2021/01/15-2021/01/17
报告题目: Arithmetic purity of strong approximation
报告人: 曹阳 教授，中国科学技术大学
摘要: Strong approximation with Brauer-Manin obstruction is defined by Colliot-Thélène and Xu to study the local-global principle for the integral points. For an algebraic variety, inspired by analytic number theory, we consider the density of integral points with coprime values in adelic space: the arithmetic purity of strong approximation.
On the other side, for a semi-simple, simply connected k-simple linear algebraic group G, it is conjectured by Wittenberg that G satisfies arithmetic purity: the complement of any codimension >=2 closed subset satisfies strong approximation. We prove this conjecture for k-isotropic groups by an analogue of fibration method and for Spin groups by using the density of rational points with almost prime polynomial values. This is joint work with Zhizhong Huang.
报告题目: Hilbert's Tenth Problem and Further Developments
报告人: 孙智伟 教授，南京大学
摘要: Hilbert's Tenth Problem (HTP) asked for an algorithm to determine wether an arbitrary polynomial equation with integer coefficients has solutions over the ring of integers. This was finally solved negatively by Yu. Matiyasevich in 1970, based on a 1961 paper by M. Davis, H. Putnam and J. Robinson.
In this three-hour talk, we introduce the theory of computability, the ingenious solution of HTP, and its further developments such as Matiyasevich's 9 unknowns theorem and the speaker's 11 unknowns theorem.
报告题目：Kloostermann refinement and large sieve inequality
报告人: 赵立璐 教授，山东大学
摘要: The circle method with Kloostermann refinement and the large sieve inequality are two important methods in analytic number theory. In this talk, we give a brief introduction to the above two methods. Some applications will be discussed.