September 24 (Tue) 2024, 14시 ~ 15시 아산이학관 525호
Title: Modularity of partition generating functions
Speaker: Soon-Yi Kang (Kangwon National University)
Abstract:
Partitions of n into parts at least d apart is one of the oldest subjects in partition theory. Euler, Rogers-Ramanujan, and Schur established partition identities for d-distinct partitions when d = 1, 2, 3, respectively. These identities were extended to a partition inequality for d-distinct partition for d > 2 by Alder, Andrews, Yee and more. In recent years, numerous researchers have developed analogous and generalized versions of Alder-type partition inequalities. Furthermore, the modularity of the generating functions for d-distinct partitions has been revealed by remarkable discoveries made by Zagier and Folsom. In this talk, we try to provide a comprehensive survey of the findings pertaining to d-distinct partitions.
September 24 (Tue) 2024, 15시 30분 ~ 16시 30분 아산이학관 525호
Title: Eichler-Selberg relations for singular moduli
Speaker: Toshiki Matsusaka (Kyushu University)
Abstract:
The Eichler–Selberg trace formula expresses the trace of Hecke operators on spaces of cusp forms as weighted sums of Hurwitz–Kronecker class numbers. We extend this formula to a natural class of relations for traces of singular moduli, where one views class numbers as traces of the constant function j_0(z) = 1. More generally, we consider the singular moduli for the Hecke system of modular functions j_m(z) := mTm (j(z) − 744). For each v ≥ 0 and m ≥ 1, we obtain an Eichler–Selberg relation. For v = 0 and m = 1,2, these relations are Kaneko’s celebrated singular moduli formulas for the coefficients of j(z). For each v ≥ 1 and m ≥ 1, we obtain a new Eichler–Selberg trace formula for the Hecke action on the space of weight 2v + 2 cusp forms, where the traces of j_m(z) singular moduli replace Hurwitz–Kronecker class numbers. These formulas involve a new term that is assembled from values of symmetrized shifted convolution L-functions. (This is a joint work with Yuqi Deng and Ken Ono)
October 4 (Fri) 2024, 10시 30분 ~ 12시 아산이학관 526호
Title: Central limit theorem for Lochs' variable
Speaker: Hae-Sang Sun (UNIST)
Abstract:
The Lochs variable measures how much two continued expansions of two distinct real numbers, are close with respect to their decimal expansions. It is Faivre who proves the Central limit theorem for the Lochs variable. In the talk, I will briefly review his proof, which is mainly an analysis on the digits of continued fraction expansions of two close decimal expansions. In the latter part, I will give a dynamical proof using the transfer operator of the Gauss map. This is joint work with Hong Kwon.
November 4 (Mon) 2024, 14시 ~ 15시
Title: Mordell-Weil groups over large algebraic extensions of number fields
Speaker: Yuichiro Taguchi (Tokyo institute of Technology)
Abstract:
We present some results on the structure of the Mordell-Weil groups of semiabelian varieties over large algebraic extensions of a number field. We consider two types of algebraic extensions; one is of extensions obtained by adjoining the coordinates of certain points of various semiabelian varieties; the other is of extensions obtained as the fixed subfield in an algebraically closed field by a finite number of automorphisms. Some of such fields turn out to be new examples of Kummer-faithful fields which are not sub-p-adic. This is a joint work with Takuya Asayama.
November 8 (Fri) 2024, 10시 30분 ~ 12시 아산이학관 525호
Title: Spectrum of the Laplacian on a compact Riemann surface with many automorphisms
Speaker: Chul-hee Lee (KIAS)
Abstract:
The study of the spectrum of the Laplacian on compact Riemann surfaces is a classical topic that continues to raise many intriguing questions. When a surface has a non-trivial automorphism group, the eigenspace associated with a given eigenvalue is stable under its action, thus forming a representation. Consequently, eigenvalues tend to appear with certain multiplicities. Despite its classical flavor, the determination of these multiplicities has not been extensively studied.
In this talk, I will review known results, highlight relevant open problems, and explain how to exploit the Selberg trace formula, with a particular focus on selecting appropriate test functions, a process that involves certain optimization problems.
November 29 (Fri) 2024, 10시 30분 ~ 12시 아산이학관 525호
Title: Representation numbers of Bell-type quadratic forms
Speaker: Yeong-Wook Kwon (Korea University)
Abstract:
The famous Lagrange's four-square theorem states that every positive integer can be represented as a sum of four squares. In 1834, Jacobi proved a quantitative version of this theorem. The sum of four squares can be regarded as an example of the so-called Bell-type quadratic forms, and Jacobi's result has been extended to certain Bell-type quadratic forms by several authors. However, most preceding results have focused on Bell-type forms of class number 1. In this talk, we derive a closed formula for the representation numbers of Bell-type quadratic forms with class numbers up to 2. This is joint work with Chang Heon Kim, Kyoungmin Kim and Soonhak Kwon.
