One Day Workshop on the Arithmetic Theory of Quadratic Forms
Date: 10월 9일 (토) 10:00 ~ 11:50, 14:00 ~ 15:50, 16:00 ~ 17:50
Place: 129-406 (SNU)
Talk 1: 10:00 ~ 11:50
Title: Universal sums of generalized polygonal numbers
Speaker: Jangwon Ju (Ulsan University)
Abstract:
The sum of generalized polygonal numbers is said to be universal if it represents all nonnegative integers. In this talk, we introduce some arithmetic method on studying representations of sums of generalized polygonal numbers. We provide effective criteria on the universalities of sums of generalized polygonal numbers with sime small order. These might be considered as a generalization of the "15-Theorem" of Conway and Schneeberger.
Talk 2: 14:00 ~ 15:50
Title: The use of modular form theory in studying quadratic forms
Speaker: Kyoungmin Kim (Hannam University)
Abstract:
In this talk, we introduce some modular form theory used in studying the number of representations of integers by quadratic forms.
Talk 3: 16:00 ~ 17:50
Title: Tight universal quadratic forms
Speaker: Mingyu Kim (Sungkyunkwan University)
Abstract:
For a positive integer $n$, let $\mathcal{T}(n)$ be the set of all integers greater than or equal to $n$. An integral quadratic form $f$ is called tight $\mathcal{T}(n)$-universal if the set of nonzero integers that are represented by $f$ is exactly $\mathcal{T}(n)$. The smallest possible rank over all tight $\mathcal{T}(n)$-universal quadratic forms is denoted by $t(n)$. In this talk, we prove that $t(n) \in \Omega(\log_2(n)) \cap \mathcal{O}(\sqrt{n})$. Explicit lower and upper bounds for $t(n)$ will be provided for some small integer $n$. We also consider the classification of all tight $\mathcal{T}(n)$-universal diagonal quadratic forms.
This is a joint work with Byeong-Kweon Oh.
