Seminar on geometry, symmetry, and physics, Spring 2020

(aka mirror symmetry/related topics)

Thursdays (usually )2:00-3:00pm in DIMACS 431

Organized by Lev Borisov, Emanuel Diaconescu, Angela Gibney, and Chris Woodward

January 30 No talk (Note the workshop on unitary duals starting Wednesday.)

February 6

February 13 Giulia Gugiatti, Imperial

Title: Mirrors of the Johnson-Kollár series

Abstract: I will construct LG mirrors for the Johnson-Kollar series of anticanonical del Pezzo surfaces in weighted projective 3-spaces. The main feature of these surfaces is that their anticanonical linear system is empty, thus they fall outside of the range of the known mirror constructions. For each of these surfaces, the LG mirror is a pencil of hyperelliptic curves. I will exhibit the regularised I-function of the surface as a period of the pencil and I will sketch how to construct the pencil starting from a work of Beukers, Cohen, and Mellit on finite hypergeometric functions. This is joint work with Alessio Corti.

March: Pandemoniacal interlude

Wall Crossings and Lagrangian Cobordisms Jeff Hicks, Cambridge Location: Platform TBA Date & time: Thursday, 16 April 2020 at 2:00PM - 3:00PM We expect that two Lagrangian submanifolds have matching Floer theoretic invariants if they are related by a Hamiltonian isotopy. For example, an appropriately weighted count of Maslov index 2 disks with boundary on a Lagrangian L should remain unchanged under Hamiltonian isotopy. If the Hamiltonian isotopy is not very nice -- -- --say that at some point the isotopy -- --passes through a Lagrangian which -- --bounds a non-regular Maslov index 0 -- --disk -- then weights used to count the disks need to be corrected by a ``wall crossing transformation'' to obtain invariance. These transformations end up playing an important role in mirror symmetry, where they are the coordinate transformations used to build a mirror space from an SYZ fibration. Lagrangian cobordisms give an equivalence relation on Lagrangian submanifolds which is weaker than Lagrangian isotopy, but is still expected to preserve Floer theory. In this talk, we look at a first example of a non-cylindrical Lagrangian cobordism providing an equivalence of Lagrangian Floer theory, and relate this to the story of wall-crossing arising from Hamiltonian isotopy.

April 23, Jack Smith, Cambridge

May 7 no talk (Yuhan Sun defense)

May 14, Umut Varolgunes, Stanford

I will start by explaining what I mean by a symplectic cluster manifold and how to represent them by certain combinatorial data called an eigenray diagram (4d only!). These manifolds admit a Lagrangian fibration over the real plane with only focus-focus singularities. They do not need to have convex boundary or exact symplectic form, but they are open and geometrically bounded. Eigenray diagrams are related to toric models and the relation will be briefly mentioned. Then, using relative symplectic cohomology and a locality statement that relies on monotonicity techniques, I will describe conjectural mirrors of symplectic cluster varieties as certain deformed (over the Novikov field) cluster varieties. I will also briefly explain the step to go from this to homological mirror symmetry, which is ongoing work. Our construction generalizes, from a purely symplectic perspective and in various directions, the works of Gross, Hacking, Keel, Kontsevich, and Siebert. This is joint work with Yoel Groman.