山大数论讨论班

星期三，15:00-16:00，2024年秋季学期（10 月开始）

中心校区明德楼 C704

组织者：山大数论组

讨论班着重于数论中的新进展。报告人为校内外学者和学生。欢迎推荐报告。

报告信息（摘要和往年报告信息见列表下方）

• 2024-10-09， 胡龙   (山东大学)

  题目： Minimal control time for 1D linear hyperbolic systems of balance laws

  摘要： In this talk, we are concerned with minimal control time for 1D linear hyperbolic systems of balance laws.  We will show how such critical quantities depend on internal and boundary couplings under various types of controllability (exact controllability (EC), null-controllability (NC), boundary controllability (BC) and internal controllability (IC)). A key aspect will be demonstrated that the minimal time can be strictly smaller than the classical one proposed by J.-M. Coron, T.T. Li and D. Russell etc. Moreover, the difference between EC and NC and equivalence between BC and IC will also be highlighted.  This talk is based on recent joint works with Guillaume Olive.

  地点：明德楼 C704

• 2024-10-16， Patrick Solé   (CNRS)

  题目： Codes and Lattices: bridge and dictionary

  摘要： Codes are vector spaces over finite fields with combinatorial properties, used in Information transmission since 1948, since their inception by Claude Shannon.

  Lattices are discrete additive subgroups of the Euclidean space with metric properties studied by classical mathematicians like Gauss, Jacobi, Minkowski. They play an important role today in post quantum crypto.

  We survey analogies between these two objects: distance vs minimum, weight enumerators vs theta series, invariant of finite groups vs modular forms.

  A direct connection is Construction A which attachs a lattice to a binary code. A variation thereof,  Construction B allows for a combinatorial proof of  the Jacobi identity.

  An abstract version is the Brou\'e -Enguehard map which is a correspondence between polynomial invariants and modular forms.
  

  地点：明德楼 C704

• 2024-10-23， 郑钧仁   (西安交通大学)

  题目： On the Brun-Titchmarsh Theorem

  摘要： Let \pi(x; q, a) denote the number of primes less than or equal to x that are congruent to a mod q. The Brun-Titchmarsh theorem gives the bound \pi(x; q, a) < (C+o(1))x / \phi(q) log x, where C depends on the ratio log x / log q. In this talk, we will present our recent work on strengthening this inequality for different ranges of log x / log q. Our approach involves various estimates for character and exponential sums, based on arithmetic exponent pairs and bilinear forms with Kloosterman sums, among other tools.

  地点：明德楼 C704

• 2024-10-30， 谢思哲   (山东大学)

  题目：Forms in prime variables and differing degrees

  摘要：Let $F_1,\ldots,F_R$ be homogeneous polynomials with integer coefficients in $n$ variables with differing degrees. Write $\boldsymbol{F}=(F_1,\ldots,F_R)$ with $D$ being the maximal degree. Suppose that $\boldsymbol{F}$ is a nonsingular system and $n\ge D^2 4^{D+6}R^5$. We prove an asymptotic formula for the number of prime solutions to $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$, whose main term is positive if (i) $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ has a nonsingular solution over the $p$-adic units $\mathbb{U}_p$ for all primes $p$, and (ii) $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ has a nonsingular solution in the open cube $(0,1)^n$. This can be viewed as a smooth local-global principle for $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ in primes with differing degrees. It follows that, under (i) and (ii), the set of prime solutions to $\boldsymbol{F}(\boldsymbol{x})=\boldsymbol{0}$ is Zariski dense in the set of its solutions. In other words, we prove Bourgain-Gamburd-Sarnak conjecture is true in this case. This is a joint work with my supervisor Professor Jianya Liu.

  地点：明德楼 C704

• 2024-11-13， 林永晓   (山东大学)

  题目：Strong subconvexity and reciprocity formula

  摘要：In this talk we will discuss the subconvexity problem for L-functions and review some recent progress, particularly quantitatively stronger bounds, that have been made on the problem. Spectral reciprocity formulas that relate two different families of L-functions, which is one key ingredient within the proof, will also be discussed.

