【廿周年院庆学术报告128】 · 【和山数学论坛第402期】

一、报告题目：Algebraic Birkhoff Factorization and Locality in Renormalization

二、报告人： 郭锂 教授

三、时  间：2024年1月12日(周五)  15:30-16:30

四、地  点：A4-216

报告摘要：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 divergency problems in mathematics.

In this talk we first give an introduction to ABF with a baby model for renormalizing Riemann integrals, in the spirit of dimensional regularization into Laurent series. To deal with multivariant regularizations, we develop a Laurent series theory for meromorphic germs with linear poles and formulate the ABF in the locality setting. The latter involves locality for various algebraic structures including those of a Hopf algebra, a Rota-Baxter algebra and a regularization map between the two algebras. We then show that if a regularization map is a locality map, then so is the corresponding renormalization map from the algebraic Birkhoff factorization. As an application in the context of the Euler-Maclaurin formula on lattice cones, we renormalize the exponential generating function which sums over the lattice points in a lattice cone. We also explore renormalization groups and revisit the analytic renormalization of Speer. Despite the analytic and physics background, the talk is mostly algebraic and combinatorial.

The talk is from joint works 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。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，并将怀尔斯文中的主猜想推广到高权模形式上。他近年来将重整化这一物理方法应用于数学研究，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。在Duke Math. J.，Comm.Math.Phy.，Adv.Math.，Trans. AMS，IMRN，Math Ann.等国际著名杂志发表论文120余篇。

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