Y.-P. Lee (Utah)
Functoriality of Gromov--Witten theory under K-equivalent
transformations
I will discuss how Gromov-Witten theory behaves under K-equivalent
birational transformations. K-equivalence includes flops, crepant
resolutions, and in some sense transitions. This is a joint program with
H.-W. Lin and C.-L. Wang.
Nick Proudfoot (Oregon)
A taste of symplectic duality
Symplectic duality is a relationship between pairs of
algebraic symplectic varieties (or, if you prefer, hyperkahler
manifolds). Like mirror symmetry, symplectic duality has a number of
different formulations, or phenomena that examples are expected to
express. These include
1) cohomological symplectic duality--a duality of vector spaces between certain ordinary and intersection cohomology groups,
2) categorical symplectic duality--Koszul duality between certain categories of sheaves,
3) Goresky-MacPherson duality--a phenomenon relating the spectra of the equivariant cohomology rings of the two varieties.
All of the examples that we understand at present arise from either combinatorics or representation theory. In the combinatorial examples, symplectic duality provides an explanation for certain symmetries of the Tutte polynomial of a matroid. In the representation theoretic examples, it relates to the study of various blocks of Category O.
This is joint work with Tom Braden, Tony Licata, and Ben Webster.
Ionut Ciocan-Fontanine (Minnesota)
Virtual classes via dg-manifolds and virtual Riemann-Roch
I will present a construction of virtual classes, both in K-theory and in Chow
groups, for "[0,1] differential-graded manifolds", which are the dg-analogues of
varieties with perfect obstruction theory. Further, I will explain certain versions
of Riemann-Roch theorems in this context and discuss some applications (cf. also a
recent paper by Fantechi and Goettsche). This is joint work with M. Kapranov.