Stony Brook 2014 Abstracts
Ron Donagi (University of Pennsylvania) Moduli of Super Riemann Surfaces.
In this talk we will explore various aspects of supergeometry, including obstruction, Atiyah, and super-Atiyah classes. This can be applied to the geometry of the moduli space of super Riemann surfaces. For genus greater than or equal to 5, this moduli space is not projected (and in particular is not split): it cannot be holomorphically projected to its underlying reduced manifold. Physically, this means that certain approaches to superstring perturbation theory that are very powerful in low orders have no close analog in higher orders. Mathematically, it means that the moduli space of super Riemann surfaces cannot be constructed in an elementary way starting with the moduli space of ordinary Riemann surfaces. It has a life of its own.
Eugene Gorsky (Columbia University) Refined Knot Invariants and Hilbert Schemes.
Motivated by string theory, Aganagic and Shakirov proposed an idea of "refined knot invariants" depending on two parameters, and gave a rigorous definition of these invariants for torus knots. Conjecturally, they are related to Poincare polynomials of Khovanov-Rozansky homology. I will explain an explicit construction of sheaves on the Hilbert scheme of points such that their equivariant Euler characteristics agree with the Aganagic-Shakirov invariants (and two parameters correspond to equivariant weights). In a special case, I will recover Haiman's construction of bigraded Catalan numbers. The talk is based on a joint work with Andrei Negut.
Sándor Kovács (University of Washington) Higher Direct Images of Logarithmic Canonical Sheaves.
The main result of the talk is an extension to the logarithmic setting of a (relatively) recent vanishing result of Chatzistamatiou and Rülling which may be considered a Grauert-Riemenschneider type vanishing theorem. This result is motivated by an application to the theory of rational pairs and thrifty resolutions in arbitrary characteristics. I will explain these notions and their basic properties and how some simple expectations fail, making simple-looking statements non-trivial and how to get around the traps laid out all over the land.
Shokurov has conjectured that the set of log discrepancies of singularities in a fixed dimension satsfies the ACC (ascending chain condition).
As part of an attempt to approach this conjecture, we propose a conjectural partial classification of log terminal singularities, which uses toric singularities as a building block.
Zsolt Patakfalvi (Princeton University) On Subadditivity of Kodaira Dimension in Positive Characteristic.
Kodaira dimension is a fundamental (if not the most fundamental) birational invariant of an algebraic variety. It assigns a number between 0 and the dimension or negative infinity to every birational equivalence class of varieties. The bigger this number is the more the variety is thought of as being "hyperbolic". Subadditivity of Kodaira dimension is a conjecture of Iitaka stating that for an algebraic fiber space f : X ➝ Y, the Kodaira dimension of the total space is at least as big as the sum of the Kodaira dimensions of the geometric generic fiber and the base. I will present a positive answer to this conjecture over an algebraically closed field of positive characteristic, when Y is of general type, f is separable and the Hasse-Witt matrix of the geometric generic fiber is not nilpotent.
Michael Thaddeus (Columbia University) The Universal Implosion and the Multiplicative Horn Problem.
The multiplicative Horn problem asks what constraints the eigenvalues of two n×n unitary matrices place on the eigenvalues of their product. The solution of this problem, due to Belkale, Kumar, Woodward, and others, expresses these constraints as a convex polyhedron in 3n dimensions and describes the facets of this polyhedron more or less explicitly. I will explain how the vertices of the polyhedron may instead be described in terms of fixed points of a torus action on a symplectic stratified space, constructed as a quotient of the so-called universal group-valued implosion.
Burt Totaro (University of California, Los Angeles) The Chow Ring of a Finite Group.
A natural class of singular varieties consists of the quotients of vector spaces by linear actions of a finite group G. We can ask what the Chow group of algebraic cycles is, for such a quotient. By taking larger and larger representations of G, we can package these Chow groups into a ring, called the Chow ring of the classifying space of G, or (for short) the Chow ring of G. It maps to the cohomology ring of G, usually not by an isomorphism.
We present the latest tools for computing Chow rings of finite groups. These tools give complete calculations for all "small" groups and many other finite groups. A surprising point is that Chow rings become "wild", in a precise sense, for some slightly larger finite groups.
Intersecting the convex cone of positive semidefinite 4 by 4 matrices with a three-dimensional affine space yields a quartic spectrahedron. It is bounded by a complex quartic hypersurface called a symmetroid. Recently, Degtyarev and Itenberg used the global Torelli Theorem for real K3 surfaces to describe the position of nodes (that is, rank-two matrices) on the real surface and spectrahedron. I will discuss this result in the context of classical algebraic geometry. In particular we will see the special structure of these surfaces and spectrahedra by looking at their projection from a node, as done classically by Cayley. This is joint work with John Christian Ottem, Kristian Ranestad, and Bernd Sturmfels.