segment of length 1 in every direction. The Kakeya conjecture says that every Kakeya set has Minkowski and Hausdorff dimensions equal to $n$. In this talk, I will discuss an improved Minkowski dimension estimate for Kakeya sets in dimensions $n\geq 5$. The improved estimate comes from using a geometric argument called a ``$k$-planebrush'', which is a higher dimensional analogue of Wolff's ``hairbrush'' argument from 1995. The $k$-planebrush argument is used in conjunction with a previously known "k-linear" result on Kakeya sets proved by Hickman-Rogers-Zhang (and concurrently by Zahl) in 2019 along with an x-ray transform estimate which is a corollary of Hickman-Rogers-Zhang (and Zahl). The x-ray transform estimate is used to deduce that the Kakeya set has a structural property called ``stickiness,'' which was first introduced in a paper by Katz-Laba-Tao in 1999. Sticky Kakeya sets exhibit a self-similar structure which is exploited by the $k$-planebrush argument.
 
 
 
 
 
===[[Changkeun]]===
 
Title: Discrete restriction estimates for manifolds avoiding a line.
 
 
 
Abstract: We identify a new way to divide the d-neighborhood of surfaces in R^3. We decompose the d-neighborhood of surfaces into a finitely-overlapping collection of rectangular boxes S. We obtain an (l^2,L^p) decoupling estimate using this decomposition, for the sharp range of exponents. The decoupling theorem we prove is new for the hyperbolic paraboloid, and recovers the Tomas-Stein restriction inequality. Our decoupling inequality leads to new exponential sum estimates where the frequencies lie on surfaces which do not contain a line. In this talk, I'll focus on explaning backgrounds and theorems rather than giving proofs. This is joint work with Larry Guth and Dominique Maldague.
 
 
 
===[[Ryan Bushlin]]===
 
Friday, December 8, 1:20-2:10 in B107
 
'''Some variational problems characterizing families of convex domains'''
 
We prove a singular integral identity for the surface measure of (n - 1)-rectifiable sets in Euclidean n-space satisfying the orientation cancellation condition. In particular, sets of finite perimeter enjoy this property, and from this observation follows a geometric inequality in which equality is attained precisely by the convex sets. More generally, the integral identity has anisotropic analogues whose corresponding inequalities characterize some geometrically simple subfamilies of the family of convex sets.
 
 
 
 
===[[Jing-Jing Huang]]===
 
 
'''Diophantine approximation on affine subspaces'''
 
We establish a clear-cut criterion for an affine subspace of R^n to be extremal (i.e. the Dirichlet exponent 1/n is best possible almost everywhere), confirming a conjecture of Kleinbock. We also extend the classical theorem of Khintchine on metric diophantine approximation to affine subspaces. Moreover, an exact formula for the Hausdorff dimension of the set of very well approximable vectors on the affine subspace is obtained. The above results are proved as consequences of our novel estimates for the number of rational points lying close to the affine subspace. The proof of this counting estimate is Fourier analytic in nature and in particular utilizes the large sieve inequality.  
 
 
 
 
 
 
[[Previous Analysis seminars|Links to previous seminars]]
 
[[Spring 2024 Analysis Seminar|Link to the analysis seminar in 2024 Spring]]

Latest revision as of 18:18, 15 November 2024

Organizers: Shengwen Gan, Terry Harris and Andreas Seeger

Emails:

  • Shengwen Gan: sgan7 at math dot wisc dot edu
  • Terry Harris: tlharris4 at math dot wisc dot edu
  • Andreas Seeger: seeger at math dot wisc dot edu

Time and Room: Wed 3:30--4:30 Van Vleck B119.

In some cases the seminar may be scheduled at different time to accommodate speakers.

