For a weighted homogeneous polynomial and a choice of a diagonal symmetry group, we define a new Fukaya category based on wrapped Fukaya category of its Milnor fiber together with monodromy information. It is analogous to the variation operator in singularity theory. As an application, we formulate a complete version of Berglund-Hübsch homological mirror symmetry and prove it for two variable cases.
Centre National de la Recherche Scientifique/Université Paris Diderot; Member, School of Mathematics
September 17, 2020
In this second talk about Broué’s Abelian Defect Group Conjecture, we will explain its geometric version in the case of finite groups of Lie type: the equivalence should be induced by the cohomology complex of Deligne-Lusztig varieties. This was actually the main motivation for the conjecture in the first place. We will illustrate those ideas with the case of SL(2,q).
In a joint work with Pierre Dehornoy and Ana Rechtman, we prove that on a closed 3-manifold, every nondegenerate Reeb vector field is supported by a broken book decomposition. From this property, we deduce that in dimension 3 every nondegenerate Reeb vector field has either 2 or infinitely periodic orbits and that on a closed 3-manifold that is not graphed, every nondegenerate Reeb vector field has positive topological entropy.