Schedule

Tentative Schedule

All talks will be in Skiles 006

Here are Orsola Capovilla-Searle’s notes from the lectures, math Q&A, and recitations:Part 1, Part 2, Part 3, and Part 4

The main part of the summer school will be 10 Lectures by Roger Casals. These lectures will be supplemented by the following active learning activities:

• Math Q&A: participants will be able to ask questions about the lecture series.
• Recitations: participants will work on problem sets with the help of an expert in the field.
• Group Work: participants will break into groups to work on the problem sets. 
• Bonus Lectures: Lenny Ng will discuss connections to holomorphic curves and augmentations (details below). 
• Wrap-up: Roger Casals will summarizing the main mathematical takeaways from the day and a brief peek at what is to come.

There will be two Research Talks on recent developments in Legendrian knot theory and microlocal sheaves. See below for details.

During lunch on two of the days there will be Panel Discussions. One will be about the job process in academia and the second will be about communication in mathematics and will discuss writing papers/grants, publishing, and giving talks.

Here are the slides about the job process: OverviewJobs2025CBMS

Here are the slides about communicating mathematics: Communicating Mathematics

Here are the list of problems discussed in the Open Problem session (thanks to Orsola Capovilla-Searle for the great notes): Open Problem Session Legendrian And Sheaves Summer School

Tuesday’s Research Talk will be by Wenyuan Li.
Title: Structural results of microlocal sheaves for Legendrians
Abstract: The category of sheaves on M x R with singular support on a Legendrian submanifold in the 1-jet bundle of M is a Legendrian isotopy invariant. We will discuss some structural results for the sheaf categories, many of which are parallel to the ones in Legendrian contact homology and Lagrangian Floer theory: (1) Lagrangian cobordisms induce functors between the categories of sheaves singularly supported on the Legendrians, and (2) morphisms of sheaves singularly supported on Legendrians satisfy a duality exact sequence coming from a relative Calabi-Yau structure. Time permitting, I will also talk about the Hochschild homology of these categories of sheaves and the relations between circle actions and Hamiltonian and Reeb orbits. Part of this is joint work with Chris Kuo.

Thursday’s Research talk will be by Orsola Capovilla-Searle.
Title: Ruling decompositions and weave decompositions agree
Abstract: Legendrian links can arise as the boundary of exact Lagrangian surfaces in the standard symplectic 4-ball. Such surfaces are called Lagrangian fillings of the link. In the last decade, our understanding of the moduli space of fillings for various Legendrians has greatly improved thanks to tools from sheaf theory, Floer theory and cluster algebras. On the one hand, Henry-Rutherford showed that the augmentation variety of a Legendrian with a fixed front, which in a way incarnates the space of Lagrangian filings, admits a decomposition into certain strata constructed using normal graded rulings. On the other hand, if we focus on Legendrians given by positive braids, the moduli space of fillings is isomorphic to a braid variety. Casals-Gorsky-Gorsky-Simental showed braid varieties admit a decomposition into certain strata constructed using algebraic weaves. The goal of the talk is to introduce these two types of decompositions and show that they in fact agree. This is joint work with Johan Asplund, James Hughes, Caitlin Leverson, Wenyuan Li, and Angela Wu.

Lenny Ng’s Bonus Lectures on Legendrian links and Floer theory.
Abstract: This pair of lectures will complement the main lecture series by discussing an alternate approach to studying Legendrian links, through Floer theory and specifically Legendrian contact homology. The Floer-theoretic story began fairly independently of the sheaf and cluster approaches described in the main lectures, but now all of these pictures are quite closely connected.

In the first lecture, I’ll introduce Legendrian contact homology and the Chekanov-Eliashberg differential graded algebra, for the setting of Legendrian links in standard contact R^3. We’ll consider both the combinatorial model for the dga and its symplectic-geometric motivation. This will culminate in the construction of the augmentation variety for a Legendrian link. In the second lecture, I’ll survey some connections between the Floer picture and topics from the main lectures, including the behavior of the dga under exact Lagrangian cobordisms, methods to distinguish fillings using augmentations, and relations between augmentations and both constructible sheaves and cluster structures.

Lecture 1

Lecture 2