Workshop

For a symplectic manifold with an anti-symplectic involution one can consider J-holomorphic maps invariant under the involution. These maps give rise to real Gromov-Witten invariants and are related to real enumerative geometry in the same spirit as their more classical counterparts. In this talk I will give an overview of the developments in real Gromov-Witten theory and discuss some properties of the invariants.

I will give an overview of results on mirror symmetry for G/P, with special emphasis on  Grassmannians and Dubrovin/Givental style mirror symmetry.

We use the A and X-cluster structures on the Grassmannian to exhibit a polytopal manifestation of mirror symmetry for Grassmannians. From a given cluster seed we obtain both an X-cluster chart and an A-cluster chart for the Grassmannian. We use the X-cluster chart and a naturally defined valuation to construct a Newton-Okounkov body, defined as the convex hull of points. Meanwhile we use the A-cluster to express the superpotential as a Laurent polynomial, and by tropicalizing this expression, we obtain a polytope, defined by inequalities. We prove that the Newton-Okounkov body and the superpotential polytope coincide. In the case that our A-cluster consists of Plucker coordinates, we also give a formula for each lattice point of these polytopes in terms of Young diagrams; using a result of Fulton-Woodward, this formula has an interpretation in terms of quantum cohomology. This is joint work with Konstanze Rietsch.