프로그램

문용석

제목: Introduction to p-adic Hodge theory

초록: TBA

최도훈

제목: 대각합 공식과 그 응용

초록: 대학합 공식 (trace formula)는 자기 동형 표현 (automorphic representation)을 연구하는 중요한 방법 중 하나이다. 이 강의에서는 기본적으로 대각합 공식을 소개하고. 대각합 공식은 그 자체로 건조한 느낌을 받기가 쉬워 그 응용으로 Jacquet-Langlands 대응의 증명에 대한 그 응용도 간략히 소개하겠다. 세부적으로 다음의 내용을 다루고자 한다.

1. 코콤팩트인 경우 대각합 공식; 사원수 (quaternion algebra) 경우

2. 코콤팩트가 아닌 경우; GL(2)

3.  Jacquet-Langlands 대응의 배경과 증명의 간략한 소개

(*이 강의는 p-진수와 유리수의 아델환에 관한 기본적은 내용을 사전 지식으로 가정하고 진행된다.)

유화종

제목: Ogg's torsion conjecture

초록: 두 시간에 걸쳐 Andrew Ogg의 torsion conjecture를 소개하고 Mazur의 증명의 개요를 살펴본다.

유명준

제목: The distribution of the cokernel of a polynomial of a random p-adic matrix

초록: A random matrix plays an important role in number theory. For example, the distribution of zeros of the Riemann zeta function (of some family of L-functions) is the same as the distribution of eigenvalues of certain random matrices. Also, Friedman--Washington computed the distribution of cokernel of a random p-adic matrix and showed it is the same as Cohen-Lenstra distribution. Recently, Wood generalized the work of Friedman--Washington considering a much more general class of measure on p-adic matrices. In this talk, we discuss the distribution of the cokernel of a polynomial of a random p-adic matrix. This is joint work with Gilyoung Cheong.  

강한빈

Title: Introduction to Arithmetic Topology

Abstract: There is an analgoue between number theory and 3-dimensional topology. We introduce the dictionary between these theories. In particular, we observe the relation between Iwasawa theory and Alexander polynomial.

오준영

Title: Distribution of low-lying zeros of a certain family of L-functions.

Abstract: We investigate the one-level density of low-lying zeros of a certain family of L-function associated with elliptic curves.

신은주

Title: Decidability for the equivalence of integral solutions of character varieties on a four-punctured sphere

Abstract: In this talk, we will consider integral solutions of character varieties on a four-punctured sphere. It is well known that the mapping class group acts on these solutions. We will then introduce trees in a one-to-one correspondence with mapping class group orbits, and finally prove that it is decidable whether or not any two integral solutions are in the same mapping class group orbit.

강민찬

Title: The Proportion of Monogenic Orders of Prime Power Indices of a Pure Cubic Field

Abstract: An order is said to be monogenic if it is generated by a single element as an algebra over the ring of integers. The monogeniety of an order can be analyzed using the index form, a homogeneous form induced by an integral basis of the order. In this talk, we investigate the proportion of monogenic orders among the orders whose indices are a power of a fixed prime in a pure cubic field. We begin by counting the number of orders whose indices are a power of a fixed prime. Then, we express the index form of an order in terms of the index form of the ring of integers and apply the finiteness result on the number of primitive solutions of a Thue-Mahler equation to obtain an upper bound on the number of monogenic orders.

윤종흔

Title: Primitively 2-universal senary integral quadratic forms

Abstract: For a positive integer n, a positive definite integral quadratic form is called primitively n-universal if it primitively represents all positive definite integral quadratic forms of rank n. Ju, Kim, Kim, Kim, and Oh proved that there are exactly 107 primitively 1-universal quaternary quadratic forms up to isometry. In this talk, we prove that the minimal rank of a primitively 2-universal integral quadratic form is six, and we prove that there are exactly 201 primitively 2-universal senary integral quadratic forms up to isometry. This is joint work with Prof. Byeong-Kweon Oh.

김민규

Title: Representations by a sum of generalized m-gonal numbers.

Abstract: For m>=3, let P_m(x)=((m-2)x^2-(m-4)x)/2. A generalized m-gonal number is an integer of the form P_m(s) for some integer s. We call a quadratic polynomial a_1P_m(x_1)+a_2P_m(x_2)+\cdots+a_kP_m(x_k) (a_i \in N) a sum of generalized m-gonal numbers. In this talk, we discuss several properties on the representation of an integer by sums of generalized m-gonal numbers.