이정인 (아주대)
제목: Distribution of the cokernels of random p-adic matrices
초록: Motivated by the Cohen-Lenstra heuristics, Friedman and Washington studied the distribution of the cokernels of random matrices over the ring of p-adic integers. This has been generalized in many directions, as well as some applications to the distribution of random algebraic objects.
In the first talk, we give an overview of random matrix theory over the ring of p-adic integers, together with their connections to conjectures in number theory. After that, we provide universality results on the distribution of the cokernels of random p-adic matrices.
In the second talk, we introduce the moment problem (uniqueness, robustness) of random abelian groups and its applications to the universality of random p-adic matrices. We also investigate the distribution of the cokernels of random p-adic matrices with given zero entries, which is based on work in progress with Gilyoung Cheong, Dong Yeap Kang and Myungjun Yu.
김지구 (연세대)
제목: Genus character L-functions of orders in quadratic function fields
초록: In this talk, we define general genus characters and related L-functions for orders in quadratic function fields. Then we give a formula on the product of class numbers of two quadratic function fields in terms of continued fraction expansions of representatives for the class group of an order in a real quadratic function field.
신규철 (성균관대)
제목: Multiplicative Hecke operators and their applications
초록: Borcherds, a Fields Medalist, established an isomorphism between the multiplicative group of meromorphic modular forms of integral weight on Γ0(1) with some conditions and the space of modular forms of weight 1/2 on Γ0(4) satisfying Kohnen plus condition. Further, he conjectured that this isomorphism commutes with the action of Hecke operators. In this talk, we first define multiplicative Hecke operators for all positive integers which act on the multiplicative group of meromorphic modular forms on Γ0(N). Next, we provide some properties about these operators. Moreover, we answer the question posed by Borcherds using multiplicative Hecke operators. This is joint work with Chang Heon Kim.
이경준 (연세대)
제목: Quotients of numerical semigroups and their genus
초록: Finding the Frobenius number and the genus of any numerical semigroup S is a well-known open problem. Similarly, it has been studied to express the Frobenius number and the genus of a quotient of a numerical semigroup. In this talk, by computing the Hilbert series of each type of numerical semigroup, we talk about an expression for the genus of a quotient of numerical semigroups generated by one of the following series: arithmetic progression, geometric series, and Pythagorean triple.
이영민 (KIAS)
제목: The number of Maass forms with exceptional eigenvalue
초록: Let $f$ be a Maass form on a congruence subgroup and $\lambda_f$ be the Laplace eigenvalue of $f$. In 1965, Selberg asserted that $\lambda_f$ is larger than or equal to $\frac{1}{4}$, which is called the Selberg eigenvalue conjecture. If $\lambda_f$ is less than $\frac{1}{4}$, then we call $\lambda_f$ an exceptional eigenvalue. In this talk, we will introduce the result of the upper bound for the number of Maass forms with exceptional eigenvalue. Moreover, by using the relation between Maass forms and automorphic representations of $\mathrm{GL}_{2}(\mathbb{A}_{\mathbb{Q}})$, we also introduce the result of Maass forms over an arbitrary number field. This work is joint with Dohoon Choi, Min Lee, and Subong Lim.
이원웅 (홍콩대)
제목: Real zeros of depth 1 quasimodular forms
초록: In this talk, we will discuss the critical points of modular forms, or more generally the zeros of quasimodular forms of depth 1. Specifically, we will focus on two classes of functions; those that belong to a neighborhood of the derivatives of the Eisenstein series, and the derivatives of modular forms with the maximal number of consecutive zero Fourier coefficients following the constant 1. This is a joint work with Bo-Hae Im.
조성윤 (POSTECH)
제목: On the Kudla-Rapoport conjecture at a place of bad reduction
초록: The Kudla-Rapoport conjecture predicts a relation between the arithmetic intersection numbers of special cycles on the unitary Shimura variety and the derivative of representation densities for hermitian forms at a place of good reduction. In this talk, I will present a variant of the Kudla-Rapoport conjecture at a place of bad reduction. Additionally, I will discuss a proof of the conjecture in several new cases in any dimension. This is joint work with Qiao He and Zhiyu Zhang.
한지영 (KIAS)
제목: Problems in Geometry of S-arithmetic Numbers with the Viewpoint of Homogeneous Dynamics
초록: Geometry of numbers is one of the most favorite areas where homogeneous dynamists love to apply their specialty. In this talk, we will recall how these applications work. One of the main tools for transferring to homogeneous spaces is the so-called Siegel transform, which provides quantitative information on lattices in the vector space. And if time permits, we will introduce their S-arithmetic analogs. This is partially joint work with Sam Fairchild.