Time：April 12-14, 2025
Venue：E10-405
Tencent Meeting: 482 2369 4187
PASSCODE: 0412
4月12日
09:00-10:15
Title：Introduction to weighted restriction estimates
Speaker：Ruixiang Zhang, University of California, Berkeley, Online
Abstract：Weighted (Fourier) restriction estimates is ubiquitous in subjects such as analysis, number theory and geometric measure theory. We will use a few examples to introduce these estimates and applications and talk about progress on a few problems. We will also compare the problems to the classical Fourier restriction estimate and discuss their key connections and differences.
10:45-12:00
Title：Some Problems at the Interface Between Harmonic Analysis and Number Theory
Speaker：Xiaochun Li, University of Illinois Urbana-Champaign, Online
Abstract：In this talk, we explore a variety of problems that connect the fields of harmonic analysis and number theory. Some key examples include Waring’s problem, the Gauss circle problem, and the Dirichlet divisor problem. These problems, though primarily from number theory, are deeply linked to decoupling theory in harmonic analysis, which helps us gain new insights into their solutions.
14:00-15:15
Title：Burgess method and van der Corput method
Speaker：Ping Xi, Xi’an Jiaotong University
Abstract：In this talk, we introduce the classical Burgess method for incomplete character sums, and the van der Corput method for analytic/algebraic exponential sums. Estimates for such sums turn out to be very crucial in analytic number theory. We would like to show how the deep works of Weil, Deligne, Laumon, Katz, et al can be adapted to deal with such (incomplete) sums, and applied further to arithmetic problems.
15:45-17:00
Title：On the dimension of Besicovitch-Rado-Kinney sets
Speaker：Xianghong Chen, Sun Yat-sen University
Abstract：A set in Rⁿ (n≥2) is called a Besicovitch-Rado-Kinney (BRK) set if it contains a sphere of every radius between 1 and 2. Kolasa and Wolff showed that every BRK set has dimension n. They also gave quantitative lower bounds for the generalized Minkowski dimension of such sets. In this talk, we will provide examples of BRK set which match their lower bound when n=3, and examples with improved upper bounds in other dimensions. Time permitting, we will also talk about generalization to Lipschitz families of curved hypersurfaces. This is based on joint work with Tongou Yang (UCLA) and Yue Zhong (SYSU).
4月13日
09:00-10:15
Title：Weighted Fourier Extension Estimates
Speaker：Xiumin Du, Northwestern University, Online
Abstract：In this talk, we will survey recent results on weighted Fourier extension estimates and their variants. These estimates concern bounds of the Fourier extension operator on sets of fractal dimensions and have applications to various problems in PDEs and geometric measure theory, including the study of divergence set of Schrodinger solutions, the spherical average Fourier decay rates of fractal measures, and Falconer’s distance set problem.
10:45-12:00
Title：Concentration problems related to the eigenfunctions
Speaker：Chuanwei Gao, Capital Normal University
Abstract：This talk explores the concentration properties of eigenfunctions of the Beltrami-Laplace operator, with a focus on their L^p norms over manifolds and their restrictions to submanifolds. We begin by surveying existing results in this area, followed by a presentation of our recent findings. Specifically, we discuss an L^p-restriction estimate for eigenfunctions on fractal sets and an improved Kakeya-Nikodym estimate, highlighting their implications for the concentration phenomena.
14:00-15:15
Title：Spectral distribution for the twisted Laplacian on hyperbolic surfaces
Speaker：Long Jin, Tsinghua University
Abstract：In this talk, we discuss the spectrum of the twisted Laplacian operator on a compact hyperbolic surface. The twisted Laplacian associated with a harmonic form is obtained by conjugating the usual Laplacian-Beltrami operator by an integral of the harmonic form. It is also the Bochner Laplacian associated with the corresponding one-dimensional representation. When the harmonic form is real, the twisted Laplacian is non-self-adjoint but still has discrete spectrum in the complex plane. We will review its spectral theory and connection with the twisted Selberg zeta function. In particular, we show that although most of the spectrum is concentrated near the real axis, there are infinite many eigenvalues away from the real axis, at least when the harmonic form is large enough. This implies the failure of the asymptotic version of Riemann hypotheses for the twisted Selberg zeta function as well as the failure of quantum unique ergodicity. This is joint work with Gong Yulin.
15:45-17:00
Title：An incidence problem arising from homogeneous dynamics
Speaker：Weikun He, AMSS, CAS
Abstract：In this talk, I will describe a problem about incidence between balls and thin tubes. It arose in a study on quantitative equidistribution of random walks in homogeneous spaces. This problem is closely related to Bourgain’s discretized projection theorem. The talk is based on a joint work with Timothée Bénard.
4月14日
09:00-10:15
Title：Kakeya set in R^3
Speaker：Hong Wang, New York University, Online
Abstract：A Kakeya set is a compact subset of R^n that contains a unit line segment pointing in every direction. Kakeya set conjecture asserts that every Kakeya set in R^n has Hausdorff dimension n. We prove this conjecture in R^3 as a consequence of a more general statement about union of tubes.
This is joint work with Josh Zahl.
10:45-12:00
Title：Soliton resolution and scattering conjecture of wave equations
Speaker：Ruipeng Shen, Tianjin University
Abstract：Soliton resolution conjecture is one of the important open problem in the research field of dispersive and wave equations. In recent years many new mathematical tools and theories are developped during the study of this problem. This talk focuses on the wave equation case and discusses the thoery, ideas and methods related to this problem. This talk also covers some materials about a simplied version of this conjecture in the defocusing case, i.e. the scattering conjecture.
14:00-15:15
Title：Some progresses regarding eigenfunctions and eigenvalues
Speaker：Renjin Jiang, Capital Normal University
Abstract：Yau conjectured that on an n-dimensional smooth manifold, the (n-1)-Hausdorff measure of nodal set of the eigenfunction is equivalent to square root of the corresponding eigenvalue. The conjecture in the analytic case had been solved by Donnelly-Fefferman in 1988. The smooth case turns out to be difficult, the work of Hardt-Simon provides an exponential bound in term of \lambda. Recently Logunov made important progress, in particular, he proved the lower bound and provide a polynomial upper bound. Some previous progresses will also be discussed. If times allows, I will also report some results regarding eigenvalue estimates.
15:45-17:00
Title：The Uncertainty Principle: Exploring Its Many Facets
Speaker：Shanlin Huang, Sun Yat-sen University
Abstract：The uncertainty principle, a fundamental concept that captures the duality between physical space and frequency space, serves as a cornerstone in harmonic analysis. This talk aims to explore its diverse applications through three aspects:
(i) Unique continuation and control of solutions to the Schrodinger equation with potentials;
(ii) The Heisenberg uniqueness pair, a concept introduced by Hedenmalm and Montes-Rodriguez in 2011, along with its variants;
(iii) Dispersive estimates for magnetic Schrodinger equation in 2D.
