Jan 15-17, 2021

**Title: **Arithmetic purity of strong approximation

**Speaker:** Prof. Yang Cao, University of Science and Technology of China

**Abstract:** 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.

**Title:** Hilbert's Tenth Problem and Further
Developments

**Speaker:** Prof. Zhi-Wei Sun, Nanjing
University

**Abstract:** 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.

**Title:** Kloostermann refinement and large
sieve inequality

**Speaker: **Prof. LiLu Zhao, Shandong
University

**Abstract: **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.

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