Difference between revisions of "Analysis Seminar"
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+ | 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 = | = Analysis Seminar Schedule = | ||
Line 9: | Line 15: | ||
!align="left" | host(s) | !align="left" | host(s) | ||
|- | |- | ||
− | | | + | |September 21, VV B139 |
+ | | Dóminique Kemp | ||
+ | | UW-Madison | ||
+ | |[[#Dóminique Kemp | Decoupling by way of approximation ]] | ||
+ | | | ||
+ | |- | ||
+ | |September 28, VV B139 | ||
+ | | Jack Burkart | ||
+ | | UW-Madison | ||
+ | |[[#Jack Burkart | Transcendental Julia Sets with Fractional Packing Dimension ]] | ||
+ | | | ||
+ | |- | ||
+ | |October 5, Online | ||
+ | | Giuseppe Negro | ||
+ | | University of Birmingham | ||
+ | |[[#Giuseppe Negro | Stability of sharp Fourier restriction to spheres ]] | ||
+ | | | ||
+ | |- | ||
+ | |October 12, VV B139 | ||
+ | |Rajula Srivastava | ||
+ | |UW Madison | ||
+ | |[[#Rajula Srivastava | Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups ]] | ||
+ | | | ||
+ | |- | ||
+ | |October 19, Online | ||
+ | |Itamar Oliveira | ||
+ | |Cornell University | ||
+ | |[[#Itamar Oliveira | A new approach to the Fourier extension problem for the paraboloid ]] | ||
+ | | | ||
+ | |- | ||
+ | |October 26, VV B139 | ||
+ | | Changkeun Oh | ||
+ | | UW Madison | ||
+ | |[[#Changkeun Oh | Decoupling inequalities for quadratic forms and beyond ]] | ||
+ | | | ||
+ | |- | ||
+ | |October 29, Colloquium | ||
+ | | Alexandru Ionescu | ||
+ | | Princeton University | ||
+ | |[[#Alexandru Ionescu | Polynomial averages and pointwise ergodic theorems on nilpotent groups]] | ||
+ | |- | ||
+ | |November 2, VV B139 | ||
+ | | Liding Yao | ||
+ | | UW Madison | ||
+ | |[[#Liding Yao | An In-depth Look of Rychkov's Universal Extension Operators for Lipschitz Domains ]] | ||
+ | | | ||
+ | |- | ||
+ | |November 9, VV B139 | ||
+ | | Lingxiao Zhang | ||
+ | | UW Madison | ||
+ | |[[#Lingxiao Zhang | Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition ]] | ||
+ | | | ||
+ | |- | ||
+ | |November 12, Colloquium | ||
+ | | Kasso Okoudjou | ||
+ | | Tufts University | ||
+ | |[[#Kasso Okoudjou | An exploration in analysis on fractals ]] | ||
+ | |- | ||
+ | |November 16, VV B139 | ||
+ | | Rahul Parhi | ||
+ | | UW Madison (EE) | ||
+ | |[[#Rahul Parhi | On BV Spaces, Splines, and Neural Networks ]] | ||
+ | | | ||
+ | |- | ||
+ | |November 30, VV B139 | ||
+ | | Alexei Poltoratski | ||
+ | | UW Madison | ||
+ | |[[#Alexei Poltoratski | Pointwise convergence for the scattering data and non-linear Fourier transform. ]] | ||
+ | | | ||
+ | |- | ||
+ | |December 7, Online | ||
+ | | John Green | ||
+ | | The University of Edinburgh | ||
+ | |[[#John Green | Estimates for oscillatory integrals via sublevel set estimates ]] | ||
+ | | | ||
+ | |- | ||
+ | |December 14, VV B139 | ||
+ | | Tao Mei | ||
+ | | Baylor University | ||
+ | |[[#Tao Mei | Fourier Multipliers on free groups ]] | ||
+ | | | ||
+ | |- | ||
+ | |February 8 | ||
| Person | | Person | ||
| Institution | | Institution | ||
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | | | + | | |
|- | |- | ||
− | | | + | |February 15 |
| Person | | Person | ||
| Institution | | Institution | ||
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | | | + | | |
|- | |- | ||
− | | | + | |February 22 |
+ | |Betsy Stovall | ||
+ | |UW Madison | ||
+ | |[[#linktoabstract | Title ]] | ||
+ | | | ||
+ | |- | ||
+ | |March 1 | ||
| Person | | Person | ||
| Institution | | Institution | ||
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | | | + | | |
|- | |- | ||
− | | | + | |March 8 |
+ | | Brian Street | ||
+ | | UW Madison | ||
+ | |[[#linktoabstract | Title ]] | ||
+ | | | ||
+ | |- | ||
+ | |March 15: No Seminar | ||
| Person | | Person | ||
| Institution | | Institution | ||
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | | | + | | |
|- | |- | ||
− | | | + | |March 23 |
| Person | | Person | ||
| Institution | | Institution | ||
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | | | + | | |
|- | |- | ||
− | | | + | |March 30 |
| Person | | Person | ||
| Institution | | Institution | ||
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | | | + | | |
|- | |- | ||
− | | | + | |April 5 |
| Person | | Person | ||
| Institution | | Institution | ||
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | | | + | | |
|- | |- | ||
− | | | + | |April 12 |
| Person | | Person | ||
| Institution | | Institution | ||
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | | | + | | |
