山大数论讨论班

星期二，15:00-16:00，2023年

中心校区明德楼 C702 或者 线上

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）

暑期专场

• 2023-02-28，曹阳 （中国科学技术大学）   10:00-11:00

  题目：Relative version of weak approximation for abelian schemes

  摘要：Over number field, weak approximation for algebraic varieties is about the density of rational points in adelic points. In this talk, for an abelian scheme over a rational variety, instead of considering the weak approximation, we consider the relative closed subgroup generated by rational points in adelic points of the total space, and study its relation with cohomological obstruction.

  地点：明德楼 C702

• 2023-02-28，刘华锋 （山东师范大学）

  题目：The values of general quadratic forms at prime arguments

  摘要：Let f_1, …, f_r be regular indefinite integral quadratic forms with s prime variables. In this talk, we discuss the values taken by real linear combinations of f_1, …, f_r. It is proved that, under certain conditions, the values of the real linear combinations of three general quadratic forms, with 6 prime variables in total, can approximate any real numbers. The previous results of this type focus on diagonal forms.

  地点：明德楼 C702

• 2023-03-07，李晓承   (北京大学)

  题目：An estimate for spherical functions on SL(3,R)

  摘要：We prove an estimate for spherical functions $\varphi_\lambda(a)$ on SL(3,R), establishing uniform decay in the spectral parameter $\lambda$ when the group variable $a$ is restricted to a compact subset of the abelian subgroup $A$. In the case of SL(3,R), it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when $\lambda$ and $a$ vary.

  地点：明德楼 C702

• 2023-03-14，Number Theory Pi Day   12:00-17:00

  地点：明德楼 C702

• 2023-03-21，Matthew P. Young   (Texas A&M University)   21:00-22:00

  题目：Large sieve inequalities for families of L-functions

  摘要：Large sieve inequalities are useful and flexible tools for understanding families of L-functions.  The quality of the bound is one measure of our understanding of the corresponding family.  For instance, they may directly give rise to good bounds for certain moments of L-functions.  In this talk, I will survey this problem and discuss recent progress for some new families.

  地点：Zoom: 825 9309 8083, password: 202303

• 2023-03-30，Xiannan Li   (Kansas State University)   10:00-11:00

  题目：Quadratic twists of modular L-functions

  摘要：The behavior of quadratic twists of modular L-functions is at the critical point is related both to coefficients of half integer weight modular forms and data on elliptic curves.  Here we describe a proof of an asymptotic for the second moment of this family of L-functions, previously available conditionally on the Generalized Riemann Hypothesis by the work of Soundararajan and Young.  Our proof depends on deriving an optimal large sieve type bound.

  地点：Zoom: 825 9309 8083, password: 202303

• 2023-04-04，NO TALK (Qingming Festival)

• 2023-04-11，Max Wenqiang Xu  徐文强   (Stanford University)  14:00-15:00

  题目：Recent progress in random multiplicative functions

  摘要：Random multiplicative functions are probabilistic models for arithmetic functions that number theorists care about. In this talk, I  will survey the subject and describe some recent progress as well as some open problems. 

  地点：Zoom: 825 9309 8083, password: 202303

• 2023-04-18，Winston Heap

  题目：Simultaneous extreme values of zeta and L-functions

  摘要：We use a modification of the resonance method to demonstrate simultaneous large values of L-functions on the critical line. The method extends to other families and can be used to show both simultaneous large and small values. Joint work with Junxian Li. 

  地点：Zoom: 825 9309 8083, password: 202303

• 2023-04-21，郭振宇   (西安交通大学)   15:00-16:00

  题目： The distribution of consecutive special primes under the Hardy-Littlewood conjecture

  摘要： In this talk, we study the distribution of pairs of consecutive Beatty or Piatetski-Shapiro primes and give conjectures with the prime counting functions. We give a heuristic argument to support the predictions which relies on a strong form of the Hardy-Littlewood conjecture. The method of the argument is highly related to Chebyshev’s bias and exponential sums.

  地点：明德楼 C702

• 2023-04-25，陈昌昊   (安徽大学)

  题目：Measure theory of Weyl sums

  摘要：I will first introduce the Weyl sums and its applications on number theory, PDE and ergodic theory. Then I will talk about the typical asymptotic bounds of Weyl sums in the sense of measures and Baire category. Joint works with R. Baker, K. Bryce, J. Maynard and I. Shparlinski.

