nLab Yiannis Vlassopoulos

Yiannis Vlassopoulos (also spelled Ioannis and also Yannis) is a mathematician interested in mathematical physics, geometry and more recently also in artificial intelligence. He was a student of David Morrison. Part of his thesis is reflected in

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

On tensor networks in machine learning of natural language:

See also:

  • Maxim Kontsevich, Alex Takeda, Yiannis Vlassopoulos, Pre-Calabi-Yau algebras and topological quantum field theories, arXiv:2112.14667
  • Maxim Kontsevich, Yannis Vlassopoulos, Natalia Iyudu, Pre-Calabi-Yau algebras and double Poisson brackets, arXiv:1906.07134
  • Maxim Kontsevich, Yannis Vlassopoulos, Natalia Iyudu, Pre-Calabi-Yau algebras as noncommutative Poisson structures, J. Algebra 567 (2021) 63–90 doi

The following is an interesting excerpt from his research project he had at IHES:

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”.

Related notions include pre-Calabi-Yau algebra.

category: people

Last revised on December 21, 2022 at 15:58:15. See the history of this page for a list of all contributions to it.