Difference between revisions of "Fall 2021 and Spring 2022 Analysis Seminars"
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− | |October 29, Colloquium | + | |October 29, Colloquium, Online |
| Alexandru Ionescu | | Alexandru Ionescu | ||
| Princeton University | | Princeton University | ||
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− | |November 12, Colloquium | + | |November 12, Colloquium, Online |
| Kasso Okoudjou | | Kasso Okoudjou | ||
| Tufts University | | Tufts University | ||
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− | | Friday, February 18, | + | |Friday, February 18, Colloquium, VVB239 |
| Andreas Seeger | | Andreas Seeger | ||
| UW Madison | | UW Madison | ||
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− | |April 12 | + | |April 12, VV B139 |
| Hongki Jung | | Hongki Jung | ||
| IU Bloomington | | IU Bloomington | ||
− | |[[# | + | |[[#Hongki Jung | A small cap decoupling for the twisted cubic ]] |
| Shaoming | | Shaoming | ||
|- | |- | ||
− | |April 15, Colloquium | + | |Friday, April 15, Colloquium, VV B239 |
| Bernhard Lamel | | Bernhard Lamel | ||
| Texas A&M University at Qatar | | Texas A&M University at Qatar | ||
− | |[[# | + | |[[#Bernhard Lamel | Convergence and Divergence of Formal Power Series Maps ]] |
| Xianghong | | Xianghong | ||
|- | |- | ||
− | |April 19 | + | |April 19, Online |
| Carmelo Puliatti | | Carmelo Puliatti | ||
| Euskal Herriko Unibertsitatea | | Euskal Herriko Unibertsitatea | ||
− | |[[#Carmelo Puliatti | | + | |[[#Carmelo Puliatti | Gradients of single layer potentials for elliptic operators |
+ | with coefficients of Dini mean oscillation-type ]] | ||
| David | | David | ||
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| | | | ||
|- | |- | ||
− | |April 25 | + | |April 25-26-27, Distinguished Lecture Series |
|Larry Guth | |Larry Guth | ||
|MIT | |MIT | ||
− | |[[#linktoabstract | | + | |[[#Larry Guth | Reflections on decoupling and Vinogradov's mean value problem. ]] |
+ | |- | ||
+ | |April 25, 4:00 p.m., Lecture I, VV B239 | ||
+ | | | ||
+ | | | ||
+ | |[[#linktoabstract | Introduction to decoupling and Vinogradov's mean value problem ]] | ||
|- | |- | ||
− | |April 26, 4:00 p.m., | + | |April 26, 4:00 p.m., Lecture II, Chamberlin 2241 |
− | | | + | | |
− | | | + | | |
− | |[[#linktoabstract | | + | |[[#linktoabstract | Features of the proof of decoupling ]] |
|- | |- | ||
− | |April 27, 4:00 p.m., | + | |April 27, 4:00 p.m., Lecture III, VV B239 |
− | | | + | | |
− | | | + | | |
− | |[[#linktoabstract | | + | |[[#linktoabstract | Open problems ]] |
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|Shaoming and Andreas | |Shaoming and Andreas | ||
+ | |- | ||
+ | |Tuesday, November 8, | ||
+ | |Robert Fraser | ||
+ | |Wichita State University | ||
+ | |[[#linktoabstract | Title ]] | ||
+ | | Shaoming and Andreas | ||
|} | |} | ||
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https://people.math.wisc.edu/~seeger/seminar/Malabika-Analysis-Seminar-2022-Title-Abstract.pdf | 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=== | ===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 | We consider a uniformly elliptic operator $L_A$ in divergence form | ||
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This is a joint work with Alejandro Molero, Mihalis Mourgoglou, and | This is a joint work with Alejandro Molero, Mihalis Mourgoglou, and | ||
Xavier Tolsa. | 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]]= | =[[Previous_Analysis_seminars]]= |
Latest revision as of 04:00, 5 July 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 | 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 | |||||
Tuesday, November 8, | Robert Fraser | Wichita State University | Title | 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.
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