Past Algebraic and Symplectic Geometry Seminar

13 February 2014
16:30
Michael Wemyss
Abstract

I will explain how, given a crepant morphism with one-dimensional fibres between 3-folds, it is possible to use noncommutative deformations to run the MMP in a satisfyingly algorithmic fashion.  As part of this, a flop is viewed homologically as the solution to a universal property, and so is constructed not by changing GIT, but instead by changing the algebra. Carrying this extra information of the new algebra allows us to continue to flop, and thus continue the MMP, without having to calculate everything from scratch. Proving things in this manner does in fact have other consequences too, and I will explain some them, both theoretical and computational.

  • Algebraic and Symplectic Geometry Seminar
13 February 2014
14:45
Tobias Dyckerhoff
Abstract

Crossed simplicial groups were introduced independently by Krasauskas and Fiedorowicz-Loday as analogues of Connes' cyclic category. In this talk, I will explain a new perspective on a certain class of crossed simplicial groups, relating them to structured surfaces. This provides a combinatorial approach to categorical invariants of surfaces which leads to known, expected, and new examples. (Based on joint work with Mikhail Kapranov.)

  • Algebraic and Symplectic Geometry Seminar
11 February 2014
15:45
Alexander Ritter
Abstract
I will explain how to compute the symplectic cohomology of a manifold $M$ conical at infinity, whose Reeb flow at infinity arises from a Hamiltonian circle-action on $M$. For example, this allows one to compute the symplectic cohomology of negative line bundles in terms of the quantum cohomology, and (in joint work with Ivan Smith) via the open-closed string map one can determine the wrapped Fukaya category of negative line bundles over projective space. In this talk, I will show that one can explicitly compute the quantum cohomology and symplectic cohomology of Fano toric negative line bundles, which are in fact different cohomology groups, and surprisingly it is actually the symplectic cohomology which recovers the Jacobian ring of the Landau-Ginzburg superpotential.
  • Algebraic and Symplectic Geometry Seminar
11 February 2014
14:00
Yohsuke Imagi
Abstract
Special Lagranigian submanifolds are area-minimizing Lagrangian submanifolds of Calabi--Yau manifolds. One can define the moduli space of compact special Lagrangian submanifolds of a (fixed) Calabi--Yau manifold. Mclean proves it has a structure of manifold (of dimension finite). It isn't compact in general, but one can compactify it by using geometric measure theory. Kontsevich conjectured a mirror symmetry, and special Lagrangians should be "mirror" to holomorphic vector bundles. By using algebraic geometry one can compactify the moduli space of holomorphic vector bundles. By "counting" holomorphic vector bundles in Calabi--Yau 3-folds Richard Thomas defined holomorphic Casson invariants (Donaldson-Thomas invariants). So far as I know it's an open question (probably very difficult) whether one can "count" special Lagrangians, or define a nice structure on the (compactified) moduli space of special Lagrangians. To do it one has to study singularities of special Lagrangians. One can smooth singularities in suitable situations: given a singular special Lagrangian, one can construct smooth special Lagrangians tending to it (by the gluing technique). I've proved a uniqueness theorem in a "symmetric" situation: given a symmetric singularity, there's only one way to smooth it (the point of the proof is that the symmetry reduces the problem to an ordinary differential equation). More recently I've studied a non-symmetric situation together with Dominic Joyce and Joana Oliveira dos Santos Amorim. Our method is based on Lagrangian Floer theory, and is effective at least for pairs of two (special) Lagrangian planes intersecting transversely. I'll give the details in the talk.
  • Algebraic and Symplectic Geometry Seminar
4 February 2014
14:00
Tom Sutherland
Abstract
The instanton corrections to the hyperkähler metric on moduli spaces of meromorphic flat SL(2,C)-connections on a Riemann surface with prescribed singularities have recently been studied by Gaiotto, Moore and Neitzke. The instantons are given by certain special trajectories of the meromorphic quadratic differentials which form the base of Hitchin's integrable system structure on the moduli space. Bridgeland and Smith interpret such quadratic differentials as defining stability conditions on an associated 3-Calabi-Yau triangulated category whose stable objects correspond to these special trajectories. The smallest non-trivial examples are provided by the moduli spaces of quaternionic dimension one. In these cases it is possible to study explicitly the periods of the Seiberg-Witten differential on the fibres of the Hitchin system which define the central charge of the stability condition and lift the period map to the space of stability conditions. This provides in particular a new categorical perspective on the original Seiberg-Witten gauge theories.
  • Algebraic and Symplectic Geometry Seminar
21 January 2014
15:45
Balazs Szendroi
Abstract
I will start by introducing Somos sequences, defined by innocent-looking quadratic recursions which, surprisingly, always return integer values. I will then explain how they can be viewed in a much larger context, that of the Laurent phenomenon in the theory of cluster algebras. Some further steps take us to the the quantum cluster positivity conjecture of Berenstein and Zelevinski. I will finally explain how, following Nagao and Efimov, cohomological Donaldson-Thomas theory leads to a proof of this conjecture in some, perhaps all, cases. This is joint work with Davison, Maulik, Schuermann.
  • Algebraic and Symplectic Geometry Seminar
26 November 2013
15:45
Gabriele Benedetti
Abstract
In this talk we aim to study periodic orbits on the energy levels of a symplectic magnetic flow on the two-sphere using methods from contact geometry. In particular we show that, if the energy is low enough, we either have two or infinitely many closed orbits. The second alternative holds if there exists a prime contractible periodic orbit. Finally we present some generalisations and work in progress for closed orientable surfaces of higher genus.
  • Algebraic and Symplectic Geometry Seminar

Pages