|- | |- | ||
− | | | + | |April 19 |
− | | | + | | No seminar |
− | | | + | | |
+ | | | ||
+ | | | ||
+ | |- | ||
+ | |April 22, Colloquium | ||
+ | |Detlef Müller | ||
+ | |University of Kiel | ||
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | | | + | | |
|- | |- | ||
− | | | + | |April 25, 4:00 p.m., Distinguished Lecture Series |
− | | | + | |Larry Guth |
− | | | + | |MIT |
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | |||
|- | |- | ||
− | | | + | |April 26, 4:00 p.m., Distinguished Lecture Series |
− | | | + | |Larry Guth |
− | | | + | |MIT |
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | |||
|- | |- | ||
− | | | + | |April 27, 4:00 p.m., Distinguished Lecture Series |
− | | | + | |Larry Guth |
− | | | + | |MIT |
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | | | + | | |
+ | |- | ||
+ | | | ||
+ | | | ||
+ | | | ||
+ | | | ||
+ | | | ||
+ | | | ||
+ | |- | ||
+ | |Talks in the Fall semester 2022: | ||
|- | |- | ||
− | | | + | |September 20, PDE and Analysis Seminar |
− | | | + | |Andrej Zlatoš |
− | | | + | |UCSD |
|[[#linktoabstract | Title ]] | |[[#linktoabstract | Title ]] | ||
− | | | + | | Hung Tran |
|- | |- | ||
+ | |||
|} | |} | ||
=Abstracts= | =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). | ||
+ | |||
+ | =[[Previous_Analysis_seminars]]= | ||
+ | |||
+ | https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars | ||
=Extras= | =Extras= | ||
+ | |||
[[Blank Analysis Seminar Template]] | [[Blank Analysis Seminar Template]] | ||
+ | |||
+ | |||
+ | Graduate Student Seminar: | ||
+ | |||
+ | https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html |
Latest revision as of 22:23, 27 November 2021
The 2021-2022 Analysis Seminar will be organized by David Beltran and Andreas Seeger. Some of the talks will be in person (room Van Vleck B139) and some will be online. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).
Zoom links will be sent to those who have signed up for the Analysis Seminar List. If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu. If you are from an institution different than UW-Madison, please send as well as an additional email to David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request.
If you'd like to suggest speakers for the fall semester please contact David and Andreas.
Analysis Seminar Schedule
date | speaker | institution | title | host(s) | |
---|---|---|---|---|---|
September 21, VV B139 | Dóminique Kemp | UW-Madison | Decoupling by way of approximation | ||
September 28, VV B139 | Jack Burkart | UW-Madison | Transcendental Julia Sets with Fractional Packing Dimension | ||
October 5, Online | Giuseppe Negro | University of Birmingham | Stability of sharp Fourier restriction to spheres | ||
October 12, VV B139 | Rajula Srivastava | UW Madison | Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups | ||
October 19, Online | Itamar Oliveira | Cornell University | A new approach to the Fourier extension problem for the paraboloid | ||
October 26, VV B139 | Changkeun Oh | UW Madison | Decoupling inequalities for quadratic forms and beyond | ||
October 29, Colloquium | 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 | 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 | ||
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 | ||
February 8 | Person | Institution | Title | ||
February 15 | Person | Institution | Title | ||
February 22 | Betsy Stovall | UW Madison | Title | ||
March 1 | Person | Institution | Title | ||
March 8 | Brian Street | UW Madison | Title | ||
March 15: No Seminar | Person | Institution | Title | ||
March 23 | Person | Institution | Title | ||
March 30 | Person | Institution | Title | ||
April 5 | Person | Institution | Title | ||
April 12 | Person | Institution | Title | ||
April 19 | No seminar | ||||
April 22, Colloquium | Detlef Müller | University of Kiel | Title | ||
April 25, 4:00 p.m., Distinguished Lecture Series | Larry Guth | MIT | Title | ||
April 26, 4:00 p.m., Distinguished Lecture Series | Larry Guth | MIT | Title | ||
April 27, 4:00 p.m., Distinguished Lecture Series | Larry Guth | MIT | Title | ||
Talks in the Fall semester 2022: | |||||
September 20, PDE and Analysis Seminar | Andrej Zlatoš | UCSD | Title | Hung Tran |
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).
Previous_Analysis_seminars
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars
Extras
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Graduate Student Seminar:
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html