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Categorical skew Howe duality and Knot invariants
Aaron D. Lauda教授(美国南加州大学数学系)
2020年12月14日、17日  Zoom会议

*主持人:胡乃红教授
*时间:2020年12月14日8:30---10:30 a.m.
2020年12月17日8:30---10:30 a.m.
*地点:Zoom 会议ID: 941 9680 3264 密码: 122133
Zoom 会议ID: 925 5399 8411 密码: 706922
*主办单位: 华东师范大学数学科学学院
承办单位: 华东师范大学数学科学学院

*讲座内容简介:
(1)Introduction to skew Howe duality:
摘要:The Reshetikhin-Turaev construction associated knot invariants to the data of a simple Lie algebra and a choice of irreducible representation. The Jones polynomial is the most famous example coming from the Lie algebra sl(2) and its two-dimensional representation. In this talk we will explain Cautis-Kamnitzer-Morrison's novel new approach to studying RT invariants associated to the Lie algebra sl(n). Rather than delving into a morass of representation theory, we will show how two relatively simple Lie theoretic ingredients can be combined with a powerful duality (Howe duality) to give an elementary and diagrammatic construction of these invariants. We will explain how this new framework solved an important open problem in representation theory, proves the q-holonomicity conjecture in knot theory (joint with Garoufalidis and Lê), and leads to a new elementary approach to categorifying' these knots invariants to link homology theories.

(2) Skein theory and computing knot invariants via Howe duality
摘要:In this talk we show how two simple Lie theoretic ingredients (the definition of the idempotent form of the quantum group for sl(m); the quantum Weyl group action for sl(m) can be combined with Howe duality to recover the sl(n)-skein theory. This includes the Kauffman bracket construction of the Jones polynomial, Kuperberg's spider for sl(3), and the MOY calculus for sl(n). The key point here is that a detailed analysis of representations is not required. We will recover the diagrammatic constructions of these knot invariants by utilizing Howe duality.

(3)Introduction to categorified quantum groups
摘要:Knot invariants like the Jones polynomial are really just the shadows of more sophisticated invariants known as knot homologies. Just as quantum knot invariants arise from the representation theory of quantum groups, we show how knot homologies arise from a categorification of quantum groups. This is a higher categorical structure that encodes an additional layer of Lie theory.

(4)Categorical skew Howe duality and Knot invariants
摘要This talk will be the culmination of the prior lectures. We will show that an identical framework that we used to understand quantum knot invariants and their diagrammatic can be used to categorify these invariants. The key idea is to lift the quantum group to the categorified quantum group, the quantum Weyl group action to a categorical analog. Then using the same machinery from the first two lectures we arrive at an elementary diagrammatic construction of sl(n) homology. This includes Khovanov's categorification as a special case.


*主讲人简介:
Aaron D. Lauda教授是美国南加州大学数学系教授、著名的量子群范畴化及链环和3-流形量子不变量专家,JPAA《纯粹和应用代数学杂志》编委。他从事领域包括:低维拓扑与拓扑量子场论,拓扑量子计算,代数结构范畴化,李代数与量子群的表示理论。2005年他获得剑桥大学Rayleigh-Knight奖,2011年获得Alfred P. Sloan 奖金,2013年获得美国国家科学基金会的“杰出青年教授奖”,2014年为首尔ICM大会“李理论与推广”部分的会议主席,2016年成为 Simons数学研究员,并在2019年成为美国数学学会会员。他参加众多特邀系列讲座,编写了《Categorification in Geometry, Topology, and Physics》与《Categorification and Higher Representation Theory》,发表论文近40篇,并获得同行广泛引用与好评。