题目：An introduction to augmentation-sheaf correspondence

报告人：高鸿灏教授 (Grenoble大学)

Lecture 1: Knots, links and braids

We review the basic concepts and constructions in knot theory and braid theory, including Reidemeister’s theorem, Alexander’s theorem and Markov’s theorem. Knot/link invariants are algebraic constructions which are invariants under knot/link isotopies. We will focus on the knot/link group as an example.

Lecture 2: Augmentations and link group representations

We introduce a non-commutative framed cord algebra as a link invariant. Augmentations are rank one representations of the framed cord algebra. We will construct a link group representation from the data of an augmentation, relating two link invariants.

Lecture 3: Microlocal sheaf theory and Legendrian invariants

This is a concise introduction to microlocal sheaf theory. The key concept is micro-support of a sheaf, which measures the propagation of the sheaf along codirections. Using micro-support at a geometric constraint, Guillermou-Kashiwara-Schapira constructed a category of sheaves, which is an invariant for smooth Legendrian submanifolds.

Lecture 4: Legendrian contact dga and its augmentations

The Floer theoretic approach to a compact Legendrian in a  contact one-jet space is thorough a differential graded algebra as a Legendrian invariant. Augmentations are dga morphisms to a trivial dga. We will walk through these constructions using an explicit example.

Lecture 5: Augmentations and sheaves for links

Given a simple sheaf microsupported along the link conormal, we construction an augmentation of the framed cord algebra of the link.

Together with a previous construction, we get a correspondence between augmentations and sheaves.

Lecture 6: Further directions

We have several options for this slot:

?Augmentation category

Given the dga associated to a Legendrian link, augmentations of the dga admit the structure of an A-infinity category. The construction of the category depends on both the dga as well as the original Legendrian. The definition is motivated by the symplectic geometry.

Using this category, we have a more “correct” moduli space of augmentations.

?Twisted augmentation-sheaf correspondence

The Floer moduli space remembers the counting of the disk interior wrapping the base manifold. These enhanced data induces augmentations deformed by a quantum (or non-exact) parameter. It is conjectured that these augmentations corresponds to twisted sheaves. We will give an example as evidence.

?Cluster mutations in the augmentation variety

The sheaf interpretation of the augmentation variety implies that there exists a cluster structure. We will explain how to capture this structure from the geometry and combinatorics of the Legendrian.

时间：5月31日（周五）、6月3日（周一）上午9:45-10：45

地点：565net必赢客户端本部教二楼 527 教室

时间：6月6日（周四） 上午9:45-10：45

地点：565net必赢客户端本部教二楼 827 教室

联系人：孙善忠

欢迎全体师生积极参加！