Shaowu Zhang, Caltech

Title: Topological Laplace transform and decomposition of nc-Hodge structures

Abstract: Non-commutative Hodge structures were introduced by Katzarkov, Kontsevich, and Pantev as a generalization of classical Hodge structures for non-commutative spaces. These structures naturally arise from the study of mirror symmetry, enumerative geometry, and singularity theory. The nc-Hodge structures consist of de Rham data and Betti data, and those from geometry are expected to satisfy a property called “exponential type.”
In this talk, we will discuss how the Betti data changes under the Fourier-Laplace transform. In particular, we will explain how to construct a perverse sheaf with vanishing cohomologies on the complex plane from a Stokes structure of exponential type. This will give rise to two equivalent descriptions of B model nc-Hodge structures associated to Landau-Ginzburg models.  We will also relate the spectral decomposition of nc-Hodge structures to the vanishing cycle decomposition after Fourier transform via certain choices of Gabrielov paths, motivated by the study on A model. This talk is based on joint work with Tony Yue Yu.TBA