时间：2022年春季学期每周三 14:00-16:30

Time: 14:00-16:30, Wednesday, the Spring Semester, 2022

地点：西湖大学云栖校区4号楼409-411教室/云谷校区待定

Venue: Room 409-411, Building 4, Yunqi Campus/ Yungu Campus TBD

授课教师: 西湖大学理学院PI 赵永强博士

Professor: Dr. Yongqiang Zhao, School of Science, Westlake University

The circle method was introduced by Ramanujan and Hardy-Littlewood to study partition functions about a century ago. It has since served as one of the most important tools in the study of Diophantine equations. Besides its various connections with other branches of mathematics, such as rational approximation, algebraic geometry, harmonic analysis, additive combinatorics and dynamical systems, it also serves as a convenient heuristic tool in many arithmetic problems. In this introductory course, we plan to cover the following topics:

· Diophantine equations over finite fields: Chevalley-Warning theorem, Lang-Weil's bound, Weil-Deligne and Hooley's bounds on complete exponential sums.

· Hensel's lemma, definition of local density, local-global principle.

· Waring's problem: Weyl differencing, Hua's lemma, singular series and integrals, updates of recent progresses.

· Diagonal equations, Artin's conjecture for diagonal forms.

· Davenport's work on cubic forms: geometry of numbers, differencing and bilinear forms, geometric alternative, cubic forms over p-adic fields.

· If time permits and there were enough interests, we plan to discuss Birch's theorem and Schmidt's important work in 1980's, which is a culmination and a vast generalization of Davenport's work on cubic forms.

Prerequisite: Basic number theory, complex analysis and real analysis is highly recommended, as is some exposure to algebraic number theory and elementary algebraic geometry.

Textbook : None.

References :

·Davenport, H., Analytic methods for Diophantine equations and Diophantine inequalities, 2^nd edition, 2005.

· Browning, T., Cubic forms and the circle method, 2021.

· Birch, B. J., Forms in many variables, Proc. Roy. Soc. Ser. A, 265, 245-263, 1961.

· Schmidt, W. M., The density of integer points on homogeneous varieties, Acta Math. 154, 243-296, 1985.

Note: Lectures start from Feb. 23.

联系人/Contact: 理学院 王老师  wangqiuhui@westlake.edu.cn