  地点：明德楼 C704

• 2024-11-20， 何伟鲲   (中国科学院数学与系统科学研究院)

  题目： 分形上的Khintchine定理

  摘要：丢番图逼近是研究实数能多好地被有理数逼近的学问。Khintchine定理是丢番图逼近的基本定理之一。给定一个期待的逼近速率，Khintchine定理根据逼近函数对应级数的收敛性判定几乎所有实数都可以或者都不可以在该速率下可逼近。这里“几乎所有”是针对Lebesgue测度的。在本报告中我将介绍与合作者Timothée Bénard与张涵获得的最新结果：在Khintchine定理中，将Lebesgue测度换成分形测度仍然成立。

  地点：明德楼 C704

• 2024-11-27， 曾衡发 （山东大学）

  题目： Ramanujan's theory of elliptic functions to the cubic base

  摘要： In this talk, I will present a new approach to the study of Ramanujan's theory of elliptic functions to the cubic base using Jacobi's theta functions. This new approach does not involve the cubic theta series discovered by J.M. Borwein and P.B. Borwein around 1991. This talk is based on joint work with Zhiguo Liu.

  地点：明德楼 C704

• 2024-12-04， 张涵 （苏州大学）

  题目： Quantitative Khintchine's theorem on fractals and Effective double equidistribution on homogeneous spaces

  摘要： Given an approximation function ψ, the classical Khintchine's theorem gives a dichotomy for the Lebesgue measure of ψ-approximable real vectors. In 1959, Schmidt proved a quantitative form of Khintchine's theorem, obtained the asymptotic number of solutions for the Diophantine inequality of typical real vectors. 

  In a recent joint work with Benard and He, we extend the classical Khintchine's theorem to self-similar measures on the real line, and  obtain a quantitative version of Khintchine's theorem for those measures. The proof is based on dynamics on homogeneous spaces and certain effective double equidistribution theorem.

  地点：明德楼 C704

• 2024-12-11， 张庆 （华中科技大学）

  题目：On local converse theorems of p-adic groups

  摘要：Let $F$ be a local field and $G$ be a quasi-split reductive group over $F$. The local converse problems asks whether a (generic irreducible smooth) representation of $G(F)$ can be uniquely determined by its twisted gamma factors. In this talk, I will report certain local converse theorems for quasi-split classical groups, which include the symplectic groups and unitary groups.

  地点：明德楼 C704

• 2024-12-18， 范洋宇 （北京理工大学）

  题目：Local Harmonic analysis and Euler systems

  摘要：In this talk, we report our new approach to the horizontal Euler system property of theta elements by the relative Satake isomorphism. This is a joint work with L. Cai and S. Lai.

  地点：明德楼 C704

• 2024-12-25， 许宾 （四川大学）

  题目：Polynomials over finite fields with small value sets and monodromy groups

  摘要：Polynomials over finite fields are extensively studied due to various motivations. One way to classify these polynomials is by examining their value sets. For instance, polynomials whose value sets encompass the entire base field are known as permutation polynomials, which have many important applications in mathematics and related fields. In this talk, we focus on analyzing the value sets of polynomials over finite fields through their monodromy groups (specific associated Galois groups). We will begin by reviewing some background and previous work, and then present some results as well as further problems for polynomials with small value sets obtained via this approach.

  地点：明德楼 C704

• 2024-12-25， 何晓光 （四川大学）

  题目：Sarnak猜想的相关研究

  摘要：在本报告中，我们将给出Sarnak猜想的相关研究。简要介绍斜积系统上Sarnak猜想的最新进展及简要证明。

  地点：明德楼 C704

往年报告：

• 2024年春季报告
  

• 2023年秋季报告    2023年春季报告

• 2022年秋季报告 2022年春季报告

• 2021年秋季报告 2021年春季报告

• 2020年秋季报告 2020年春季报告

Contact us at: brhuang "at" sdu dot edu dot cn,   Office: Mingde C701,   Tel: 883-69786