Spring 2023 Analysis Seminar: Difference between revisions
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Abstract: Let $A\subset \mathbb{F}^3_p$ with $\# A=p^a$. For any direction $\theta$ in $\mathbb{F}^3_p$, define $\pi_{\theta}(A)$ to be the set of lines in direction $\theta$ and passing through $A$. Define the exceptional set $E_s(A):=\{\theta: \# \pi_\theta (A)<p^s \}$. Falconer-type estimate gives $\# E_s(A)\lesssim p^{2+s-a} $. I will talk about a new result: If $s<\frac{a+1}{2}$, then $\# E_s(A)\lesssim p^{2+2s-2a}$. | Abstract: Let $A\subset \mathbb{F}^3_p$ with $\# A=p^a$. For any direction $\theta$ in $\mathbb{F}^3_p$, define $\pi_{\theta}(A)$ to be the set of lines in direction $\theta$ and passing through $A$. Define the exceptional set $E_s(A):=\{\theta: \# \pi_\theta (A)<p^s \}$. Falconer-type estimate gives $\# E_s(A)\lesssim p^{2+s-a} $. I will talk about a new result: If $s<\frac{a+1}{2}$, then $\# E_s(A)\lesssim p^{2+2s-2a}$. | ||
===[[Yuqiu Fu]]=== | |||
Title: Incidence estimates for tubes and balls with dimensional spacing condition in R^2. | |||
Abstract: We will discuss essentially sharp incidence estimates in R^2 for a collection of tubes of dimension \delta \times 1 and a collection of balls of radius \delta, which satisfy some dimensional spacing condition. An application of these estimates is a new lower bound on the Hausdorff dimension of a (s,t) – Furstenberg set in R^2 when t > 1 + \epsilon(s,t) and s + t/2 \geq 1, where \epsilon is small depending on (s,t). This is joint work with Kevin Ren. | |||
Revision as of 16:07, 28 February 2023
Organizer: Shaoming Guo
Email: shaomingguo (at) math (dot) wisc (dot) edu
Time: Tuesdays, 4-5pm
Room: Van Vleck B139
All talks will be in-person unless otherwise specified.
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+join (at) g-groups (dot) wisc (dot) edu
| Date | Speaker | Institution | Title | Host(s) |
|---|---|---|---|---|
| Jan. 24 | ||||
| Jan. 31 | ||||
| Feb. 7 | Shaoming Guo | UW Madison | Hörmander's generalization of the Fourier restriction problem | Analysis group |
| Feb. 14 | Diogo Oliveira e Silva | Instituto Superior Técnico (Lisboa) | The Stein-Tomas inequality: three recent improvements | Betsy Stovall, Andreas Seeger |
| Feb. 21 | Jack Burkart | UW Madison | Sobolev Spaces for General Metric Spaces | Analysis group |
| Feb. 28 | Shengwen Gan | MIT | Exceptional set estimates in finite field | Analysis group |
| Mar. 7 | Yuqiu Fu | MIT | Incidence estimates for tubes and balls with dimensional spacing condition in R^2. | Zane Li |
| Mar. 14 | Spring break | |||
| Mar. 21 | Zhiren Wang | Penn State | Shaoming Guo, Chenxi Wu | |
| Mar. 28 | ||||
| Apr. 4 | Liding Yao | Ohio State | Brian Street | |
| Apr. 11 | Dominique Maldague | MIT | Betsy Stovall, Andreas Seeger | |
| Apr. 18 | David Beltran | Universitat de València. | Andreas Seeger | |
| Apr. 25 | Herve Gaussier | Institut Fourier | Xianghong Gong, Andy Zimmer | |
| May 2 | Lisa Naples | Macalester College | Jack Burkart |
Abstracts
Shaoming Guo
Title: Hormander's generalization of the Fourier restriction problem
Abstract: Hörmander 1973 proposed to study a generalized Fourier extension operator, and asked whether the generalized operator satisfies the same L^p bounds as that of the standard Fourier extension operator. Surprisingly, Bourgain 1991 gave a negative answer to Hörmander’s question. In this talk, I will discuss a modification of Hörmander’s question whose answer may be affirmative. This is a joint work with Hong Wang and Ruixiang Zhang.
Diogo Oliveira e Silva
Title: The Stein-Tomas inequality: three recent improvements
Abstract: The Stein-Tomas inequality dates back to 1975 and is a cornerstone of Fourier restriction theory. Despite its respectable age, it is a fertile ground for current research. The goal of this talk is three-fold: we present a recent proof of the sharp endpoint Stein-Tomas inequality in three space dimensions; we present a variational refinement and withdraw some consequences; and we discuss how to improve the Stein-Tomas inequality in the presence of certain symmetries.
Jack Burkart
Title: Sobolev Spaces for General Metric Spaces
Abstract: Sobolev spaces are classically defined in Euclidean space as L^p functions possessing weak derivatives (of some order). Recently, there has been interest in doing analysis and developing a theory of calculus on general metric spaces. A natural question one might ask is how can one define Sobolev spaces in an arbitrary metric space? In this talk, I'll discuss some ways we can generalize concepts like the Poincare inequalty to an arbitrary metric space and showcase some alternative definitions that can be used in more general settings. After discussing some known results in this area, I'll spend the latter part of the talk discussing some of my own ongoing research involving establishing Poincare inequalities in domains in Euclidean space that are not necessarily W^{1,p} extension domains and some other questions we are currently considering. This talk features joint and ongoing work with Ryan Alvarado, Lisa Naples, and Benham Esmayli.
Shengwen Gan
Title: Exceptional set estimates in finite field
Abstract: Let $A\subset \mathbb{F}^3_p$ with $\# A=p^a$. For any direction $\theta$ in $\mathbb{F}^3_p$, define $\pi_{\theta}(A)$ to be the set of lines in direction $\theta$ and passing through $A$. Define the exceptional set $E_s(A):=\{\theta: \# \pi_\theta (A)<p^s \}$. Falconer-type estimate gives $\# E_s(A)\lesssim p^{2+s-a} $. I will talk about a new result: If $s<\frac{a+1}{2}$, then $\# E_s(A)\lesssim p^{2+2s-2a}$.
Yuqiu Fu
Title: Incidence estimates for tubes and balls with dimensional spacing condition in R^2.
Abstract: We will discuss essentially sharp incidence estimates in R^2 for a collection of tubes of dimension \delta \times 1 and a collection of balls of radius \delta, which satisfy some dimensional spacing condition. An application of these estimates is a new lower bound on the Hausdorff dimension of a (s,t) – Furstenberg set in R^2 when t > 1 + \epsilon(s,t) and s + t/2 \geq 1, where \epsilon is small depending on (s,t). This is joint work with Kevin Ren.
[1] Previous Analysis Seminars
[2] Fall 2022 Analysis Seminar