Past Algebraic and Symplectic Geometry Seminar

11 March 2014
15:45
Ben Davison
Abstract
I will discuss some very well studied cohomology groups that turn out to be captured by the machinery of critical CoHAs, for example the compactly supported cohomology of singular quiver varieties and untwisted character varieties. I will explain the usefulness of this extra CoHA structure on these groups, starting with a new proof of the Kac conjecture, and discuss a conjectural form for the CoHA associated to untwisted character varieties that provides a new way to think about the conjectures of Hausel and Rodriguez-Villegas. Finally I will discuss an approach to purity for the compactly supported cohomology of quiver varieties and a related approach to a conjecture of Shiffmann and Vasserot, analogous to Kirwan surjectivity for the stack of commuting matrices.
  • Algebraic and Symplectic Geometry Seminar
11 March 2014
14:00
Ben Davison
Abstract
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.
  • Algebraic and Symplectic Geometry Seminar
4 March 2014
15:45
Damien Calaque
Abstract
We will start with a recollection on factorization algebras and factorization homology. We will then explain what fully extended TFTs are, after Jacob Lurie. And finally we will see how factorization homology can be turned into a fully extended TFT. This is a joint work with my student Claudia Scheimbauer.
  • Algebraic and Symplectic Geometry Seminar
4 March 2014
14:00
Damien Calaque
Abstract
We will explain how the result of Pantev-Toën-Vaquié-Vezzosi, about shifted symplectic structures on mapping stacks, can be extended to relative mapping stacks and Lagrangian structures. We will also provide applications in ordinary symplectic geometry and topological field theories.
  • Algebraic and Symplectic Geometry Seminar
25 February 2014
15:45
Pranav Pandit
Abstract
The talk will focus on how the asymptotic behavior of the Riemann-Hilbert correspondence (and, conjecturally, the non-abelian Hodge correspondence) on a Riemann surface is controlled by certain harmonic maps from the Riemann surface to affine buildings. This is part of joint work with Katzarkov, Noll and Simpson, which revisits, from the perspective afforded by the theory of harmonic maps to buildings, the work of Gaiotto, Moore and Neitzke on spectral networks, WKB problems, BPS states and wall-crossing.
  • Algebraic and Symplectic Geometry Seminar
25 February 2014
14:00
Kevin Walker
Abstract

The classical Deligne conjecture (now a theorem with several published proofs) says that chains on the little disks operad act on Hochschild cohomology.  I'll describe a higher dimensional generalization of this result.  In fact, even in the dimension of the original Deligne conjecture the generalization has something new to say:  Hochschild chains and Hochschild cochains are the first two members of an infinite family of chain complexes associated to an arbitrary associative algebra, and there is a colored, higher genus operad which acts on these chain complexes.  The Connes differential and Gerstenhaber bracket are two of the simplest generators of the homology of this operad, and I'll show that there exist additional, independent generators.  These new generators are close cousins of Connes and Gerstenhaber which, so far as I can tell, have not been described in the literature.

  • Algebraic and Symplectic Geometry Seminar
18 February 2014
14:00
Susama Agarwala
Abstract
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.
  • Algebraic and Symplectic Geometry Seminar

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