Nov.11, 2023
9:30-10:30
Speaker：Ma Luo
Title：Iterated integrals of modular forms, I
Abstract：
Iterated integrals was introduced and developed by Kuo-Tsai Chen, to study fundamental groups and higher homotopy groups. This theory has now become an important tool not only in topology, but also in algebraic geometry and in number theory. To illustrate the versatility of iterated integrals, various properties, examples, and applications will be presented in my first talk. In the second talk, we focus on a special class of numbers called multiple zeta values, which appear in number theory, in the study of motives, and in quantum field theory. They can be written as iterated integrals due to Kontsevich's observation. Recently, it was shown that they can be written differently as iterated integrals of modular forms. All of these are subsumed in a program called Galois theory of periods, which we will describe and discuss.
11:00-12:00
Speaker：Ce Xu
Title：Alternating Multiple Mixed Values
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 Q[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 Lu Yan and Jianqiang Zhao.
14:00-16:00
Speaker：Zigang Zhu
Title：Computer Algebra in the study of Automorphism Groups of Smooth Hypersurfaces
Abstract：
In this talk, we introduce notions of partitionability and characteristic sets of homogeneous polynomials and give a complete classification of groups faithfully acting on smooth cubic fivefolds and fourfolds. The classification is computer-aided with GAP (Groups, Algorithms, Programming), SageMath, and Mathematica. We demonstrate the functionalities of GAP, SageMath, and Mathematica, and discuss their interactions. We illustrate the usage of these mathematical softwares in the classification process through specific examples. This talk is based on a joint work with Song Yang and Xun Yu.
Nov.12, 2023
9:30-10:30
Speaker：Ma Luo
Title：Iterated integrals of modular forms, II
Abstract：
Iterated integrals was introduced and developed by Kuo-Tsai Chen, to study fundamental groups and higher homotopy groups. This theory has now become an important tool not only in topology, but also in algebraic geometry and in number theory. To illustrate the versatility of iterated integrals, various properties, examples, and applications will be presented in my first talk. In the second talk, we focus on a special class of numbers called multiple zeta values, which appear in number theory, in the study of motives, and in quantum field theory. They can be written as iterated integrals due to Kontsevich's observation. Recently, it was shown that they can be written differently as iterated integrals of modular forms. All of these are subsumed in a program called Galois theory of periods, which we will describe and discuss.
11:00-12:00
Speaker：Ce Xu
Title：On Some Unramified Families of Motivic Euler Sums
Abstract：
It is well known that sometimes Euler sums (i.e., alternating multiple zeta values) can be expressed as Q-linear combinations of multiple zeta values (MZVs). In her thesis Glanois presented a criterion for motivic Euler sums (MES) to be unramified, namely, expressible as Q-linear combinations of motivic MZVs. By applying this criterion we present a few families of such unramified MES in two groups. In one such group we can further prove the concrete identities relating the MES to the motivic MZVs, determined up to rational multiple of a motivic Riemann zeta value by a result of Brown. This is a joint work with Jianqiang Zhao.
14:00-16:00
Discussions
