Archived Events

I will describe recent work, joint with G.

The problem of description of the moduli space $I_n$ of mathematical instanton vector bundles on the projective space $P_3$ has been a challenging problem since 70's. It was conjectured by R.Hartshorne in 1976 that $I_n$ is irreducible for an arbitrary second Chern class $n>0$. This problem has an affirmative solution for small values of $n$, up to $n=5$. Namely, the cases $n=1,2,3,4$ and 5 were settled by Barth (1977), Hartshorne (1978), Ellingsrud-Stromme (1981), Barth (1981) and Coanda-Tikhomirov-Trautmann (2003), respectively.

Suppose that the circle acts on a symplectic manifold in a Hamiltonian fashion. Under certain "minimal'' conditions of the fixed point set of the action, we classify the integral cohomology ring and the Chern classes of the manifold, and we classify the circle action.

In its crudest form the generalized Birch and Swinnerton-Dyer relates the order of vanishing of an L-function at the center of the critical strip to the rank of a Chow group. This talk describes the conjecture and an attempt to put it to a modest test in the case of motives of the form Sym$^3H^1(E)$, where E is an elliptic curve over Q. This is joint work with Joe Buhler and Jaap Top.