咨询委员会：刘建亚、苗长兴、田刚
组织委员会：曹桢斌、陈学志、高传伟、关力凡
时间：2024年5月8日-5月10日
地点：西湖大学云谷校区E5-329
ZOOM ID: 993 6665 2284
PASSCODE: 362506
5月8日
9:00-10:15
主持人：北京应用物理与计算数学研究所 苗长兴
主讲人: 威斯康星麦迪逊分校、南开大学 郭少明
主题: 解析数论中的TDI系统
摘要: 报告的内容是关于解析数论中TDI系统的解的个数估计问题。TDI是translation- and dilation-invariant的缩写。TDI系统最典型的例子是Vinogradov方程组。2015年Bourgain, Demeter and Guth 用调和分析中的decoupling 方法给出了一维Vinogradov方程组解的个数的最优上界估计。在报告中我们会讨论高维Vinogradov方程组以及更一般的TDI系统解的个数的上界估计。
10:45-12:00
主持人：清华大学 张城
主讲人: 河南省科学院 曹桢斌
主题: Decoupling and exponential sum estimates
摘要: Exponential sum estimate is an important problem in Number theory, which implies Lindelof hypothesis, Vinogradov's mean value theorem and so on. In 2015, Bourgain-Demeter showed sharp decoupling for the paraboloid, which can offer abundant results on exponential sum estimates. In this talk, we introduce the history of decoupling, related tools developed in Harmonic analysis, and several open problems.
14:00-15:15
主持人：复旦大学 贺丹青
主讲人: 山东大学、中国科学技术大学 赵立璐
主题: The quadratic analogy of Roth's theorem
摘要: In 1953, Roth proved that a subset of natural numbers with positive density contains nontrivial three-term arithmetic progressions. In recent years, the quadratic analogues of Roth's theorem have been widely studied. In this talk, we shall briefly give a survey on this topic.
15:45-17:00
主持人：中山大学 宋亮
主讲人: 中国科学技术大学 郭经纬
主题: Weyl's law, an example at the intersection of harmonic analysis, number theory and spectral geometry
摘要: In 1911 Hermann Weyl started the study of the asymptotic formula describing distribution of eigenvalues of the Dirichlet Laplacian in a bounded domain. Since then, the study of the asymptotics of eigenvalue counting functions has always been an interesting topic in spectral geometry. In particular, people are interested in improving the remainder estimate of such asymptotics.
I will talk about the asymptotics for the Dirichlet/Neumann Laplacian associated with several Euclidean domains with symmetry (e.g. balls and spherical shells) and explain how to transfer the associated eigenvalue counting problems into certain lattice point counting problems, to which tools from harmonic analysis and analytic number theory can be applied.
5月9日
9:00-10:15
主持人：天津大学 沈瑞鹏
主讲人: 清华大学 金龙
主题: Selberg zeta 函数，迹公式和一些相关问题
摘要: Selberg在1956年对双曲曲面引入了Riemann zeta函数的类比，其中素数以素闭测地线替代。由Selberg迹公式，Selberg zeta函数亚纯延拓到整个复平面。理解Selberg zeta函数的零点和极点分布对双曲几何和数论都是重要问题。对紧曲面，Selberg zeta函数是一个整函数并且零点对应于Laplace算子的特征值，对非紧曲面，Lax-Phillips关于自守形式的工作说明Selberg zeta函数与散射理论紧密相关。我们讨论一些该领域的最新进展以及一些为解决的问题。如果时间允许，我们还讨论双曲动力系统中的推广。
10:45-12:00
主持人：浙江师范大学 徐甜
主讲人: 南方科技大学 刘博辰
主题: An aspect on Diophantine approximation in geometric measure theory
摘要: We construct a new family of sets of given Hausdorff dimension via Diophantine approximations, among which the classical Jarnik example is the endpoint case. Then we are able to show a recent sharp result of Ren and Wang on orthogonal projections is also sharp on the sum-product set A+cB. If time allows, I will also discuss higher dimensional cases. This is joint work with Longhui Li.
14:00-15:15
主持人：北京应用物理与计算数学研究所 郑继强
主讲人: 浙江大学 席亚昆
主题: 流形上的调和分析问题
摘要: In this talk, we survey some recent progress in harmonic analysis problems on Riemannian manifolds, centered around the study of eigenfunctions of the Laplacian. These problems include Weyl type asymptotics, Lp estimates, restrction estimates, period integrals, and related inverse spectral problems. We shall highlight along the way how these problems are interesting to experts in different areas in mathematics, including harmonic analysis, differential geometry, number theory, mathematical physics, and dynamical systems.
15:45-17:00
主持人：北京航空航天大学 张安
主讲人: 安徽大学 陈昌昊
主题: Montgomery conjecture and Kakeya
摘要: Bourgain (1991) proved that Motgomery conjecture implies Kakeya conjecture.In this talk, I will introduce these two conjectures and show how they are related to each other.
5月10日
9:00-10:15
主持人：西湖大学 王子鹏
主讲人: 西湖大学 关力凡
主题: On recent work in integral distance sets
摘要: In this talk, I will try to sketch the recent work of Greenfeld-Iliopoulou-Peluse on the structure of sets with mutual integral distance. Their study of this problem is motivated by some research in harmonic analysis. More interestingly, their method can be compared with the polynomial method.
10:45-12:00
主持人：大连理工大学 李俊峰
主讲人: 南京晓庄学院 许绍镇
主题: 局部zeta函数、振荡积分与多项式同余问题
摘要: 本报告将介绍局部zeta函数在退化相位振荡积分渐近估计中的应用以及与多项式同余问题之间的联系，其中涉及代数几何中的奇点解消以及Newton多面体。
14:00-17:00
主持人：北京应用物理与计算数学研究所 苗长兴
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