Past Algebraic and Symplectic Geometry Seminar

18 October 2016
15:45
Emilie DuFresne
Abstract

The study of separating invariants is a new trend in Invariant Theory and a return to its roots: invariants as a classification tool. For a finite group acting linearly on a vector space, a separating set is simply a set of invariants whose elements separate the orbits o the action. Such a set need not generate the ring of invariants. In this talk, we give lower bounds on the size of separating sets based on the geometry of the action. These results are obtained via the study of the local cohomology with support at an arrangement of linear subspaces naturally arising from the action.

(Joint with Jack Jeffries)

  • Algebraic and Symplectic Geometry Seminar
11 July 2016
16:30
Andrei Okounkov
Abstract

This is will be a progress report on our long-ongoing joint work with Bezrukavnikov on lifting the monodromy of the quantum differential equation for symplectic resolutions to automorphisms of their derived categories of coherent sheaves. I will attempt to define the ingredient that go both into the problem and into its solution.
 

  • Algebraic and Symplectic Geometry Seminar
11 July 2016
14:45
Nigel Hitchin
Abstract

Following an idea of Gaiotto, a symplectic representation of a complex Lie group G defines a complex Lagrangian subvariety inside the moduli space of G-Higgs bundles. The talk will discuss the case of G=SL(2) and its link with determinant  divisors, or equivalently Brill-Noether loci, in the moduli space of semistable SL(2)-bundles. 

  • Algebraic and Symplectic Geometry Seminar
11 July 2016
13:30
Dominic Joyce
Abstract

Given a (-1)-shifted symplectic derived scheme or stack (X,w) over C equipped with an orientation, we explain how to construct a perverse sheaf P on the classical truncation of X so that its hypercohomology H*(P) can be regarded as a categorification of (or linearisation of) X. Given also a Lagrangian morphism L -> X equipped with a relative orientation, we outline a programme in progress to construct a natural morphism of constructible complexes on the truncation of L from the (shifted) constant complex on L to a suitable pullback of P to L. The morphisms and resulting hypercohomology classes are expected to satisfy various identities under products, composition of Lagrangian correspondences, etc. This programme will have interesting applications, such as proving associativity of a Kontsevich-Soibelman type COHA multiplication on H*(P) when X is the derived moduli stack of coherent sheaves on a Calabi-Yau 3-fold Y, and defining Lagrangian Floer cohomology and the Fukaya cat!
 egory of an algebraic or complex symplectic manifold S.

  • Algebraic and Symplectic Geometry Seminar
14 June 2016
15:45
Alexandru Oancea
Abstract

I will present a definition of symplectic homology groups for pairs of Liouville cobordisms with fillings, and explain how these fit into a formalism of homology theory similar to that of Eilenberg and Steenrod. This construction allows to understand form a unified point of view many structural results involving Floer homology groups, and yields new applications. Joint work with Kai Cieliebak.

  • Algebraic and Symplectic Geometry Seminar
7 June 2016
15:45
Constantin Teleman
Abstract

For a holomorphic function (“superpotential”)  W: X —> C on a complex manifold X, one defines the (2-periodic) matrix factorisation category MF(X;W), which is supported on the critical locus Crit(W) of W. At a Morse singularity, MF(X;W) is equivalent to the category of modules over the Clifford algebra on the tangent space TX. It had been suggested by Kapustin and Rozansky that, for Morse-Bott W, MF(X;W) should be equivalent to the (2-periodicised) derived category of Crit(W), twisted by the Clifford algebra of the normal bundle. I will discuss why this holds when the first neighbourhood of Crit(W) splits, why it fails in general, and will explain the correct general statement.

  • Algebraic and Symplectic Geometry Seminar
31 May 2016
15:45
Greg Berczi
Abstract
I will start with a short report on recent progress in constructing quotients by actions of non-reductive algebraic groups and extending Mumford's geometric invariant theory to a wide class of non-reductive linear algebraic groups which we call graded groups. I will then explain how certain components of the Hilbert scheme of points on smooth varieties can be described as non-reductive quotients and why this description is especially efficient to study the topology of Hilbert schemes. In particular I will explain how equivariant localisation can be used to develop iterated residue formulae for tautological integrals on geometric subsets of Hilbert schemes and I present new formulae counting curves on surfaces (and more generally hypersurfaces in smooth varieties) with given singularity classes. This talk is based on joint works with Frances Kirwan, Thomas Hawes, Brent Doran and Andras Szenes. 
  • Algebraic and Symplectic Geometry Seminar
10 May 2016
15:30
to
17:00
Ben Davison
Abstract
The integrality conjecture is one of the central conjectures of the DT theory of quivers with potential, which itself is a key tool in understanding the local calculation of DT invariants on moduli spaces of coherent sheaves, as well as having deep links to geometric representation theory, noncommutative geometry and algebraic combinatorics.  I will explain some of the ingredients of the proof of this conjecture by myself and Sven Meinhardt.  In fact the proof gives much more than the original conjecture, which ultimately concerns identities in a Grothendieck ring of mixed Hodge structures associated to moduli spaces of representations, and proves that these equalities categorify to isomorphisms in the category of mixed Hodge structures.  I'll explain what this all means, as well as giving some applications of the categorified version of the theory.
  • Algebraic and Symplectic Geometry Seminar

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