Algebraic and Symplectic Geometry Seminar

In 1992 (or thereabouts) Bloch and Kriz gave the first explicit definition of the category of mixed Tate motives (MTM). Their definition relies heavily on the theory of algebraic cycles. Unfortunately, traditional methods of representing algebraic cycles (such as in terms of formal linear combinations of systems of polynomial equations) are notoriously difficult to work with, so progress in capitalizing on this description of the category to illuminate outstanding conjectures in the field has been slow. More recently, Gangl, Goncharov, and Levin suggested a simpler way to understand this category (and by extension, algebraic cycles more generally) by relating specific algebraic cycles to rooted, decorated, planar trees. In our talks, describing work in progress, we generalize this correspondence and attempt to systematize the connection between algebraic cycles and graphs. We will construct a Lie coalgebra L from a certain algebra of admissible graphs, discuss various properties that it satisfies (such as a well defined and simply described realization functor to the category of mixed Hodge structures), and relate the category of co-representations of L to the category MTM. One promising consequence of our investigations is the appearance of alternative bases of rational motives that have not previously appeared in the literature, suggesting a richer rational structure than had been previously suspected. In addition, our results give the first bounds on the complexity of computing admissibility of algebraic cycles, a previously unexplored topic.

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.

The cohomological Hall algebra of vanishing cycles associated to a quiver with potential is a categorification of the refined DT invariants associated to the same data, and also a very powerful tool for calculating them and proving positivity and integrality conjectures. This becomes especially true if the quiver with potential is "self dual" in a sense to be defined in the talk. After defining and giving a general introduction to the relevant background, I will discuss the main theorem regarding such CoHAs: they are free supercommutative.

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.

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.

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.