Multiple zeta values, WZ-pairs and some infinite sums
Kam Cheong Au (区锦昌)
University of Bonn, Germany
Abstract: The technique of Wilf-Zeilberger pair (WZ-pair), developed in 1970s, is known for its ability to prove many identities. On the other hand, in 1980s, researches about algebraic structure of multiple zeta values (MZVs) emerged; application of it towards classical problems like infinite sum has only been recently developed. Its techniques and applicable range are both very different from WZ method.
We combine these two powerful methods (WZ pairs and multiple zeta values) to prove a large number of series identities due to Z.-W. Sun, many of them have been long standing conjectures. Mathematica implementations of above procedures will also be mentioned. Moreover, we shall also discuss some interesting observations and further potential applications of these techniques.
Multiple L-values of conductor four, Entringer numbers, and modular forms
Masanobu Kaneko (金子昌信)
Faculty of Mathematics, Kyushu University
744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
Abstract: We study the multiple L-values (MLVs) with characters of conductor four, and find an unexpected connection to the so-called Entringer numbers in combinatorics. We also mention a conjectural relation between MLVs of depth 2 and period polynomials associated to modular forms of level four. This is a joint work with Hirofumi Tsumura.
Many p-adic odd zeta values are irrational
Li Lai (赖力)
Beijing International Center for Mathematical Research, Peking University
Beijing 100871, P. R. China
Abstract: Let p be any prime number. The p-adic zeta function $\zeta_p(\cdot)$ is obtained by the p-adic interpolation of special values of the Riemann zeta function $\zeta(\cdot)$. In 2005, Calegari proved that both $\zeta_2(3)$ and $\zeta_3(3)$ are irrational. In this talk, we show that for any $\varepsilon> 0$ and any sufficiently large odd integer $s \geqslant s_0(p,\varepsilon)$, the number of irrational numbers among $\zeta_p(3)$, $\zeta_p(5)$, $\zeta_p(7)$, $\ldots$, $\zeta_p(s)$ is at least $(c_p-\varepsilon)\sqrt{s/\log s}$, where $c_p> 0$ is a constant depending only on p. This is a joint work with Johannes Sprang.
Relations of multiple t-values
Zhonghua Li (李忠华)
School of Mathematical Sciences, Tongji University
Shanghai 200092, P. R. China
Abstract: As level 2 variants of multiple zeta values, Hoffman introduced the multiple t-values, which have many relations similar to the multiple zeta values. In this talk, we will focus on the relations of the multiple t-values. In particular, we will talk about some evaluation formulas of the multiple t-values.
Multiple zeta values: From arithmetic to physics
Cezar Lupu
Yanqi Lake Beijing Institute of Mathematical Sciences and Applications
Beijing 101408, P. R. China
Abstract: In this talk, we give a survey on multiple zeta values with an emphasis on their applications ranging from arithmetic properties of odd zeta values to periods of zig-zag graphs in quantum field theory. We explore various identities involving different families of multiple zeta values and we emphasize their importance in proving some very important conjectures.
Series with denominators of summandscontaining binomial coefficients
Zhi-Wei Sun (孙智伟)
Department of Mathematics, Nanjing University
Nanjing 210093, P. R. China
Abstract: In this talk, we give a survey of infinite series identities with denominators of summands containing binomial coefficients.Besides various known results, we also mention some open series identities conjectured by the speaker.
Levinson's method and zeros of Dirichlet L-functions
Xiaosheng Wu (吴小胜)
School of Mathematics, Hefei University of Technology
Hefei 230009, P. R. China
Abstract: We will introduce the classical Levinson's method and its application in the distribution of zeros of Dirichlet L-functions. Some processes on simple zeros are also enclosed.
