강연 제목과 초록

마지막 업데이트 : 2020/02/09 17:30

권재훈(서울대학교)

Title : Quantum groups and canonical basis

Abstract : In this talk we give a brief introduction to canonical basis of representations of quantum groups. We focus on the quantum group of type A_\infty and explain how the theory of (dual) canonical basis is related to the characters problem of simple modules in a certain module category of simple Lie algebra of type A_n.

김경민(성균관대학교)

Title : Quadratic forms with a strong regularity property on the representations of squares

Abstract :

A (positive definite and non-classic integral) quadratic form is called strongly s-regular if it satisfies a strong regularity property on the number of representations of squares of integers. In this talk, we prove that for any integer k≥2, there are only finitely many isometry classes of strongly s-regular quadratic forms with rank k if the minimum of the nonzero squares that are represented by them is fixed. This is a joint work with Byeong-Kweon Oh.

김대준(서울대학교)

Title : Representations of finite number of quadratic forms with same rank

Abstract : In this talk, we introduce some results pertaining to the existence of a quadratic form of rank n representing any k quadratic forms of rank m given in advance, where k,m,n are positive integers with m≤n.

We also discuss some related topics. This is a joint work with professor Byeong-Kweon Oh.

김도형(서울대학교)

Title : Selmer objects, and rational points on curves

Abstract : We will review the notion of Selmer groups. There are two equivalent formulations; one giving rise to a discrete object while the other a compact one. They are related by the Pontryagin duality. Their role in the arithmetic study of elliptic curves will be recalled, and then we will move on to the study of rational points on hyperbolic curves. Some basic facts from Lie theory will be covered and applied to the construction of Selmer sets which generalize Selmer groups.

김민규(성균관대학교)

Title : Diagonal odd regular ternary quadratic forms

Abstract : A (positive definite primitive integral) quadratic form is called odd regular if it represents every odd positive integer which is locally represented. In this talk, we prove that there are at most 147 diagonal odd regular ternary quadratic forms and prove the odd regularity of all but 6 candidates.

선해상(울산과학기술원)

Title : Algebraic independence of the Kubota-Leopoldt L-functions

Abstract : The Kubota-Leopoldt p-adic L-function is regarded as a p-adic avatar of the Dirichlet L-function, in that it shares various analogous properties for special values. For example, the Kronecker limit formula holds for both complex and p-adic L-functions. I am going to give an expository introduction to another interesting topic, namely algebraic independence of the L-functions. The complex version is a consequence of universality of the functions. In the lecture, I will discuss algebraic independence of the p-adic L-functions including the mod p reduction of the p-adic L-functions.

오병권(서울대학교)

Title : Representations of quadratic forms with low ranks

Abstract :

In this talk, we present some results on the representations of quadratic forms with lower rank.

In particular, we focus on the representations of quadratic forms of rank 2 and 3 by quadratic forms.

We explain some conjectures related with this subject.

우르노아끼또(서울대학교)

Title : Representations of symmetric groups

Abstract : In this talk, we review the new approach to the theory of complex representations of the finite symmetric groups. We will obtain the following well-known results: labeling the irreducible -modules by partitions of n, construction of Young's orthogonal bases in irreducible modules, explicit description of matrices of simple transpositions with respect to these bases, and the Murnaghan-Nakayama formula for irreducible characters.

유명준(고등과학원)

Title : Introduction to the theory of elliptic curves

Abstract : The importance of the theory elliptic curves in number theory can not be overemphasized. For example, Andrew Wiles proved Fermat's last theorem by showing a certain property of elliptic curves. In my two lectures, I will explain the basic theory of elliptic curves. Especially, the Mordell-Weil theorem and the Selmer groups will be covered.

유정우(서울대학교)

Title : Integral representations and cyclotomic Hecke algebras

Abstract : In this talk, i will consider only integral representations of $H_n$ and explain the reason why we can restrict ourselves to the category. Moreover, I will introduce the two main theorems: Basis theorem and Mackey theorem.

유화종(서울대학교)

Title : Various congruence formula for Ramanujan's tau function

Abstract :

In this two talks, we introduce all known congruence formula for Ramanujan's tau function.

We exploit the hidden idea for such congruence formula: reducible Galois representation.

We then study various properties on the Galois representations associated to Ramanujan's delta function.

이계식(한경대학교)

Title : 파이썬을 활용한 이차형식의 표현에 관한 다양한 프로그램

Abstract : 이 강의에서는 이차형식의 표현과 관련한 다양한 파이썬 기반 프로그램을 소개한다. 또한 이러한 프로그램을 사용하는 방법에 대하여 안내하며, 이차형식의 연구와 관련한 프로그램을 어떻게 제작할 수 있는지 소개한다.

이신명(서울대학교)

Title : Representations of symmetric groups in characteristic p: from Hecke algebras' point of view

Abstract : Besides the categorification of the quantum group of type A at q=1, representations of Hecke algebras describe the behavior at q=0 as well. In other words, irreducible representations equipped with the crystal operators realize the Kashiwara's crystal $B(\infty)$ and $B(\lambda)$. Specifically focusing on $B(\Lambda_0)$, we recover the representation theory of the symmetric groups in any characteristic p. This crystal-governing approach would provide us a deep understanding of the recursive nature of the theory.

이정원(서울대학교)

Title : Minimal S-universality criterion sets

Abstract : We introduce definition and various properties on minimal S-universality criterion sets. When S is a subset of positive integers, we show that the minimal S-universality criterion set is unique. For higher rank cases, we prove that a minimal S-universality criterion set is not unique when S is the set of all quadratic forms of rank n with n ≥ 9. We also define a recoverable Z-lattice and provide some necessary conditions, and some sufficient conditions for Z-lattices to be recoverable. This is a joint work with Byeong-Kweon Oh.

이현세(서울대학교)

Title : Degenerate affine Hecke algebra and characters

Abstract : We will review basic results on degenerate affine Hecke algebra and its representations. Basis theorem, center of the algebra and Mackey type filtration will be given. Among these results, we focus on Mackey thorem which allow us calculate formal characters of representations. Kato modules and its structure theorem will be given as fundemental representations of the algebra.

장일승(서울대학교)

Title : Construction of the positive part of universal enveloping algebra using Hecke algebras

Abstract : In this talk, I will explain a construction of the positive part of the universal enveloping algebra for the affine Kac-Moody algebra of type $A_{\ell}^{(1)}$ (or type $A_{\infty}$) in terms of integrable representations of affine Hecke algebras

주장원(울산대학교)

Title : Ternary quadratic forms representing same integers

Abstract : In 1997, Kaplansky conjectured that if two positive definite ternary quadratic forms with integer coefficients have perfectly identical integral representations, then they are isometric, both regular, or included either of two families of ternary quadratic forms. In this talk, we prove the existence of pairs of ternary quadratic forms representing same integers which are not in the Kaplansky's list.

최도훈(고려대학교)

Title : Introduction to adelic trace formula

Abstract : I will give two introductory lectures on adelic trace formula.

허태혁(서울대학교)

Title : Restriction and induction functors

Abstract : In this talk, we define the analogues of $e_i, f_i, \varepsilon_i$, and $\varphi_i$ and show their properties. They depend on the cyclotomic Hecke algebra and we will describe the meaning of those in terms of crystal theory.

황병학(서울대학교)

Title : Crystal operators and character calculations

Abstract : In this talk, we construct crystal operators on representations of the affine Hecke algebra. Also we present a useful tool for studying irreducible modules, with small rank examples.