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Number Theory Lunch Seminar
Counting integral matrices with a given characteristic polynomial
Counting integral matrices with a given characteristic polynomial
演讲者
时间
2025年12月11日 12:15 至 13:00
地点
A4-1
摘要
Let $P(x)\in \mathbb{Z}[x]$ be a monic irreducible polynomial of degree $n$ and $M_n(\mathbb{Z})$ be the space of $n\times n$ integral matrices.
Let $V=\{X\in M_n(\mathbb{Z}): \det(xI-X)=P(x)\}$ and $B_T$ be the Euclidean ball centered at $0$ of radius $T$ in $M_n(\mathbb{R})$.
In this talk, I will explain the asymptotic formula of Eskin, Mozes and Shah:
\[
\lim_{T\rightarrow \infty}\frac{\# (V\cap B_T)}{T^{n(n-1)/2}}=C_P
\]
for some constant $C_P>0$. If time permits, I will explain the interpretation of the constant $C_P$ in terms of orbital integrals by work of Yuchan Lee base on work of Dasheng Wei and Fei Xu. No new results will be discussed in this talk.
Let $V=\{X\in M_n(\mathbb{Z}): \det(xI-X)=P(x)\}$ and $B_T$ be the Euclidean ball centered at $0$ of radius $T$ in $M_n(\mathbb{R})$.
In this talk, I will explain the asymptotic formula of Eskin, Mozes and Shah:
\[
\lim_{T\rightarrow \infty}\frac{\# (V\cap B_T)}{T^{n(n-1)/2}}=C_P
\]
for some constant $C_P>0$. If time permits, I will explain the interpretation of the constant $C_P$ in terms of orbital integrals by work of Yuchan Lee base on work of Dasheng Wei and Fei Xu. No new results will be discussed in this talk.
演讲者介绍
邓太旺博士于2022年11月加入BIMSA，担任助理研究员。他的研究兴趣是朗兰兹纲领（广义上说，是朗兰兹纲领的算术、分析和表示方面）。他在巴黎十三大学获得了数学博士学位。此前，他曾在波恩大学、马克斯•普朗克数学研究所和清华大学担任博士后。