  地点：Zoom: 825 9309 8083, password: 202303

• 2023-05-02，Matthew de Courcy-Ireland   (Stockholm University)

  题目：Six-dimensional sphere packing and linear programming

  摘要：This talk is based on joint work with Maria Dostert and Maryna Viazovska. We prove that the Cohn--Elkies linear programming bound is not sharp for sphere packing in dimension 6. This is in contrast to Viazovska's sharp bound in dimension 8, even though it is believed that closely related lattices achieve the optimal densities in both dimensions. The proof uses modular forms to construct feasible points in a dual program, generalizing a construction of Cohn and Triantafillou to the case of odd weight and non-trivial Dirichlet character. Non-sharpness of linear programming is demonstrated by comparing this dual bound to a stronger upper bound obtained from semidefinite programming by Cohn, de Laat, and Salmon. Our construction has vanishing Fourier coefficients along an arithmetic progression, which can be explained using skew self-adjointness of Hecke operators.

  地点：Zoom: 825 9309 8083, password: 202303

• 2023-05-09，李时璋   (中国科学院)

  题目：On the cohomology of classifying stacks

  摘要：我们将从有限平坦交换群概型说起，这些对象由迪厄多内模所分类，我们将看到如何从这些群概型的分类叠的上同调中得到迪厄多内模，最终我们将陈述一些关于这些分类叠的上同调的较为一般的结构性定理：这是我在与Kubrak以及Mondal的合作中得到的一个结果。

  地点：明德楼 C702

• 2023-05-16，华晟昊   (山东大学)

  题目： The distribution of central values of quadratic twisted L-functions

  摘要： We introduce some recent progress on distribution of central values of $L$-functions in quadratic twists families, including the determination of GL(3) automorphic forms, the Keating--Snaith conjecture, and the Farmer--Gonek--Hughes conjecture. This talk is based on joint works with Bingrong Huang and a joint work in progress with Mingyue Fan and Sizhe Xie.

  地点：明德楼 C702

• 2023-05-23，杨鹏宇   (中国科学院)

  题目：The distribution of periodic torus orbits in homogeneous spaces

  摘要：The distribution of periodic torus orbits in homogeneous spaces is related to several problems in number theory, such as distribution of integral points on spheres and distribution of CM points on modular curve. In this talk, we first recall the history of this subject, and then give an entropy bound for limit distributions of periodic torus orbits in the case of $\rm PGL_4$. Joint work with A. Wieser.

  地点：明德楼 C702

• 2023-05-30，Claire Burrin   (University of Zurich)

  题目：Pairs of saddle connections

  摘要：There is a collection of classical problems at the intersection of number theory and analysis that concern the distribution of lattice or primitive lattice points in their ambient space. In two dimensions, we can think of the set of primitive integer vectors in the plane as describing the set of primitive simple closed geodesics (up to free homotopy) on the two-dimensional torus. In this talk I will explain how counting saddle connections on Veech surfaces stands as a natural ‘higher genus’ analogue to the classical primitive lattice point counting problem. I will then discuss a recent result (in joint work with Jon Chaika and Samantha Fairchild) on counting pairs of saddle connections.

  地点：Zoom: 825 9309 8083, password: 202303

• 2023-06-06，黄治中   (中国科学院)

  题目： 二次超曲面有理点的分布

  摘要： 作为经典Hasse—Minkowski定理的一个实效版本，我们将利用Heath-Brown发展的delta圆法给出二次超曲面上满足适当局部条件的有理点增长率的估计，并讨论最优误差项与丢番图逼近之间的联系。本次报告基于与 D. Schindler (Göttingen) 和 A. Shute (Bristol) 最近的合作工作。

  地点：明德楼 C702

• 2023-06-13，王好武   (武汉大学)    15:00-16:00

  题目：The classification and construction of singular-weight automorphic products

  摘要：Automorphic products are modular forms with infinite product expansions and special zeros. For example, the denominator of the Weyl-Kac character formula of any affine Lie algebra defines an automorphic product of singular (i.e. minimal) weight as a Jacobi form. In this talk, we discuss automorphic products of singular weight on orthogonal groups $\rm O(n,2)$. This type of modular form is often the denominator of some Borcherds-Kac-Moody Lie algebra, and is related to vertex operator algebras. It was conjectured that there are only finitely many such modular forms. We present some results on the classification and construction of these exceptional modular forms. This talk is based on joint work with Brandon Williams.

  地点：明德楼 C702

• 2023-06-13，吴涵   (中国科学技术大学)    16:00-17:00

  题目：Some uniform Weyl subconvex bounds based on distributional Motohashi's formula

  摘要：We presented a distributional Motohashi's formula at this number theory seminar 3 years ago. We will briefly present a refined/published version of this formula. Based on this version, applications to the uniform Weyl-type subconvex bounds for some $PGL(2)$ $L$-functions, as well as the limitation of the method, will be discussed. These results are joint works with Balkanova--Frolenkov and Ping Xi, respectively. If time permits, we will present a further application to the partition function, joint work with Nicholas Andersen.

  地点：明德楼 C702

  
  

• 2023-06-16，肖玄玄   (澳门科技大学)    09:00-10:00

  题目：Large Values of L-functions and the Resonance Method

  摘要：In this report, we give an introduction of the resonance method. The method can be used in the study of large values of L-functions. With the design of the resonator, we give new results for Dirichlet L-functions and Rankin-Selberg L-functions.

