Ioannis Vlassopoulos

**Ioannis Vlassopoulos** was a student of D. Morrison. Part of his thesis is reflected in

*Quantum cohomology and Morse theory on the loop space of toric varieties*, math.AG/0203083

The following is an interesting excerpt from his research project he had at IHES (for longer account see here —this link does not seem to work)

Givental has conjectured, that the U(1)-equivariant Floer cohomology of the universal covering of the loop space, of asymplectic manifold, should have the structure of a D-module, over the Heisenberg algebra of first order differential operators on a complex torus and that this should be the same as the quantum cohomology D-module of the manifold. I intent to study this conjectured equality and its implications in computing the quantum D-module. This implies also computation of the quantum ring, as the later is the semi-classical limit of the former. There are three concrete directions of research. First, note that there is a “Fourier transform” of equivariant cycles that transforms relations in the D-module to differential operators. If we could compute this transform, then we could compute the D-module. Because of the infinite dimensionality of the loop space though, there are problems with computing the integral involved I have managed to compute it in the case of positive toric manifolds, but this method relies on the Fourier expansion and doesn’t seem to generalize to non-toric manifolds. For the case of general semi-positive symplectic manifolds, I propose a totally different method, which relies on using localization techniques and a certain exact sequence arising from Morse theory of the simplistic action functional, in order to regularize the ratios of relevant equivariant Euler classes. The second program I propose, is to use the model of Getzler, Jones and Petrack (pdf), for the equivariant cohomology of the loop space. They identify it with a version of the cyclic bar complex, involving Connes’s operator B and this could be used to compute the relevant “Fourier transform”.

category: people

Revised on June 8, 2011 15:12:31
by Zoran Škoda
(161.53.130.104)