March 20 (Fri) 2026, 10:30 ~ 11:30 (Asan science building 525)
Title: Markoff and Lagrange spectra for Hecke groups
Speaker: Byungchul Cha (Hongik University)
Abstract:
For an integer $q\ge 3$, \emph{the Hecke group $G_q$ of index $q$} is the subgroup generated by $z\mapsto -1/z$ and $z\mapsto z + 2\cos(\pi/q)$ of the orientation-preserving isometry group of the upper half plane. We study the structure of Markoff and Lagrange spectra of $G_q$. In particular, we characterize the initial discrete parts of these spectra. This mirrors classical results on the Markoff and Lagrange spectra of the real line. This is joint work with Dong Han Kim.
April 10 (Fri) 2026, 11:00 ~ 12:00 (Asan science building 525)
Title: Goldfeld conjecture for genus 2 curves
Speaker: Keunyoung Jeong (Chonnam National University)
Abstract:
Goldfeld's conjecture predicts that the average rank in the family of quadratic twists of an elliptic curve over the rationals is 1/2. Naive analogues of Goldfeld's conjecture have also been studied for elliptic curves over number fields and for genus 2 curves, together with the phenomenon of disparity. However, these analogues have mainly been considered in the setting of hyperelliptic twists. In this talk, we investigate a non-hyperelliptic analogue of Goldfeld's conjecture for a genus 2 curve.
May 29 (Fri) 2026, 10:30 ~ 11:30 (Asan science building 525)
Title: Serre weight conjectures and modularity lifting for GSp4
Speaker: Heejong Lee (KIAS)
Abstract:
Given a Galois representation attached to a regular algebraic cuspidal automorphic representation, the Hodge--Tate weight of the Galois representation is matched with the weight of the automorphic representation. Serre weight conjectures are mod p analogue of such a correspondence, relating ramification at p of a mod p Galois representation and Serre weights of mod p algebraic automorphic forms. In this talk, I will discuss how to understand Serre weight conjectures and modularity lifting as a relationship between representation theory of finite groups of Lie type (e.g. GSp4(Fp)) and the geometry of p-adic local Galois representations. Then I will explain the proof idea in the case of GSp4. This is based on a joint work with Daniel Le and Bao V. Le Hung.
June 5 (Fri) 2026, 11:30 ~ 12:00 (Asan science building 525)
Title: Dimensions of spaces of modular forms
Speaker: Min Lee (University of Bristol)
Abstract:
Martin conjectured that every non-negative integer can be expressed as the dimension of the space of newforms of weight 2 and level N for some positive integer N. Recently, Ross disproved this conjecture by showing that 67846 is not attained in this way, and proposed a counter-conjecture that the set of such dimensions has density zero among the non-negative integers. We prove a general form of Ross’ conjecture. This is joint work with Andrew R. Booker.
June 8 (Mon) 2026, 14:00 ~ 15:00 (Asan science building 526)
Title: Mathematics, AI, and Formalization
Speaker: Seewoo Lee (UC Berkeley)
Abstract:
AI is now impacting mathematics through tools for generating conjectures, searching examples, proving theorems, and formalizing theorems, but it is often unclear what “AI doing mathematics” actually means. This talk surveys recent developments and uses examples to distinguish these modes of assistance, emphasizing current capabilities, limitations, and practical workflows.