Organizer: Shaoyun Yi

Email: yishaoyun926@xmu.edu.cn

Office: 616

Tentative Schedule:

Week No.	Date	Topics	Speaker
1	2/28	From Riemann Hypothesis to Langlands Program, Part I (Abstract) & Organizational Meeting	Shaoyun Yi
2	3/6	Gröbner Bases	Yueheng Wang
3	3/13	From Riemann Hypothesis to Langlands Program, Part II (Abstract)	Shaoyun Yi
4	3/20	Counting Rational Points on and near Manifolds (Abstract)	Lin Feng
5	3/27	Mollification and its applications	Xiaode Xu
6	4/3	Introduction to Lebesgue measure	Xiaode Xu
7	4/10	No Talk
8	4/17	General discussions
9	4/24	No Talk
10	5/1	No Talk
11	5/8	Introduction to measurable functions	Xiaode Xu
12	5/15	Picard's Little Theorem	Jingtian Xu
13	5/22	Introduction to algebraic geometry	Chuhui Gao
14	5/29	Introduction to module theory	Chuhui Gao
15	6/5	An Invitation to Modular Forms (Abstract)	Shaoyun Yi
15	6/5	Some conjectures in analytic number theory (Abstract) Postpone	Biao Wang

Abstracts

Shaoyun Yi - From Riemann Hypothesis to Langlands Program

In this talk, we will first give a brief introduction to the Riemann Hypothesis, which is about the well-known Riemann zeta function. Then we will move on to the discussions of its generalizations, the so-called $L$-functions, which are the central objects in the Langlands Program. If time permits, we will also talk about some famous examples in the Langlands Program and explore how they help the developments of modern number theory in different areas.

Lin Feng - Counting Rational Points on and near Manifolds

The topic is located on the interface of number theory, harmonic analysis, dynamics, and Diophantine geometry. The motivation for counting the rational points on or near the equation is to prove some Diophantine equation have no solution.
Let $M\subset \mathbb{R}^n$ denote a compact $C^{\infty}$-manifold of dimension $d$. Choose an integer $Q\ge 1$ (height bound) and a (thickness) parameter $\delta >0$ which always assumed to be sufficiently small. Define the counting function \[ \mathcal{N}_M(\delta,Q):=\#\set{(q,\textbf{a})\in\mathbb{Z}^{n+1}: q\le Q, \mathrm{dist}(\textbf{a}/q,M)\le \delta/q}. \] The main of the talk is to prove that: for some type of $M\subset \mathbb{R}^2$ with dimension $1$, if $\epsilon>0$, then \[ \mathcal{N}_M(\delta,Q)=c\delta Q^2+O\left(Q^{3/2+\epsilon}\right) \] for any $Q\ge 1$ and $\delta>0$ small enough. Besides, we will introduce some results for Hypersurfaces given by JingJing Huang in 2018, and the idea to counting the rational points by using lattices.
Prerequisites: Some knowledge of Manifolds and Harmonic Analysis is useful but not necessary.

Shaoyun Yi - An Invitation to Modular Forms

In this talk, we mainly give an introduction to classical modular forms with various examples and applications. Then we will briefly report some recent results on dimension formulas of spaces of Siegel modular forms of degree $2$.

Biao Wang - Some conjectures in analytic number theory

In this talk, we will introduce some conjectures in analytic number theory, including the Riemann Hypothesis, Siegel zeros, Twin prime conjecture, Subconvexity problem, Quantum Unique Ergodicity, and Sarnak's conjecture.