Prof. Carlos Simpson, CNRS, Université Côte d’Azur, Nice, will give a series of lectures on Hodge theory and local systems in the period March 3-5, 2020. The goal of this educational workshop is to introduce a new cutting edge areas in modern Mathematics and to survey some of the directions of VICHREN GRANT CKGA.
The workshop is supported by the Simons Collaborative Grant – HMS, NRU HSE, RF government grant, ag. 14.641.31.000, Simons Principle Investigator Grant, Bulgarian National Scientific Program “VIHREN”, Project № KP-06-DV-7.
Prof. Carlos Simpson will cover the following topics:
- Kähler manifolds, Laplacians and the Kähler identities
- the Hodge decomposition on cohomology
- smooth morphisms or families of varieties
- monodromy, the Gauss-Manin connection, motivic local systems
- variation of Hodge structures
- Zucker’s theorem: cohomology with coefficients in a VHS
- Norm estimates, nilpotent and SL2-orbit theorems
- character varieties, harmonic bundles and Higgs bundles
- C*-action and deformation to a VHS, spectral curves and the Hitchin fibration
- Dolbeault cohomology with VHS coefficients, Higgs bundle coefficients
- rigid local systems and motivicity conjecture
Other lecturers will be Prof. Alexander Efimov (Steklov Mathematical Institute of the Russian Academy of Sciences, National Research University Higher School of Economics), Prof. Ludmil Katzarkov (University of Miami, USA; Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences; National Research University Higher School of Economics, Russian Federation), Prof. Viсtor Przyjalkowski (Steklov Mathematical Institute of the Russian Academy of Sciences, National Research University Higher School of Economics), Prof. Dmitry Kaledin (Steklov Mathematical Institute of the Russian Academy of Sciences and National Research University Higher School of Economics, Russian Federation).
They will give lectures on the following topics:
Prof. Viсtor Przyjalkowski: Geometry of Landau-Ginzburg Models
Abstract: We discuss geometric and numerical properties of Landau-Ginzburg models of Fano varieties that reflect geometric and numerical properties of the initial Fano varieties. The main example is the threefold case.
Prof. Alexander Efimov: Overview of Non-commutative Hodge to de Rham Degeneration and its Generalizations
Abstract: We will recall the statement of Hodge to de Rham degeneration conjecture (which is now Kaledin’s theorem) and our counterexamples to its generalized versions formulated by Kontsevich.
Prof. Ludmil Katzarkov: Some Applications of Non-commutative Hodge Structures
Abstract: We will consider some applications of non-commutative Hodge structures to birational geometry.
Prof. Dmitry Kaledin: Motivic Structures in Non-commutative Geometry
Abstract: I am going to give an overview of various motivic structures that appear in non-commutative geometry with an emphasys on recent results (especially in p-adic Hodge theory).