Schedule
Wednesday [05.06.2019]
16:00-17:00 Kuznetsov I
17:30-18:30 Oblomkov I
19:00-20:00 Dinner
Thursday [06.06.2019]
9:45-10:45 Kuznetsov II
11:15-12:15 Oblomkov II
12:30-13:30 Lunch
16:00-17:15 Viazovska I
17:45-19:00 Viazovska II
19:00-20:00 Dinner
Friday [07.06.2019]
9:45-10:45 Kuznetsov III
11:15-12:15 Oblomkov III
12:30-13:30 Lunch
16:00-17:15 Oprea I
17:45-19:00 Zvonkine I
19:00-20:00 Dinner
Saturday [08.06.2019]
9:30-10:45 Oprea II
11:00-12:15 Zvonkine II
12:30-13:30 Lunch
Titles and Abstracts
A. Kuznetsov (Steklov Institute)
Derived categories and moduli spaces
n the lectures I will try to show how information about the derived
category of coherent sheaves of a variety helps understanding moduli spaces of
sheaves on it. I will start with a short introduction into derived categories.
After that I will discuss construction of symplectic forms on moduli spaces,
such as the Beauville-Donagi form on the Hilbert scheme of lines of a cubic
fourfold. Finally, I will talk about Hilbert schemes of curves on Fano threefolds and their Abel-Jacbi maps.
A. Oblomkov (UMass Amherst)
Matrix factorizations and knot homology
I will explain how one can use matrix factorizations
to compute homology of torus links that are
links of the homogeneous plane curve singularities.
The computation is one of the outcomes of a sequence of
the joint papers with Lev Rozansky. In these papers
we explain how one can construct a coherent
sheaf on the Hilbert scheme of points on the plane
for every braid and the homology of the closure of
the braid is a space of global sections of the sheaf.
No prior knowledge of matrix factorizations or knot
invariant theory is assumed, all necessary topological
and algebraic constructions will be explained in the course.
D. Oprea (UC San Diego)
Virtual invariants of Quot schemes of surfaces
The Quot schemes of smooth projective surfaces parametrizing quotients of dimension 0 and 1 admit natural 2-term perfect obstruction theories. I will explain several results concerning the associated virtual invariants, with emphasis on the virtual Euler characteristics. This is based on joint work with Rahul Pandharipande.
M. Viazovska (EPFL)
The universal optimality of the E8 and Leech lattices
In a recent joint work with Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Danylo Radchenko we have proven that the E8 and Leech lattices minimize energy of every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians). This theorem implies recently proven optimality of E8 and Leech lattices as sphere packings and broadly generalizes it to long-range interactions. The key ingredient of the proof is sharp linear programming bounds. To construct the optimal auxiliary functions attaining these bounds, we prove a new interpolation theorem.
D. Zvonkine (Versailles/CNRS)
Cohomological field theories
I. Definition and examples
Cohomological field theories are families of cohomology classes on moduli spaces of curves. They were introduced by Kontsevich and Manin to axiomatize properties of Gromov-Witten invariants. Since then many examples unrelated to these invariants were discovered.
II. Givental's group action
A large group, called the twisted symplectic group, acts on CohFTs. It allows one to construct more complicated CohFTs from simpler ones.
Depending on time and the audience's wishes we can discuss tautological relations, the localization formula, or Teleman's classification
of semisimple CohFTs.
