March 14 (Fri) 2025, 10시 ~ 11시 아산이학관 526호
Title: Sums of generalized polygonal numbers of almost prime length
Speaker: Daejun Kim (Korea University)
Abstract:
Extending the Lagrange's four-square theorem, it is expected that every sufficiently large integer congruent to 4 modulo 24 can be written as a sum of four squares of prime numbers. It has been shown that such integers can be expressed as a sum of four squares of integers, each with fewer than five prime factors. In this talk, we discuss an analogous problem concerning sums of three generalized m-gonal numbers, where parameters are restricted to integers with a bounded number of prime divisors. With some restriction modulo 30, we show that a density one set of integers can be represented as such a sum, where the parameters are restricted to have at most 6361 prime factors. This is a joint work with Soumyarup Banerjee and Ben Kane.
March 14 (Fri) 2025, 11시30분 ~ 12시30분 아산이학관 526호
Title: Non-vanishing mod p of derived Hecke algebra of the multiplicative group over number field
Speaker: Jaesung Kwon (Seoul National University)
Abstract:
We investigate the derived Hecke action on the cohomology of an arithmetic manifold associated to the multiplicative group over a number field. The degree one part of the action is proved to be non-vanishing modulo under mild assumptions. The main ingredient is the Grunwald--Wang theorem. This work is joint with Dohyeong Kim.
April 4 (Fri) 2025, 10시30분 ~ 11시30분 아산이학관 526호
Title: Dynamic coloring and list dynamic coloring of planar graphs
Speaker: Sang June Lee (Kyung Hee University)
Abstract:
A dynamic coloring of a graph G is a proper coloring of the vertex set V (G) such that for each vertex of degree at least 2, its neighbors receive at least two distinct colors. The dynamic chromatic number χd(G) is the smallest number k such that there is a dynamic coloring of G with k colors. It is known that the gap χd(G) − χ(G) could be arbitrarily large for some graphs. Based on the Four Color Theorem, one of the most interesting problems about dynamic chromatic numbers is to find upper bounds of χd(G) for planar graphs G. Lin and Zhao (2010) and Fan, Lai, and Chen (recently) showed that for every planar graph G, we have χd(G) ≤ 5, which is best possible because χd(C5) = 5. Also, it was conjectured that χd(G) ≤ 4 if G is a planar graph without a C5 as a component. As a partial answer, Kim and Park (2011) showed that χd(G) ≤ 4 if G is a planar graph with girth at least 7.
In this talk we settle the above conjecture that χd(G) ≤ 4 if G is a planar graph without a C5 as a component. We also study the corresponding list coloring called a list dynamic coloring. This is joint work with Seog-Jin Kim and Won-Jin Park.
May 2 (Fri) 2025, 10:30 ~ 11:30 아산이학관 526호
Title: Arithmetic of generalized Hecke operators
Speaker: Chang Heon Kim (Sungkyunkwan University)
Abstract:
Generalized Hecke operators originated from the replication formula in Monstrous Moonshine by Conway, Norton, Koike, Ferenbaugh, etc. In this talk we will extend these operators to apply to harmonic Maass functions on modular curves of higher genera. Then these operators can be applied to weakly holomorphic modular forms of weight 0, deriving numerous arithmetic properties of Fourier coefficients (e.g. congruences of Fourier coefficients, Zagier dualities, etc). Further, these operators can be extended to weakly holomorphic modular forms of nonzero weights. Additionally, we will identify the conditions under which the action of these operators preserves holomorphicity. This is a joint work with Daeyeol Jeon, Soon-Yi Kang and Gyeong Seok Min.
May 30 (Fri) 2025, 10:00 ~ 11:00 아산이학관 526호
Title: On the Kudla-Rapoport conjecture at a place of bad reduction
Speaker: Sungyoon Cho (POSTECH)
Abstract:
The Kudla-Rapoport conjecture predicts a relation between the arithmetic intersection numbers of special cycles on a 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. This talk is partially based on joint work with Qiao He, Chul-hee Lee, and Zhiyu Zhang.
May 30 (Fri) 2025, 11:200 ~ 12:20 아산이학관 526호
Title: Distribution of Hecke eigenvalues for holomorphic Siegel modular forms
Speaker: Henry H. Kim (University of Toronto)
Abstract:
We discuss two results on the distribution of Hecke eigenvalues of holomorphic Siegel modular forms. The first is the average Sato-Tate distribution, and the second is the Gaussian central limit theorem. The main tool is the vertical Sato-Tate theorem proved using Arthur's invariant trace formula.