Abstracts

We introduce the Riemann zeta function and proves its analytic properties. After that, we prove the Prime Number Theorem.

강연 자료 (1/30, 1/31)

We introduce Montgomery's 1974 paper on the pair correlation of zeros of the Riemann zeta function. This is the beginning of a new field of study, statistical understanding of the zero distribution via the random matrix theory.

강연 자료

We introduce Katz and Sarnak’s conjecture on low-lying zeros of L-functions in a nice family. After that, we explain what we have to calculate to compute one-level density. Lastly, we will review what the known results are.

Oppenheim conjecture is the problem asking the condition of a quadratic form Q having arbitrarily small value Q(v), where v is chosen in the integral lattice. The conjecture was completely proved by Margulis showing that values of a quadratic form at integer points are dense in the real line if and only if the form is irrational under the condition that the form is non-degenerate, indefinite, and the rank is at least 3. In this talk, I will introduce some historical results related to this topic, so-called Oppenheim conjecture-typed problem, and how homogeneous dynamics contributes. We are particularly interested in studies about the distribution of joint values of integral vectors under a pair of a quadratic form and a linear form. This is joint work with Seonhee Lim and Keivan Mallahi-Karai.

In 1992, Claudia Spiro asked which set E determines an arithmetic function f uniquely in some set S of arithmetic functions under the condition f(a+b)=f(a)+f(b) for all a, b in E and she called E an additive uniqueness set for S. That is, she showed that the set of primes is an additive uniqueness set for multiplicative functions which do not vanish at some prime numbers. Since her intriguing result, many mathematicians have been solving enormously various problems related to Spiro’s paper. In this talk, we introduce several examples of additive uniqueness sets and related results.

강연 자료

We study the distribution of eigenvalues of random matrices. In particular, we show how to compute n-level density for unitary matrices and compare it with n-level density for low-lying zeros of primitive Dirichlet L-functions.

강연 자료

As an application of low-lying zeros of L-function, we explain how to bound on the average analytic rank of elliptic curves.

강연 자료

There is a well-known way of constructing a 1-dimensional rational distribution from circular units, called a circular distribution. We generalize this construction to a higher dimensional case, i.e. we construct an isomorphism between distributions on an n-dimensional rational vector space V and the n-th Milnor K-group of a circular algebra on V. Using this construction, we explicitly construct a GL_n-modular symbol with values in the n-th Milnor K-group-valued distributions on V, which we call the universal Dedekind-Eisenstein modular symbol. The logarithmic derivative of this modular symbol gives rise to an Eisenstein cocycle which contains key informations on periods of Eisenstein series and consequently the zeta values of totally real fields of degree n. The talk is based on the joint work with Glenn Stevens.

강연 자료

In this talk, we will review dynamics of SL2-action on hyperbolic spaces. We will then introduce Hurwitz complex continued fractions and dynamical properties of the complex Gauss map, in particular the finite range property. (Part of the talk is based on joint work with Jungwon Lee and Dohyeong Kim.)