September 24 (Tue) 2024, 14시 ~ 15시 아산이학관 525호
Title: Modularity of partition generating functions
Speaker: Soon-Yi Kang (Kangwon National University)
Abstract:
Partitions of n into parts at least d apart is one of the oldest subjects in partition theory. Euler, Rogers-Ramanujan, and Schur established partition identities for d-distinct partitions when d = 1, 2, 3, respectively. These identities were extended to a partition inequality for d-distinct partition for d > 2 by Alder, Andrews, Yee and more. In recent years, numerous researchers have developed analogous and generalized versions of Alder-type partition inequalities. Furthermore, the modularity of the generating functions for d-distinct partitions has been revealed by remarkable discoveries made by Zagier and Folsom. In this talk, we try to provide a comprehensive survey of the findings pertaining to d-distinct partitions.
September 24 (Tue) 2024, 15시 30분 ~ 16시 30분 아산이학관 525호
Title: Eichler-Selberg relations for singular moduli
Speaker: Toshiki Matsusaka (Kyushu University)
Abstract:
The Eichler–Selberg trace formula expresses the trace of Hecke operators on spaces of cusp forms as weighted sums of Hurwitz–Kronecker class numbers. We extend this formula to a natural class of relations for traces of singular moduli, where one views class numbers as traces of the constant function j_0(z) = 1. More generally, we consider the singular moduli for the Hecke system of modular functions j_m(z) := mTm (j(z) − 744). For each v ≥ 0 and m ≥ 1, we obtain an Eichler–Selberg relation. For v = 0 and m = 1,2, these relations are Kaneko’s celebrated singular moduli formulas for the coefficients of j(z). For each v ≥ 1 and m ≥ 1, we obtain a new Eichler–Selberg trace formula for the Hecke action on the space of weight 2v + 2 cusp forms, where the traces of j_m(z) singular moduli replace Hurwitz–Kronecker class numbers. These formulas involve a new term that is assembled from values of symmetrized shifted convolution L-functions. (This is a joint work with Yuqi Deng and Ken Ono)
October 4 (Fri) 2024, 10시 30분 ~ 12시 아산이학관 526호
Title: Central limit theorem for Lochs' variable
Speaker: Hae-Sang Sun (UNIST)
Abstract:
The Lochs variable measures how much two continued expansions of two distinct real numbers, are close with respect to their decimal expansions. It is Faivre who proves the Central limit theorem for the Lochs variable. In the talk, I will briefly review his proof, which is mainly an analysis on the digits of continued fraction expansions of two close decimal expansions. In the latter part, I will give a dynamical proof using the transfer operator of the Gauss map. This is joint work with Hong Kwon.
November 4 (Mon) 2024, 14시 ~ 15시
Title: Mordell-Weil groups over large algebraic extensions of number fields
Speaker: Yuichiro Taguchi (Tokyo institute of Technology)
Abstract:
We present some results on the structure of the Mordell-Weil groups of semiabelian varieties over large algebraic extensions of a number field. We consider two types of algebraic extensions; one is of extensions obtained by adjoining the coordinates of certain points of various semiabelian varieties; the other is of extensions obtained as the fixed subfield in an algebraically closed field by a finite number of automorphisms. Some of such fields turn out to be new examples of Kummer-faithful fields which are not sub-p-adic. This is a joint work with Takuya Asayama.
November 8 (Fri) 2024, 10시 30분 ~ 12시 아산이학관 525호
Title: Spectrum of the Laplacian on a compact Riemann surface with many automorphisms
Speaker: Chul-hee Lee (KIAS)
Abstract:
The study of the spectrum of the Laplacian on compact Riemann surfaces is a classical topic that continues to raise many intriguing questions. When a surface has a non-trivial automorphism group, the eigenspace associated with a given eigenvalue is stable under its action, thus forming a representation. Consequently, eigenvalues tend to appear with certain multiplicities. Despite its classical flavor, the determination of these multiplicities has not been extensively studied.
In this talk, I will review known results, highlight relevant open problems, and explain how to exploit the Selberg trace formula, with a particular focus on selecting appropriate test functions, a process that involves certain optimization problems.
November 29 (Fri) 2024, 10시 30분 ~ 12시 아산이학관 525호
Title: Representation numbers of Bell-type quadratic forms
Speaker: Yeong-Wook Kwon (Korea University)
Abstract:
The famous Lagrange's four-square theorem states that every positive integer can be represented as a sum of four squares. In 1834, Jacobi proved a quantitative version of this theorem. The sum of four squares can be regarded as an example of the so-called Bell-type quadratic forms, and Jacobi's result has been extended to certain Bell-type quadratic forms by several authors. However, most preceding results have focused on Bell-type forms of class number 1. In this talk, we derive a closed formula for the representation numbers of Bell-type quadratic forms with class numbers up to 2. This is joint work with Chang Heon Kim, Kyoungmin Kim and Soonhak Kwon.