Bridges between symplectic geometry and combinatorics of polytopes

20.06.2018, 14:00  –  Haus 9, Raum 2.22
Institutskolloquium

Silvia Sabatini, Milena Pabiniak (Uni Köln)

14:00 Silvia Sabatini (Köln): 12, 24 and beyond: A bridge from reflexive Polytopes to symplectic Geometry
15:00 Tea and Coffee break
15:30 Milena Pabiniak (Köln): How to distinguish between symplectic toric manifolds?

Abstracts:

Silvia Sabatini (Köln): 12, 24 and beyond: A bridge from reflexive Polytopes to symplectic Geometry
Mathematics finds itself divided and subdivided into hyper-specialized areas of study, each of them with its own internal beauty. However, what I find most fascinating is when one can build a bridge between two of these seemingly isolated theories. For instance, (symplectic) geometry and combinatorics have a very strong connection, due to the existence of some special manifolds admitting a torus symmetry. The latter is encoded in a map, called moment map, which "transforms" the manifold into a convex polytope. Hence many combinatorial properties of (some special types of) polytopes can be studied using symplectic techniques. In this talk I will focus on the class of reflexive polytopes, which was introduced by Batyrev in the context of mirror symmetry, and explain how the "12" and "24" phenomenon in dimensions 2 and 3 can be generalized to higher dimensions using symplectic geometry.

Milena Pabiniak (Köln): How to distinguish between symplectic toric manifolds?
The main protagonists in this talk are symplectic toric manifolds, i.e. manifolds equipped with a closed, non-degenerate 2-form, called a symplectic structure, and "many" symmetries, coming from an action of a torus of big dimension. I will explain how such manifolds are classified by convex polytopes (Delzant classification), building a bridge between symplectic geometry and combinatorics of convex polytopes.
Delzant classification is a classification up to equivariant symplectomorphisms, i.e. diffeomorphisms respecting not only the symplectic form but also the torus action. A classification of symplectic toric manifolds up to diffeomorphisms or symplectomorphisms remains mysterious. We still don't have tools to recognize when two different polytopes correspond to the same symplectic manifold equipped with two different toric actions. Masuda and Suh suggested that maybe the integral cohomology ring could classify symplectic toric manifolds up to diffeomorphisms.
In the second part of the talk I will discuss my joint work with Sue Tolman where we investigate this suggestion. I will explain how one calculates the integral cohomology ring of a symplectic toric manifold from the associated polytope. I will further discuss how one can use a construction borrowed from algebraic geometry, called toric degeneration, to construct desired symplectomorphisms. In particular, we will see how such non-equivariant symplectomorphisms affect polytopes associated to the symplectic toric manifolds in question. There will be lots of pictures.