走向现代数学学术报告 - Daniel K. Nakano教授（No. 792)
报告题目：An Introduction to Monoidal Triangular Geometry
报告专家：Daniel K. Nakano 教授 (University of Georgia)
邀 请 人：徐斐 教授
报告时间：2025年5月19日 14:30
报告地点：E2-401, GTIIT South Campus
报告摘要：In your undergraduate mathematics education, one of the first algebraic objects that you were introduced to was the concept of a group or a ring. Rings are natural objects with an addition, multiplication and unit. Later on, in your graduate education, you probably took a course on commutative ring theory, where you learned that one can form a topological space via the prime ideals of the ring. In this setting the connections between algebra and geometry become more transparent.
In this talk, I will introduce the ideas of tensor triangular geometry and monoidal triangular geometry. Balmer introduced tensor triangular geometry, in 2005, as a way to study symmetric tensor categories using the additive and multiplicative structures on the objects. That is, one can view symmetric monoidal (triangulated) categories like a commutative ring, and study its spectrum. The presenter along with Vashaw and Yakimov have recently investigated monoidal triangular geometry where the multiplicative structure is not necessarily commutative. Our setting is more conducive to the study of arbitrary finite-dimensional Hopf algebras and finite tensor categories.
After introducing the basic concepts, I will demonstrate how to compute the Balmer spectrum using support data techniques. Examples will be shown that demonstrate how natural geometric objects can be realized via the Balmer spectra. At the end of the talk, I will discuss the Chinese remainder theorem for rings that dates back to the 5th century. A 21st century version of the Chinese remainder theorem will then be presented for monoidal triangulated categories that involves localization functors.
Many of the results in this talk involve joint work with Kent Vashaw and Milen Yakimov.
个人简历：Daniel K. Nakano is a Distinguished Research Professor of Mathematics at the University of Georgia. He is an expert in categorical and homological methods in representation theory, including cohomology and support varieties for algebraic and finite groups, algebraic and geometric methods in Lie theory, and monoidal triangulated categories. Nakano graduated from the University of California, Berkeley in 1986, and earned a doctorate in mathematics from Yale University in 1990 under the supervision of George B. Seligman. After temporary positions at Auburn University and Northwestern University, he became an assistant professor at Utah State University in 1994 and moved to the University of Georgia in 2001. In 2010, Nakano was named Distinguished Research Professor. In 2012, he became one of the inaugural fellows of the American Mathematical Society. In 2016, he received the Lamar Dodd Award Creative Research Award. He is currently an editor for the Transactions/Memoirs of the American Mathematical Society, and was on the editorial boards of Communications in Algebra, Journal of Pure and Applied Algebra.
