The regular time and place for the seminar is Tuesday at 3:00 – 4:00 pm in Math Tower (MW) 154. If you are interested, please contact Liding Yao and Valentin Kunz.
If you are interested to topics of mathematical analysis related to operator theory, take a look to the Analysis and Operator Theory Seminar.
| date | speaker | institution | title |
|---|---|---|---|
| Sep 10 | Joe Rosenblatt | UIUC | (On 2pm, at MW 152, joint with ergodic seminar) Approximate identity using singular measures |
| Sep 17 | Taeyong Ahn | Inha University (Korea) | Equidistribution of inverse images of analytic subsets for holomorphic endomorphisms of compact K\”ahler manifolds |
| Sep 24 | Song-Ying Li | UC Irvine | Supnorm Estimates for Solution of Cauchy-Riemann Equations |
| Oct 1 | Mayuresh Londhe | Indiana Bloomington | An effective version of Fekete’s theorem |
| Oct 8 | Yifan Jing | OSU | Sidon sets and sum-product phenomenon |
| Oct 15 | Valentin Kunz | OSU | Several Complex Variables and the Quarter-Plane Problem: An Overview |
| Nov 12 | John D’Angelo | UIUC | Positivity Conditions in Complex Analysis |
| Nov 14 | Armin Schikorra | Pittsburgh | (Joint with AOTS seminar) On Calderon-Zygmund theory for the p-Laplacian |
| Nov 19 | Tanya Firsova | Kansas State | Critical loci for automorphisms of C^2 |
| Dec 3 | Samantha Sandberg | OSU | Arithmetic Progressions in Fractal Sets of Sufficient Thickness |
| Dec 10 | Liding Yao | OSU | TBD |
| Jan 6 | David Cruz-Uribe | UAlabama | (Joint with AOTS seminar) The fine properties of Muckenhoupt weights in the variable Lebesgue spaces |
| Apr 1 | Nathan Wagner | Brown | TBD |
Joe Rosenblatt
Note: Joe’s talk is on 2-3 pm, and the location is MW 152.
Title: Approximate identity using singular measures
Abstract: We investigate the almost everywhere convergence of sequences of convolution operators given by probability measures $\mµ_n$ on $\Bbb R$. Assume that this sequence of operators constitutes an $L^p$-norm approximate identity for some $1\le p<\infty$. We ask, under what additional conditions do we have almost everywhere convergence for all $f\in L^p(\Bbb R)$.
We focus on the particular case of a sequence of contractions $C_{t_n}\mu$ of a single Borel probability measure $\mu$, with $t_n\to0$, so that that the sequence of operators is an $L^p$-norm approximate identity. If $\mu$ is discrete, then no sequence of such contractions can give a.e. convergence for all of $L^p(\Bbb R)$. If $\mu$ is absolutely continuous with respect to Lebesgue measure, then there is a sequence $(t_n)$ such that a.e. convergence holds on all of $L^1(\Bbb R)$.
But when the measure µ is continuous and singular to Lebesgue measure, obtaining a.e. results for some sequence $(t_n)$, is more challenging. Such results can always be obtained on $L^2(\Bbb R)$ when $\mu$ is a Rajchman measure. For non-Rajchman measures obtaining a.e. results on $L^2(\Bbb R)$ is sometimes possible, but not easy. In fact, it may be the case that there is a continuous, singular probability measure $\mu$ for which there is no sequence $(t_n)$ tending to zero with $C_{t_n}\mu\ast f\to f$ a.e., even just for all $f\in L^\infty(\Bbb R)$.
Taeyong Ahn
Title: Equidistribution of inverse images of analytic subsets for holomorphic endomorphisms of compact K\”ahler manifolds
Abstract: In this talk, we discuss the limit of inverse images of analytic subsets under a given holomorphic endomorphism of a compact K\”ahler manifold. We expect that the limit sits inside the Julia set of the given map in a reasonable sense under reasonable conditions. To this end, we will briefly talk about currents and superpotentials as methods. We will investigate a sufficient condition for this convergence. If time permits, we will discuss the difference between the complex projective space and a general compact K\”ahler manifold in this topic and in particular, the regularization of a positive closed currents and semi-regular transforms of currents.
Song-Ying Li
Title: Supnorm estimates for solution of Cauchy-Riemann equations
Abstract: In this talk, I will present some recent developments on the estimate of the solutions of Cauchy-Riemann equation $\overline\partial u=f$ on bounded domains $\Omega$ in $\mathbf C^n$ with non-smooth boundary, in particular for the product case $\Omega=D^n$ where $D\subset\mathbf C$. This include my recent result on the supnorm estimate when $D$ is has $C^{1,\alpha}$ boundary.
I will also present the most recent joint work with Long and Luo. Assuming that $D\subset\mathbf C$ is a bounded Lipschitz domain, we construct a new solution operator $Tf$ on $(0,1)$-form $f$ being $\overline\partial$-closed and $T:L^p_{(0,1)}(D^n)\to L^p(D^n)$ for $1<p\le\infty$.
Mayuresh Londhe
Title: An effective version of Fekete’s theorem
Abstract: A classical result of Fekete gives necessary conditions on a compact set in the complex plane so that it contains infinitely many sets of conjugate algebraic integers. In this talk, we discuss an effective version of Fekete’s theorem in terms of a height function. As an application, we give a lower bound on the growth of the leading coefficient of certain polynomial sequences, generalizing a result by Schur. Lastly, if time permits, we discuss an upper bound on minimal asymptotics of height over sequences of algebraic numbers. This talk is based on a joint work with Norm Levenberg.
