Conference: CBMS Conference: Legendrian Links and the Microlocal Theory of Sheaves

This award provides support to a CBMS conference on Legendrian links and the microlocal theory of sheaves. This will be a week-long workshop to be held June 9-16, 2025 at the Georgia Institute of Technology. The workshop will be centered around a mini-course consisting of ten lectures by Dr.

Roger Casals and will be supplemented with several talks by other experts. There will also be sessions where participants can engage with the material by working on problems and discussing the lectures with experts. While the organizers hope to have broad participation from across the country, they will focus on participants from the Southeast.

There are currently many universities in the region with active research groups in symplectic and contact geometry, such as the University of Georgia, Duke University, the Georgia Institute of Technology, the University of Alabama, and Louisiana State University, but few have experts in the topic of this workshop. So this will be a valuable opportunity to expand the tools available to researchers in the Southeast.

This workshop should be of interest to a wide range of graduate students and researchers in nearby fields.

In more detail, workshop will focus on recent developments in the study of Legendrian links in contact 3-manifolds. Specifically, the newly discovered connection between Lagrangian fillings and cluster algebras, as well as its method of proof, using the microlocal theory of sheaves. This is a topic of current research interest and there has been significant activity in the area, such as establishing that the sheaf moduli associated with any Legendrian (-1)-closure of a positive braid admits a cluster structure.

Several open questions have now been solved based on this result and the study of weaves, including the existence of infinitely many Lagrangian fillings, the existence of cluster structures on Richardson varieties, and the Fock-Goncharov duality conjecture for braid varieties, among others. Several generalizations and related results are also being studied, including a conjectural classification of Lagrangian fillings. The conference website is https://sites.gatech.edu/ttss2025/.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.