If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+subscribe (at) g-groups (dot) wisc (dot) edu

Links to previous seminars

Link to Spring 2025 Analysis Seminar


Date Speaker Institution Title Host(s) Notes (e.g. unusual room/day/time)
We, 9-11 Gevorg Mnatsakanyan UW Madison Almost everywhere convergence of the Malmquist Takenaka series
We, 9-18 Lars Niedorf UW Madison Restriction type estimates and spectral multipliers on Métivier groups
Th, 9-26, 2:25-3:25, VV B215 Niclas Technau University of Bonn Rational points on/near homogeneous hyper-surfaces Andreas Note changes of time/date/room, joint with Number Theory Seminar
We, 10-2 Sergey Denisov UW Madison Applications of inverse spectral theory for canonical systems to NLS.
We, 10-9 Shukun Wu Indiana University On almost everywhere convergence of planar Bochner Riesz means Shengwen
We, 10-16 Nathan Wagner Brown University Dyadic shifts, sparse domination, and commutators in the non-doubling setting Andreas
We, 10-23 Betsy Stovall UW Madison Title: Incidences among flows
We, 10-30 Burak Hatinoglu Michigan State University Norm estimates of Chebyshev polynomials Alexei
We, 11-6 Bingyuan Liu University of Texas Rio Grande Valley The Diederich--Fornaess index and the dbar-Neumann problem Xianghong
We, 11-13 Maxim Yattselev Indiana University (Indianapolis) On rational approximants of multi-valued functions Sergey
We, 11-20, 2:25-3:25, Sterling 1333 Li Ji Macquarie University Hardy spaces associated to the multiparameter flag structure Brian


Abstracts

Gevorg Mnatsakanyan

Title: Almost everywhere convergence of the Malmquist Takenaka series

Link to Abstract: [1]

Lars Niedorf

Title: Restriction type estimates and spectral multipliers on Métivier groups

Abstract: We present a restriction type estimate for sub-Laplacians on arbitrary two-step stratified Lie groups. Although weaker than previously known estimates for the subclass of Heisenberg type groups, these estimates turn out to be sufficient to prove an Lp-spectral multiplier theorem with sharp regularity condition s > d|1/p-1/2| for sub-Laplacians on Métivier groups.

Niclas Technau

Title: Rational points on/near homogeneous hyper-surfaces

Abstract: How many rational points are on/near a compact hyper-surface? This question is related to Serre's Dimension Growth Conjecture. We survey the state of the art, and explain a standard random model. Furthermore, we report on recent joint work with Rajula Srivastava (Uni/MPIM Bonn). Our arguments are rooted in Fourier analysis and, in particular, clarify the role of curvature in the random model.

Sergey Denisov

Title: Applications of inverse spectral theory for canonical systems to NLS

Abstract: For nice initial data, NLS can be integrated using the inverse scattering theory for the Dirac equation on the line. We will discuss the connection of the Dirac equation to canonical systems and use the recent characterization of the Szegő class of measures on the real line to obtain a new semi-conserved quantity for the NLS. The bounds for the negative Sobolev norms will be presented as an application for the L2 NLS solutions (based on joint work with Roman Bessonov).

Shukun Wu

Title: On almost everywhere convergence of planar Bochner Riesz means

Abstract: We prove that the planar Bochner Riesz mean converges almost everywhere for any L^p function in the optimal range, for 5/3<p<2. Our approach is based on a weighted L^2 estimate, which may be of independent interest. For example, up to an epsilon loss, we can reprove Cordoba's L^4 orthogonality by solely considering L^2 space and using L^2 orthogonality. This is a joint work with Xiaochun Li.

