Abstracts
Given any covariant "homology" theory on algebraic varieties, the bivariant machinery of Fulton and MacPherson constructs a "operational" bivariant theory, which formally includes a contravariant "cohomology" component. Taking the homology theory to be Chow homology, this is how the Chow cohomology of singular varieties is defined. In this talk, I will describe joint work with Sam Payne in which we study the operational K-theory associated to the K-homology of coherent sheaves. Remarkably, despite its formal definition, the operational theory has many properties which make it easier to understand than the K-theory of vector bundles or perfect complexes. This is illustrated most vividly by singular toric varieties, where relatively little is known about K-theory of vector bundles, while the operational equivariant K-theory has a simple description in terms of the fan, directly generalizing the smooth case.
The map that sends a rational curve in a Grassmann variety to its kernel-span pair can be used to express any (3 point, genus zero) Gromov-Witten invariant of a Grassmannian as a classical triple intersection number of Schubert classes on a two-step flag variety. A conjecture of Allen Knutson asserts that any such triple intersection number is equal to the number of triangular puzzles that can be constructed using a list of puzzle pieces, such that the border of the puzzles is determined by the intersection problem. I will discuss a recent proof of this conjecture and how it leads to a positive combinatorial formula for Grassmannian Gromov-Witten invariants. I will also discuss generalizations to equivariant Gromov-Witten invariants. This talk is based on papers with A. Kresch, L. Mihalcea, K. Purbhoo, and H. Tamvakis.
I report on joint work with Luca Migliorini at Bologna. If you have a map of complex projective manifolds, then the rational cohomology of the domain splits into a direct sum of pieces in a way dictated by the singularities of the map. By Poincaré duality, the corresponding projections can be viewed as cohomology classes (projectors) on the self-product of the domain. These projectors are Hodge classes, i.e. rational and of type (p,p) for the Hodge decomposition. Take the same situation after application of an automorphism of the ground field of complex numbers. The new projectors are of course Hodge classes. On the other hand, you can also transplant, using the field automorphism, the old projectors into the new situation and it is not clear that the new projectors and the transplants of the old projectors coincide. We prove they do, thus proving that the projectors are absolute Hodge classes, i.e. their being of Hodge type survives the totally discontinuous process of a field automorphism. We also prove that these projectors are motivated in the sense of Andre.
The concept of a matrix factorization was originally introduced by Eisenbud to study syzygies over local rings of singular hypersurfaces. More recently, interactions with mathematical physics, where matrix factorizations appear in quantum field theory, have provided various new insights. I will explain how matrix factorizations can be studied in the context of noncommutative algebraic geometry based on differential graded categories. We will see the relevance of the noncommutative analogue of de Rham cohomology in terms of classical singularity theory. Finally, I will outline how the Kapustin-Li formula for the noncommutative Serre duality pairing (originally computed via path integral methods) can be mathematically explained using a combination of homological perturbation theory and local duality. This talk is partly based on joint work with Daniel Murfet.
The Mori Dream Space Conjecture asserts that the nef cone of the moduli space \overline M0,n of stable n-pointed rational curves is finitely generated and that every element is a semi-ample divisor. Conformal blocks divisors form full dimensional sub-cones of semi-ample divisors in the nef cone of \overline M0,n. In this talk I will discuss different ways to approach understanding whether or not these cones of conformal blocks divisors are finitely generated, and how that might relate to the Mori Dream Space Conjecture.
If the Tate conjecture is true, and if étale cohomology with finite coefficients is computable, then the Néron-Severi group of any smooth projective variety is computable, as is the group of numerical equivalence classes of codimension p cycles.
Jason Starr (Stony Brook), Rational curves and rational points of varieties over global function fields
For a variety defined over a global function field K, e.g., Fp(t), the simplest cohomological obstruction is the "elementary obstruction" of Colliot-Thélène and Sansuc. Assuming this vanishes, what additional geometric hypotheses imply existence of a rational point? Combining "rational simple connectedness" in characteristic 0 with work of Esnault on rational points over finite fields, Chenyang Xu and I give new, uniform proofs of three classical results: Lang's proof that K is C2, a theorem of Brauer-Hasse-Noether that twisted forms of Grassmannians over K admit rational points, and the split case of Harder's theorem / Serre's "Conjecture II" over K. I will focus on the geometry of rational curves over C, and how one
"transports" this to results over a field such as K.