Yifan Jing
Title: Sidon sets and sum-product phenomenon
Abstract: Given natural numbers s and k, we say that a finite set X of integers is an additive B_s[k] set if for any integer n, the number of solutions to the equation n = x_1+x_2+ … +x_s, with x_1, x_2, …, x_s lying in X, is at most k. Here we consider two such solutions to be the same if they differ only in the ordering of the summands. We define a multiplicative B_s[k] set analogously. These sets have been studied thoroughly from various different perspectives in combinatorial and additive number theory. In this talk, we will discuss sum-product phenomena for these sets. We show that there is either an exceptionally large additive B_s[k] set, or an exceptionally large multiplicative B_s[k] set. This is joint work with Akshat Mudgal.
Valentin Kunz
Title: Several Complex Variables and the Quarter-Plane Problem: An Overview
Abstract: The ‘quarter-plane problem’ refers to the boundary value problem which models the interaction of a monochromatic plane-wave, e.g., an acoustic pressure field, with the tip of an infinitely thin corner in three spatial dimensions, e.g., the tip of a turbofan blade. In this talk, we will outline how the theory of several complex variables can help us gain a better understanding of such physical phenomena. We are particularly interested in finding an asymptotic formula for the field observed at a great distance from the quarter-plane, which, in turn, requires knowledge of the singularity structure of some two-complex-variable spectral functions.
John D’Angelo
Title: Positivity Conditions in Complex Analysis
Abstract: This talk surveys some positivity conditions in Several Complex Variables and CR geometry by way of Hermitian symmetric functions. Most of the talk will be accessible to graduate students in all fields of mathematics. After discussing a decisive example in detail, we make some connections to Complex Geometry and to Kohn’s approach to subelliptic multipliers.
Armin Schikorra
Note: Armin’s talk is on Thursday 3-4pm, with usual location.
Title: On Calderon-Zygmund theory for the p-Laplacian
Abstract: I will discuss how to disprove a conjecture by Iwaniec from 1983 about Calderon-Zygmund estimates for L^r where r is close to p-1.
Tanya Firsova
Title: Critical loci for automorphisms of C^2
Abstract: For one-dimensional holomorphic maps, the dynamics of the map is largely determined by the orbits of the critical points. Automorphisms of C^2 are invertible and, as such, do not have critical points. Critical loci, as defined by E. Bedford, J. Smillie, and J. Hubbard, are the sets where dynamically defined foliations or laminations exhibit tangencies. They often serve as a good analog of the critical points. We’ll introduce critical loci within various dynamically significant regions, explaining their interactions and relationship to the system’s dynamics. We’ll describe the known topological models of the critical locus in the escape region. In particular, the model for complex Hénon maps in an HOV region, the first one developed in a non-perturbative setting. This is a joint work with R. Radu and R. Tanase.
Samantha Sandberg
Title: Arithmetic Progressions in Fractal Sets of Sufficient Thickness
Abstract: We consider the conditions required on a set that guarantee it contains arithmetic progressions. Szemeredi proved the existence of arithmetic progressions in subsets of the natural numbers with positive upper density. In the fractal setting, it is known by Maga and Keleti that full Hausdorff dimension is not enough to guarantee the existence of a 3-term arithmetic progression in subsets of d-dimensional Euclidean space; however, it turns out that Fourier decay coupled with nearly full Hausdorff dimension is sufficient for the existence of arithmetic progressions, as shown by Laba and Pramanik. In this talk, we consider another notion of size: Newhouse thickness. It is known that thickness larger than 1 is enough in the real line to guarantee the existence of a 3-term arithmetic progression. In higher dimensions, Yavicoli showed that it takes thickness larger than 10^8, along with some additional assumptions, to guarantee a 3-point configuration. We give the first result in higher dimensions showing the existence of 3-term arithmetic progressions in sets of thickness larger than 2/(1-2r), where r is a constant dependent on the set.
David Cruz-Uribe
Note: David’s talk is on Monday.
Title: The fine properties of Muckenhoupt weights in the variable Lebesgue spaces
Abstract: The theory of Muckenhoupt A_p weights, 1≤p<∞, has been an important area of harmonic analysis for more than 50 years. As part of this, a rich structure theory for these weights has been developed, including the reverse Hölder inequality, left-openness, Jones factorization, and Rubio de Francia extrapolation.
In the past 15 years this theory has been extended to the variable Lebesgue spaces, with the introduction of the A_{p(⋅)} weights by Cruz-Uribe, Fiorenza, and Neugebauer. Given an exponent function p(⋅), we say that a weight w is in A_{p(⋅)} if [w]_{A_{p(⋅)}}=\sup_Q |Q|^{-1}\|w\|_{L^{p(⋅)}(Q)}\|w^{-1}\|_{L^{p'(⋅)}(Q)} is finite.
These weights give sufficient (and in some cases necessary) conditions for maximal operators, singular integrals, and other operators of classical harmonic analysis to be bounded on the weighted variable Lebesgue space L^{p(⋅)}(w).
However, unlike in the constant exponent case, very little is known about the structural properties of A_{p(⋅)} weights, beyond a theory of extrapolation developed by Cruz-Uribe and Wang. In this talk I will discuss recent progress in this area, proving a reverse Hölder inequality for weights in A_{p(⋅)}. As an application, I will use it to prove left and right-openness of these weight classes, a result which we can also prove for matrix weights. If time permits I will also discuss other partial results on the structure of A_{p(⋅)} weights.
This research is joint with my PhD student, Michael Penrod.