Friday, March 2, 2018
03/02/2018 - 2:30pm to 3:30pm
A Tropical QFT is a functor from a category of tropical curves up to type (as opposed to bordisms up to homotopy) to a linear category. Ruddat and I define this and construct a particular example whose target category consists of spaces of mirror polyvector fields. We show that this TrQFT elegantly computes the multiplicities which appear in tropical Gromov-Witten theory. I will sketch this and relate it to an expected isomorphism between a conjectural log/tropical quantum cohomology ring and the mirror ring of polyvector fields. In particular, I'll outline my proof of the degree 0 case of this for cluster varieties, a.k.a., the Frobenius structure conjecture, in which canonical ``theta functions" on cluster varieties are defined in terms of certain mirror logGW numbers. Finally, I'll discuss ideas on quantum and motivic refinements of this.
03/02/2018 - 4:00pm to 5:00pm
Arthur Coble constructed a family of rational surfaces with a smooth rational curve invariant under every surface automorphism (the curve isbi-anticanonical), with a very large group of symmetries(a lattice in SO(9,1)). Coble conjectured that generically thisgroup is faithful on the rational curve, which would be quiteinterestingeven from a purely lattices-in-Lie-groups perspective.We will discuss some related ideas, including a particular surface for which the symmetry group can be worked outexactly\; while it is far from discrete in PGL2(C), it turns out tobe a lattice in PGL2(Q3). (Joint work with Igor Dolgachev.)