多重zeta值的洗牌Hopf代数
Hongyu Xiang (向洪玉)
School of Mathematics, Sichuan University
Chengdu 610064, P. R. China
摘要:在这个报告中, 我们主要讨论如何在多元zeta值shuffle代数上构造与shuffle乘法相容的余乘结构. 我们的主要技巧是采用类似涨开(blowing-up)的技术将shuffle代数结构提升到陈分式构成的Locality代数. 利用陈分式的偏导数算子, 我们定义了陈分式上与Locality乘法相容的余乘, 然后我们把这一余乘搬到多元zeta值shuffle代数上, 得到和shuffle乘法相容的余乘. 这是在张斌老师的指导下完成的工作.
The Hopf algebra of weak quasisymmetric functions
Houyi Yu (喻厚义)
School of Mathematics and Statistics, Southwest University
Chongqing 400715, P. R. China
Abstract: As a natural basis of the Hopf algebra of quasisymmetric functions, monomial quasisymmetric functions are formal power series parameterized by compositions. In this talk, we extend such a definition for weak compositions and obtain the Hopf algebra of weak quasisymmetric functions. This Hopf alghebra gives the free commutative Rota-Baxter algebra a power series realization, in support of a suggestion of Rota that Rota-Baxter algebra should provide a broad context for generalizations of symmetric functions. This is a joint work with Li Guo, J.-Y. Thibon and Bin Zhang.
带线性极点的实亚纯函数芽的局部化Galois理论
Bin Zhang (张斌)
Yangtze Center for Mathematics, Sichuan University
Chengdu 610064, P. R. China
摘要: 在这个报告中, 我们主要讨论带线性极点的实亚纯函数芽的一类特殊的变换群, 即局部化的Galois群, 希望以此来描述重整化方法应用中出现的重整化群. 这一探索中关键概念是我们引入的局部化结构, 我们将利用“依赖空间”的概念, 仔细讨论带线性极点的实亚纯函数芽的局部化关系, 进而描述由一些分式生成的局部化子代数的结构. 这些局部化子代数的固定全纯函数芽且保持留数的局部化自同构群就是局部化Galois群. 在两个特殊的局部化子代数上, 通过对局部化子代数结构的探索, 我们具体计算了其局部化Galois群. 作为应用之一, 我们给出了Speer对Feynman振幅的解析重整化的数学解释, 证明了这时局部化Galois群在这类重整化上作用是传递的. 这是和郭锂教授, Paycha教授合作的结果.
Alternating multiple mixed values
Jianqiang Zhao (赵健强)
Department of Mathematics, The Bishop's School
La Jolla, CA 92037, United States of America
Abstract: In this talk we define and study a variant of multiple zeta values (MZVs) of level four, called alternating multiple mixed values (AMMVs), forming a $\mathbb{Q}[\mathrm{i}]$-subspace of the colored MZVs of level four. This variant includes the alternating version of Hoffman's multiple t-values, Kaneko-Tsumura's multiple T-values, and the multiple S-values studied by the authors previously as special cases. We exhibit nice properties of AMMVs similar to the ordinary MZVs such as the duality, integral shuffle and series stuffle relations and then establish some other explicit relations among them. We will also discuss some conjectures concerning the dimensions of the above-mentioned subspaces of AMMVs. These conjectures hint at a few very rich but previously overlooked algebraic and geometric structures associated with these vector spaces. This is a joint work with Ce Xu and Lu Yan.
Hyper-Mahler measures and binomial harmonic sumsvia Goncharov--Deligne cyclotomy
Yajun Zhou (周亚俊)
Academy of Advanced Interdisciplinary Studies (AAIS), Peking University
Beijing 100871, P. R. China
Abstract: The hyper-Mahler measures $m_k(1+x_1+x_2), k\in\mathbb Z_{>1}$ and $m_k(1+x_1+x_2+x_3),k\in\mathbb Z_{>1}$are evaluated in closed form via Goncharov--Deligne periods, namely$\mathbb Q$-linear combinations of multiple polylogarithms at cyclotomic points (complex-valued coordinates that are roots of unity). Some infinite series involving binomial coefficients and harmonic numbers are also explicitly representable as Goncharov--Deligne periods of levels 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18 and 24, which find applications to some recent conjectures of Z.-W. Sun.