Nathan Wagner

Title: Dyadic shifts, sparse domination, and commutators in the non-doubling setting

Abstract: In this talk, we will discuss a dyadic variant of the Hilbert transform, which is a useful model of its continuous counterpart and the prototypical example of a so-called "Haar shift". After discussing some background and motivation in the Lebesgue measure case, we will turn to the situation where the L2 Haar functions are defined with respect to a locally finite Borel measure μ, which may not satisfy the dyadic doubling condition. In this more general setting, Lopez-Sanchez, Martell, and Parcet identified a weak regularity condition on the measure μ which characterizes weak-type and Lp estimates for this dyadic Hilbert transform. I then will discuss joint work with Jose Conde Alonso and Jill Pipher, where we obtain a domination of the dyadic Hilbert transform (and more generally, Haar shifts) by a modified sparse form. As an application, we characterize the class of weights where the dyadic Hilbert transform and related operators are bounded. A surprising novelty is that the usual (dyadic) Muckenhoupt A2 condition is necessary, but no longer sufficient in the non-doubling setting, and our modified weight condition reflects the "complexity" of the underlying Haar shift. Finally, we will examine a different dyadic Haar shift model of the Hilbert transform and its relationship to BMO (bounded mean oscillation) functions via commutators in the non-doubling setting (joint with Tainara Borges, Jose Conde Alonso, and Jill Pipher).

Betsy Stovall

Title: Incidences among flows

Abstract: In this talk, we will discuss a family of combinatorial problems that may be viewed as discretized and (possibly) perturbed versions of various continuum incidence problems in harmonic analysis. Namely, given a finite collection L of integral curves of some “small” family of “nice” vector fields, how many incidences can be formed under various definitions of “small,” “nice,” and “incidence”? Well-studied special cases include the Joints Problem in R^n, the Szeméredi—Trotter Theorem for point-line intersections in the plane, and work of Magyar—Stein—Wainger, Pierce, and others, in which the discretization is to the integers. In this talk, we will discuss results and examples both old and new; the new results are joint with Huang and Tammen.

Burak Hatinoglu

Title: Norm estimates of Chebyshev polynomials

Abstract: The Chebyshev polynomial of a compact set in the complex plane is the unique monic polynomial of a given degree minimizing the sup-norm among the monic polynomials of the same degree. As a classical object in the approximation theory, its history and connections with potential theory go back to the first half of the 20th century, Fekete, Szego and Faber. However, in the last 20 years many remarkable improvements were made in the study of Chebyshev polynomials. In this talk, after reviewing basics and classical theorems, I will consider recent results on norm estimates of Chebyshev polynomials, discuss a Cantor-type set construction approach, and an application to the spectral theory of periodic operators.

Bingyuan Liu

Title: The Diederich--Fornaess idex and the dbar-Neumann problem

Abstract: Introduced in 1977, the Diederich–Fornaess index was developed to help construct bounded plurisubharmonic functions on bounded pseudoconvex domains. For the past three decades, it is believed that the Diederich–Fornaess index is linked with the global regularity of the dbar-Neumann operator. A longstanding open question has been whether a Diederich–Fornaess index of 1 implies this global regularity. In this talk, we will overview the background and present and sketch a proof of a recent theorem, jointly proved by Emil Straube and me. This theorem answers this open question affirmatively for (0, n-1) forms.

Maxim Yattselev

Title: On Rational Approximants of Multi-Valued Functions

Abstract: Let D be a bounded Jordan domain and A be its complement on the Riemann sphere. The asymptotic behavior in D of the best rational approximants in the uniform norm on A of functions holomorphic on A that admit a multi-valued continuation to quasi every point of D with finitely many possible branches will be discussed.

Ji Li

Title: Hardy spaces associated to multiparameter flag structures

The theory of multi-parameter flag singular integral originates from the study of the ∂ ̄-problem on the Heisenberg group by D. Phong and E.M. Stein. In our recent work, we established a complete flag Hardy space theory on the Heisenberg group, including characterisations via Littlewood–Paley area function, square function, non-tangential and radial maximal functions, atoms, and the flag Riesz transforms. It provided a unified approach for proving the L^p boundedness of different types of singular integrals, and led to the endpoint L\log L\to L^{1,\infty} estimates. The representations of flag BMO functions are also provided.

Retrieved from "https://wiki-dev.math.wisc.edu/index.php?title=Fall_2024_Analysis_Seminar&oldid=27591"