星光直播 星光直播 报告[2025] 062号

(高水平大学建设系列报告1084号)

报告题目: Masser’s Conjecture on Equivalence of Integral Quadratic Forms

报告人：李涵 副教授  维思大学（Wesleyan University）

报告时间：2025年07月01日，15:00-16:00

报告地点： 汇文楼1420

报告内容：In the realm of quadratic forms theory, a fundamental problem lies in determining the equivalence of two given integral quadratic forms. This problem, framed in terms of matrices, seeks to ascertain whether there exists a unimodular integral matrix X that satisfies the equation A=X’BX, where A and B are given symmetric n-by-n integral matrices, and X’ denotes the transpose of X. While a straightforward decision procedure exists for definite forms, the situation is more complex for indefinite forms. Surprisingly, it wasn't until the early 1970s, with the work of C. L. Siegel, that a solution was found for indefinite forms. In the late 1990s, D. W. Masser conjectured the existence of a polynomial search bound for X in terms of the heights of A and B, for n greater than or equal to 3. The goal of this presentation is to discuss our resolution of this conjecture, achieved jointly with Professor Gregory A. Margulis. Our approach involved translating the problem into one concerning the actions of Lie groups on homogeneous spaces, and subsequently solving it using tools from ergodic theory, harmonic analysis, and representation theory. The talk should be largely accessible to undergraduate students familiar with abstract algebra and functional analysis.

报告人简介：李涵副教授的研究兴趣涵盖李群、离散子群、齐性空间上的动力学，及其与数论和几何的交叉领域。他于2014年获得耶鲁大学数学博士学位。在加入卫斯理安大学之前，他曾在德克萨斯大学奥斯汀分校和伯克利数学科学研究所（MSRI）进行为期一年的博士后研究。

欢迎师生参加！

邀请人：肖维

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                     2025年6月30日