Forthcoming events in this series


Tue, 20 Oct 2015

15:45 - 16:45
L4

Generating the Fukaya categories of Hamiltonian G-manifolds

Yanki Lekili
(King's College London)
Abstract

Let $G$ be a compact Lie group and $k$ be a field of characteristic $p\ge 0$ such that $H^*(G)$ does not have $p$-torsion. We show that a free Lagrangian orbit of a Hamiltonian $G$-action on a compact, monotone, symplectic manifold $X$ split-generates an idempotent summand of the monotone Fukaya category over $k$ if and only if it represents a non-zero object of that summand. Our result is based on: an explicit understanding of the wrapped Fukaya category through Koszul twisted complexes involving the zero-section and a cotangent fibre; and a functor canonically associated to the Hamiltonian $G$-action on $X$. Several examples can be studied in a uniform manner including toric Fano varieties and certain Grassmannians. 

Tue, 13 Oct 2015

15:45 - 16:45
L4

D-modules from the b-function and Hamiltonian flow

Travis Schedler
(Imperial College London)
Abstract

Given a hypersurface, the Bernstein-Sato polynomial gives deep information about its singularities.  It is defined by a D-module (the algebraic formalism of differential equations) closely related to analytic continuation of the gamma function. On the other hand, given a hypersurface (in a Calabi-Yau variety) one can also consider the Hamiltonian flow by divergence-free vector fields, which also defines a D-module considered by Etingof and myself. I will explain how, in the case of quasihomogeneous hypersurfaces with isolated singularities, the two actually coincide. As a consequence I affirmatively answer a folklore question (to which M. Saito recently found a counterexample in the non-quasihomogeneous case): if c$ is a root of the b-function, is the D-module D f^c / D f^{c+1} nonzero? We also compute this D-module, and for c=-1 its length is one more than the genus (conjecturally in the non-quasihomogenous case), matching an analogous D-module in characteristic p. This is joint work with Bitoun.
 

Tue, 09 Jun 2015

13:30 - 14:30
L4

(COW SEMINAR) Uniformizing the moduli space of abelian 6-folds

Valeri Alexeev
(University of Georgia)
Abstract

By classical results of Mumford and Donagi, Mori-Mukai, Verra, the moduli spaces A_g of principally polarized abelian varieties of dimension g are unirational for g≤5 and are of general type for g≥7. Answering a conjecture of Kanev, we provide a uniformization of A6 by a Hurwitz space parameterizing certain curve covers. Using this uniformization, we study the geometry of A6 and make advances towards determining its birational type. This is a joint work with Donagi-Farkas-Izadi-Ortega.

Tue, 05 May 2015

15:45 - 16:45
L4

Tropical schemes

Diane Maclagan
(University of Warwick)
Abstract

Tropicalization replaces a variety by a polyhedral complex that is a "combinatorial shadow" of the original variety.  This allows algebraic geometric problems to be attacked using combinatorial and
polyhedral techniques.  While this idea has proved surprisingly effective over the last decade, it has so far been restricted to the study of varieties and algebraic cycles.  I will discuss joint work with Felipe Rincon, building on work of Jeff and Noah Giansiracusa, to understand tropicalizing schemes, and more generally the concept of a tropical scheme.

Tue, 28 Apr 2015

15:45 - 16:45
L4

Motives over Abelian geometries via relative power structures

Andrew Morrison
(ETH Zurich)
Abstract

We describe the cohomology of moduli spaces of points on schemes over Abelian varieties and give explicit calculations for schemes in dimensions less that three. The construction of Gulbrandsen allows one to consider virtual motives in dimension three. In particular we see a new proof of his conjectures on the Euler numbers of generalized Kummer schemes recently proven by Shen. Joint work in progress with Junliang Shen.

Tue, 28 Apr 2015

14:00 - 15:00
L4

On the proof of the S-duality modularity conjecture for the quintic threefold

Artan Sheshmani
(Ohio State)
Abstract

I will talk about recent joint work with Amin Gholampour, Richard Thomas and Yukinobu Toda, on an algebraic-geometric proof of the S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. Our strategy is to first use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over relative Hilbert scheme of points on surfaces and then prove modularity; More precisely, together with Gholampour we have proven that the generating series, associated to the top intersection numbers of the Hibert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson for absolute Hilbert schemes. These intersection numbers, together with the generating series of Noether-Lefschetz numbers, will provide the ingrediants to prove modularity of the above DT invariants over the quintic threefold.

Tue, 21 Apr 2015

15:45 - 16:45
C1

Donaldson-Thomas theory for Calabi-Yau 4-folds

Yalong Cao
(Hong Kong)
Abstract

Donaldson-Thomas theory for Calabi-Yau 3-folds is a complexification of Chern-Simons theory. In this talk, I will discuss joint work with Naichung Conan Leung on the complexification of Donaldson theory.

Tue, 03 Mar 2015

15:45 - 16:45
L4

The closed-open string map for S^1-invariant Lagrangians

Dmitry Tonkonog
(Cambridge)
Abstract

Given a Lagrangian submanifold invariant under a Hamiltonian loop, we partially compute the image of the loop's Seidel element under the closed-open string map into the Hochschild cohomology of the Lagrangian. This piece captures the homology class of the loop's orbits on the Lagrangian and can help to prove that the closed-open map is injective in some examples. As a corollary we prove that $\mathbb{RP}^n$ split-generates the Fukaya category of $\mathbb{CP}^n$ over a field of characteristic 2, and the same for real loci of some other toric  varieties.

Tue, 24 Feb 2015

15:45 - 16:45
L4

The exponential map based at a singularity

Daniel Grieser
(Oldenberg)
Abstract
We study isolated singularities of a space embedded in a smooth Riemannian manifold from a differential geometric point of view. While there is a considerable literature on bi-lipschitz invariants of singularities, we obtain a more precise (complete asymptotic) understanding of the metric properties of certain types of singularities. This involves the study of the family of geodesics emanating from the singular point. While for conical singularities this family of geodesics, and the exponential map defined by them, behaves much like in the smooth case, the situation is very different in the case of cuspidal singularities, where the exponential map may fail to be locally injective. We also study a mixed conical-cuspidal case. Our methods involve the description of the geodesic flow as a Hamiltonian system and its resolution by blow-ups in phase space. 
 
This is joint work with Vincent Grandjean.
Tue, 03 Feb 2015

15:45 - 16:45
L4

Homological projective duality

Richard Thomas
(Imperial)
Abstract
I will describe a little of Kuznetsov's wonderful theory of Homological projective duality, a generalisation of classical projective duality that relates derived categories of coherent sheaves on different algebraic varieties. I will explain an approach that seems simpler than the original, and some applications that occur in joint work with Addington, Calabrese and Segal.