2019 봄학기 정수론 세미나
7월 19일 (금) 16:00~17:00, 27동 116호
Speaker : Peter Junho Whang (MIT)
Title : Diophantine analysis on moduli of local systems
Abstract : Moduli spaces of local systems on surfaces are widely studied in geometry. Focusing on the special linear rank two case, after motivating our Diophantine study we use mapping class group dynamics and differential geometric tools to establish a structure theorem for the integral points on the moduli spaces, generalizing work of Markoff (1880). We also give an effective analysis of integral points for nondegenerate algebraic curves on these spaces. Along the way, we present other related results connecting the geometry and arithmetic of the moduli spaces to elementary observations on surfaces
6월 14일 (금) 16:00~18:00, 27동 325호
Speaker : Kyoung-Seok Lee (IBS-CGP)
Title : Rational points and motives of moduli spaces of vector bundles on curves
Abstract : After the seminal work of Harder and Narasimhan, there have been intensive works studying relationships between rational points and cohomology groups of moduli spaces of vector bundles on curves. Being motivated by these works, people also have studied motives of moduli spaces of vector bundles on curves. In this talk, I will review these developments and discuss rational points and motives of moduli spaces of vector bundles on curves.
5월 31일 (금) 16:00~18:00, 27동 325호
Speaker : Jaebum Sohn (Yonsei University)
Title : Combinatorial proofs between some classes of partition functions
Abstract : In this talk, we introduce some basic concepts about partition functions which include the meaning of hook length and t-core. After that, I will give combinatorial proofs between some classes of partition functions.
5월 24일 (금) 16:00~17:00, 27동 325호
Speaker : Serin Hong (University of Michigan, Ann Arbor)
Title : Classification of quotient bundles over the Fargues-Fontaine curve
Abstract : Vector bundles on the Fargues-Fontaine curve play a pivotal role in recent development of p-adic Hodge theory and related fields, as they provide geometric interpretations of many constructions in these fields. The most striking example is the geometrization of the local Langlands correspondence due to Fargues where the correspondence is stated in terms of certain sheaves on the stack of vector bundles over the Fargues-Fontaine curve. In this talk, we give several classification theorems regarding vector bundles over the Fargues-Fontaine curve. Our main result is a complete classification of all quotient bundles of a given vector bundle. As its consequences, we also get a complete classification of globally generated vector bundles and a classification of almost all sub-bundles of a given bundle. Our proof is based on dimension counting of certain moduli spaces of bundle maps using Scholze's theory of diamonds.
5월 24일 (금) 17:00~18:00, 27동 325호
Speaker : Yong Suk Moon (Purdue University)
Title : p-adic analogue of Riemann-Hilbert correspondence
Abstract : We will first talk about the recent result of Diao-Lan-Liu-Zhu on the p-adic analogue of Riemann-Hilbert correspondence, and explain how it is linked with problems related to p-adic local Langlands, such as Fontaine-Mazur conjecture. Then we will talk about our joint work with Tong Liu proving that every relative crystalline representation with Hodge-Tate weights in [0, 1] arises from a p-divisible group if the ramification is small, and explain its application to studying the correspondence.
5월 3일 (금) 16:00~17:00, 27동 325호
Speaker : Bo-Hae Im (KAIST)
Title : Infinite rank of abelian varieties over certain infinite extensions
Abstract : In this talk, we will give some probabilistic result on the infinite multiplicity of irreducible subrepresentations of the free group coming from nontorsion points of abelian varieties.
4월 16일 (화) 16:00~18:00, 27동 325호
Speaker : Joachim König (KAIST)
Title : On the inverse Galois problem with ramification conditions
Abstract : The central problem in inverse Galois theory is to understand the structure of the absolute Galois group of a given field, such as $\mathbb{Q}$. It is generally expected that significant information about $G_{\mathbb{Q}}$ can be deduced from Galois theory over the local fields $\mathbb{Q}_p$, which is much better understood. A guiding principle is to construct, or count extensions of $\mathbb{Q}$ with a prescribed Galois group, and with prescribed local behaviour at some (finite or infinite!) set of primes. Specific instances of this approach lead to several problems of interest in number theory, such as minimal ramification problem, Grunwald problem, etc. In this talk I will focus on Galois realizations with conditions on the inertia subgroups.
I will discuss recent progress on two specific instances: Firstly, the construction of extensions with prescribed Galois group and "small" ramification indices. This is directly related to the construction of low-degree number fields with unramified $G$-extensions and leads to generalized Cohen-Lenstra heuristics.
Secondly, the construction of $G$-extensions with "powerfree" discriminant. This generalizes previous extensive investigations about fields with squarefree discriminants, corresponding to the special case $G=S_n$.
If time allows, I will also discuss some work in progress about Galois realizations with prescribed decomposition groups.
4월 12일 (금) 16:00~18:00, 27동 325호
Speaker : Yeonsu Kim (Chonnam National University)
Title : Classification of strongly positive representations of GU(n,n) and its application
Abstract : The classification of discrete series is one important subject with numerous application in the harmonic analysis and in the theory of automorphic forms. The classification consists of two steps, whose first step is the classification of strongly positive representations. With Ivan Matic (University of Osijek, Croatia), we obtain the first step, i.e., classification of strongly positive representations of a quasi-split similitude unitary groups GU(n, n) defined with respect to a quadratic extension of non-archimedean local fields. This is second result of the project "classification of discrete series of all similitude classical groups". If time permits, we are going to explain briefly the second step of the classification, which are in progress.
4월 5일 (금) 16:00~18:00, 27동 325호
Speaker : Jungwon Lee (UNIST)
Title : Dynamics of continued fractions and conjecture of Mazur-Rubin
Abstract : Mazur and Rubin established several conjectural statistics for modular symbols. We show that the conjecture holds on average. We plan to discuss the approach based on dynamical analysis of transfer operator associated to a certain skew-product Gauss map and subsequent result on mod p non-vanishing of modular L-values with Dirichlet twists (joint with Hae-Sang Sun).
3월 29일 (금) 17:00~18:00, 27동 325호
Speaker : Yoonbok Lee (Incheon National University)
Title : On the zeros of Epstein zeta functions near the critical line
3월 15일 (금) 16:00~18:00, 27동 325호
Speaker : Chan-Ho Kim (KIAS)
Title : A quantitative analysis of the level lowering congruences
Abstract : This is joint work in progress with Kazuto Ota. We discuss how much congruence ideals of Hecke algebras vary in various new quotients. (The first part will be more or less an ''RTG-style" expository talk.)