Monday, June 20, 2022

2022-06-27 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: Radial projections in finite space

by Ben Lund(IBS 이산수학그룹)

Given a set $E$ and a point $y$ in a vector space over a finite field, the radial projection $\pi_y(E)$ of $E$ from $y$ is the set of lines that through $y$ and points of $E$. Clearly, $|pi_y(E)|$ is at most the minimum of the number of lines through $y$ and $|E|$. I will discuss several results on the general question: For how many points $y$ can $|\pi_y(E)|$ be much smaller than this maximum? This is motivated by an analogous question in fractal geometry. The Hausdorff dimension of a radial projection of a set $E$ in $n$ dimensional real space will typically be the minimum of $n-1$ and the Hausdorff dimension of $E$. Several recent papers by authors including Matilla, Orponen, Liu, Shmerikin, and Wang consider the question: How large can the set of points with small radial projections be? This body of work has several important applications, including recent progress on the Falconer distance conjecture. This is joint with Thang Pham and Vu Thi Huong Thu.

2022-06-23 / 14:00 ~ 15:30

SAARC 세미나 - SAARC 세미나:

by 지홍창()

In this talk, we discuss the fluctuation of f(X) as a matrix, where X is a large square random matrix with centered, independent, identically distributed entries and f is an analytic function. In particular, we show that for a generic deterministic matrix A of the same size as X, the trace of f(X)A is approximately Gaussian which decomposes into two independent modes corresponding to tracial and traceless parts of A. We also briefly discuss the proof that mainly relies on Hermitization of X and its resolvents.

2022-06-27 / 15:00 ~ 16:00

학과 세미나/콜로퀴엄 - 정수론: Effective joint equidistribution of rational points on expanding horospheres

by 이민(영국 브리스톨 대학)

In this talk, we study the behaviour of rational points on the expanding horospheres in the space of unimodular lattices. The equidistribution of these rational points is proved by Einsiedler, Mozes, Shah and Shapira (2016). Their proof uses techniques from homogeneous dynamics and relies particularly on measure-classification theorems due to Ratner. We pursue an alternative strategy based on Fourier analysis, Weil's bound for Kloosterman sums, recently proved bounds (by M. Erdélyi and Á. Tóth) for matrix Kloosterman sums, Roger's formula, and the spectral theory of automorphic functions. Our methods yield an effective estimate on the rate of convergence for a specific horospherical subgroup in any dimension. This is a joint work with D. El-Baz, B. Huang, J. Marklof and A. Strömbergsson.

Events for the 취소된 행사 포함