Activities This Week

The classification of hypersurface singularities aims at writing down a normal form of the defining power series with respect to some equivalence relation, and to give list of normal forms for a distinguished class of singularities. Arnold’s famous ADE-classification of singularities over the complex numbers had an enormous influence on singularity theory and beyond. I will report on some of the impact of his work to other disciplines and to some real-life applications of the classification. Stimulated by Arnold’s work, the classification has been carried on to singularities over fields of positive characteristic, partly with surprising differences. I will report on recent results about this classification and about related problems.

Open Gromov-Witten (OGW) invariants should count pseudoholomorphic maps from curves with boundary to a symplectic manifold, with Lagrangian boundary conditions and various constraints on boundary and interior marked points. The presence of boundary of real codimension 1 poses an obstacle to invariance. In a joint work with J. Solomon (2016-2017), we defined genus zero OGW invariants under cohomological conditions. The construction is rather abstract. Nonetheless, in a recent work, also joint with J. Solomon, we find that the generating function of OGW has many properties that enable explicit calculations. Most notably, it satisfies a system of PDE called open WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equation. For the projective space, this PDE generates recursion relations that allow the computation of all invariants. Furthermore, the open WDVV can be reinterpreted as an associativity of a suitable version of a quantum product.

No prior knowledge of any of the above notions will be assumed.