学术报告一

报告题目：Algebraic Birkhoff factorization and locality in renormalization

报告人：郭锂 (Rutgers University–Newark)

报告时间：2021年5月5日（星期三）8:00-9:30

腾讯会议ID： 203 228 476 会议密码：210505

参加人员：本科生、研究生、教师

摘要：The Algebraic Birkhoff Factorization (ABF) of Connes and Kreimer gives an algebraic formulation of the renormalization process in quantum field theory. Their ABF is an factorization of an algebra homomorphism from a Hopf algebra to a Rota-Baxter algebra. This algebraic formulation facilitates the mathematical study in renormalization and allows the renormalization method to be applied to problems in mathematics.

In this talk we first give an introduction to ABF with regularization in a Rota-Baxter algebra. This is the case when the regularization lands in one variable Laurent series. For multi-variable regularizations, partially defined operations are needed. This is given in the setup of locality, including Hopf algebras and locality Rota-Baxter algebras. The notion of locality is an interpretation of the locality principle in renormalization, stating that a locality property is preserved in the process of renormalization. We show that if a regularization map is a locality map, then so is the corresponding renormalization map from the algebraic Birkhoff factorization. Group actions on ABF are also considered.

This is a joint work with P. Clavier, S. Paycha and B. Zhang.

报告人简介：郭锂，美国罗格斯大学纽瓦克分校教授。郭锂博士于兰州大学获学士学位，于武汉大学获硕士学位，于华盛顿大学获博士学位。 并在俄亥俄州立大学、普林斯顿高等研究院和佐治亚州大学作博士后。现任罗格斯大学数学与计算机科学系系主任。郭锂博士的数论工作为怀尔斯证明费马大定理的文章所引用，并将重整化这一物理方法应用于数学研究。他近年来推动Rota-Baxter代数及相关数学和理论物理的研究，应邀为美国数学会在“What Is”栏目中介绍Rota-Baxter代数，并出版这个领域的第一部专著。研究涉及结合代数，李代数，Hopf代数，operad，数论，组合，计算数学，量子场论和可积系统等数学和理论物理的广泛领域。

学术报告二

报告题目：凸多面体锥的剖分

报告人：张斌(四川大学)

报告时间：2021年5月5日（星期三）9:30-11:00

腾讯会议ID： 203 228 476 会议密码：210505

参加人员：本科生、研究生、教师

摘要：凸多面体锥的剖分在凸多面体锥的应用中扮演重要角色，而凸多面体锥的剖分的集合则富含结构。这个报告中将介绍我们发现的剖分的结构，比如洗牌结构，和分式的对应，导子结构等等。这是基于和郭锂教授，Paycha教授合作的结果。

报告人简介：张斌，四川大学数学学院教授，主要从事几何和数学物理方向研究，探索重整化方法的相关应用。

学术报告三

报告题目：Some algebraic structures on rooted trees

报告人：高兴(兰州大学)

报告时间：2021年5月5日（星期三）14:00-15:00

腾讯会议ID： 203 228 476 会议密码：210505

参加人员：本科生、研究生、教师

摘要：In this talk, we first recall some classical Hopf algebras on rooted trees, including Connes-Kreimer Hopf algebra and Loday-Ronco Hopf algebra. Then we give a combinatorial description of the coproduct of the Loday-Ronco Hopf algebra, and construct an infinitesimal version of the Connes-Kreimer Hopf algebra. Finally, a dendriform-Nijenhuis bialgebra is built on top of decorated planar rooted trees.

报告人简介：高兴，博士，兰州大学萃英教授、博士生导师。于2010年7月在兰州大学数学与统计学院获得博士学位，留校工作至今。在2015年8月至2016年8月间，在美国Rutgers大学交流访问。主要从事Rota-Baxter代数和代数组合等领域的研究， 在Journal of Algebra、 Journal of Pure and Applied Algebra、J. Algebraic Combin. 等国际期刊上发表SCI学术论文四十余篇。主持数学天元基金、青年科学基金、国家自然科学基金面上项目和甘肃省自然科学基金项目, 获甘肃省自然科学奖二等奖，出版教材一本。

学术报告四

报告题目：An algebraic study of Volterra integral equations and their operator linearity

报告人：黎允楠(广州大学)

报告时间：2021年5月5日（星期三）15:00-16:00

腾讯会议ID： 203 228 476 会议密码：210505

参加人员：本科生、研究生、教师

摘要：The algebraic study of special integral operators led to the notions of Rota-Baxter operators and shuffle products which have found broad applications such as iterated integrals. In this talk we point out that there are rich algebraic structures underlying Volterra integral operators and the corresponding equations.

First Volterra integral operators with separable kernels can produce a matching twisted Rota-Baxter algebra satisfying twisted integration-by-parts operator identities. To provide a universal space to express general integral equations, we construct free (relative) operated algebras in terms of bracketed words or rooted trees with decorations on the vertices and edges.

Utilizing the free construction of matching Rota-Baxter algebras by Gao-Guo-Zhang, further explicit constructions of the free objects in the category of matching twisted Rota-Baxter algebras are given, providing a universal space for separable Volterra equations. As an application, we show that any separable Volterra integral equation is operator linear in the sense that it can be simplified to a linear combination of iterated integrals.

This is joint work with Li Guo and Richard Gustavson.

报告人简介：黎允楠，广州大学数学与信息科学学院副教授，硕士生导师，博士毕业于华东师范大学数学系，研究方向为李代数、量子群与代数组合。现与合作者在国际知名数学期刊Mathematische Zeitschrift, Journal of Combinatorial Theory, Series A., Journal of Algebra, Journal of Algebraic Combinatorics等发表论文数篇，完成国家自然科学基金青年基金项目，主持和参与国家自然科学基金面上项目各1项。2015年成为美国数学会数学评论网评论员，2018年国家公派访问学者前往美国罗格斯大学研修1年，曾受邀为Advances in Mathematics, European Journal of Combinatorics, Journal of Algebraic Combinatorics, The Ramanujan Journal等国际知名数学期刊审稿。

学术报告五

报告题目：Term Rewriting Systems on Free Modules

报告人：郑上华（江西师范大学)

报告时间：2021年5月5日（星期三）16:00-17:00

腾讯会议ID： 203 228 476 会议密码：210505

参加人员：本科生、研究生、教师

摘要：Rewriting rules are used in symbolic computations on vector spaces, by which a basis element appearing in a vector is replaced by another vector. For example, (term) rewriting rules are used to compute Grobner bases for polynomial ideals. Often the replacement vector, when expressed in terms of the basis, does not involve the basis element being replaced. We call a rewriting system simple when this holds for all rewriting rules. In this paper, we give multiple characterizations of the confluence property of a simple rewriting system on a free module over a commutative unitary ring in terms of binary relations derived from it. We further study the properties of simple rewriting systems when the basis is equipped with a compatible linear order, as well as the inheritance of these properties to free submodules.

报告人简介：郑上华，理学博士,江西师范大学副教授，硕士生导师。2015年6月毕业于兰州大学，获得理学博士学位。主要从事罗巴代数，项重写系统和Grobner-Shirshov基理论等领域的研究。主持国家自然科学基金2项，江西省教育厅科技项目1项。在中国科学、J. Lie Theory、Comm. Algebra、Pacific J. Math.等著名期刊发表学术论文7余篇。