  地点：明德楼 C702

• 2023-06-16，魏志宁   (俄亥俄州立大学)    10:00-11:00

  题目：The refined strong multiplicity one and its applications

  摘要：In this talk, I will introduce several refined strong multiplicity one theorems for Siegel paramodular forms for symplectic groups. As an applicationI will give a criterion for determining a Siegel cusp form by twisted L-values. This is a joint work with Xiyuan Wang, Pan Yan and Shaoyun Yi.

  地点：明德楼 C702

• 2023-07-03，张瑞祥   (美国加州大学伯克利分校)    09:00-10:00

  题目：A stationary set method for estimating oscillatory integrals

  摘要：Given a polynomial $P$ of constant degree in $d$ variables and consider the oscillatory integral $$I_P = \int_{[0,1]^d} e(P(\xi)) \mathrm{d}\xi.$$ Assuming $d$ is also fixed, what is a good upper bound of $|I_P|$? In this talk, I will introduce a ``stationary set'' method that gives an upper bound with simple geometric meaning. The proof of this bound mainly relies on the theory of o-minimal structures. As an application of our bound, we obtain the sharp convergence exponent in the two dimensional Tarry's problem for every degree via additional analysis on stationary sets. Consequently, we also prove the sharp $L^{\infty} \to L^p$ Fourier extension estimates for every two dimensional Parsell-Vinogradov surface whenever the endpoint of the exponent $p$ is even. This is joint work with Saugata Basu, Shaoming Guo and Pavel Zorin-Kranich.

  地点：明德楼 C702

• 2023-07-03，杨道道   (格拉茨工业大学)    10:00-11:00

  题目：Variants of GCD sums and extreme values of derivatives of the Riemann zeta function

  摘要：GCD sums naturally appear in a Diophantine approximation problem considered by Hardy and Littlewood in 1922.  It was until 1949 that  Gál solved their  problem.  Since  Gál's fundamental theorem, there have been new developments in the past decade. I will review these results and report some new results on the limit of GCD sums and the log-type GCD sums.   Another topic of the talk concerns large values of derivatives of zeta and L-functions. Large values of  zeta and L-functions are classical topics in analytic number theory, which can be dated back to a result of Bohr and Landau in 1910. Resonance methods are modern tools to produce large values of zeta and L-functions. GCD sums are one of important ingredients. I will talk on producing large values of derivatives of zeta and L-functions via resonance methods. The link between  log-type GCD sums and derivatives of the Riemann zeta function will also be discussed.

  地点：明德楼 C702

• 2023-08-08，侯家齐  (University of Wisconsin-Madison)  08:30-09:20

  题目：Periods of eigenfunctions on arithmetic hyperbolic 3-manifolds

  摘要：Let $X$ be a compact congruence arithmetic hyperbolic 3-manifold, and $f$ a Hecke--Maass form on $X$ with spectral parameter tending to infinity. In this talk, I will present a bound for the period of $f$ along a hyperbolic surface in $X$, which improves the local bound. This result is based on the method of arithmetic amplification.

  地点：明德楼 C702

• 2023-08-08，孙宇宸  (图尔库大学)  09:40-10:30

  题目：On divisor bounded multiplicative functions in short intervals

  摘要：In 2016, the celebrated Matomaki--Radziwi\l\l theorem showed that  there are cancellations for 1-bounded multiplicative functions in almost all short intervals. Our recent work proved that Matomaki--Radziwi\l\l theorem can be extended to divisor bounded multiplicative functions. Especially, we proved that for any sufficiently large $X$, $\epsilon>0$ and $h \geq (\log X)^{(1+\epsilon)k \log k- k +1}$, we have

       $$\frac{1}{h} \sum_{x<n \leq x+h}d_k(n) - \frac{1}{x}  \sum_{x<n \leq 2x}d_k(n) = o(\log^{k-1}x)$$

  for almost all $x \in [X,2X]$, where $d_k(n) = \sum_{m_1 \cdots m_k = n} 1$.

  地点：明德楼 C702

• 2023-08-08，Max Wenqiang Xu 徐文强  (Stanford University)  10:40-11:30

  题目：Positive definite quadratic characters

  摘要：We call a quadratic character $\chi$ mod $d$ positive definite if its partial sums at all different truncations are nonnegative. It is known that such characters are rare due to an earlier result of Baker, Montegomery and a recent result of Kalmynin.  We give a new upper bound, which is conjectured to be close to optimal based on a probabilistic model. The proof is based on establishing a $L$-function analogue of Fydorov--Hiary--Keating conjecture.

  This is joint work in progress with R. Angelo and K. Soundararajan.

  地点：明德楼 C702

往年报告：

• 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