Forthcoming events in this series


Tue, 05 Nov 2013

15:45 - 16:45
L4

Delooping and reciprocity

Michael Groechenig
((Imperial College, London))
Abstract

The Contou-Carrère symbol has been introduced in the 90's in the study of local analogues of autoduality of Jacobians of smooth projective curves. It is closely related to the tame symbol, the residue pairing, and the canonical central extension of loop groups. In this talk we will a discuss a K-theoretic interpretation of the Contou-Carrère symbol, which allows us to generalize this one-dimensional picture to higher dimensions. This will be achieved by studying the K-theory of Tate objects, giving rise to natural central extensions of higher loop groups by spectra. Using the K-theoretic viewpoint, we then go on to prove a reciprocity law for higher-dimensional Contou-Carrère symbols. This is joint work with O. Braunling and J. Wolfson.

Tue, 29 Oct 2013

15:45 - 16:45
L4

Quasimaps, wall-crossings, and Mirror Symmetry II

Ionut Ciocan-Fontanine
(Minnesota)
Abstract

Quasimaps provide compactifications, depending on a stability parameter epsilon, for moduli spaces of maps from nonsingular algebraic curves to a large class of GIT quotients. These compactifications enjoy good properties and in particular they carry virtual fundamental classes. As the parameter epsilon varies, the resulting invariants are related by wall-crossing formulas. I will present some of these formulas in genus zero, and will explain why they can be viewed as generalizations (in several directions) of Givental's toric mirror theorems. I will also describe extensions of wall-crossing to higher genus, and (time permitting) to orbifold GIT targets as well.
The talk is based on joint works with Bumsig Kim, and partly also with Daewoong Cheong and with Davesh Maulik.

Tue, 29 Oct 2013

14:00 - 15:00
L4

Quasimaps, wall-crossings, and Mirror Symmetry I

Ionut Ciocan-Fontanine
(Minnesota)
Abstract

Quasimaps provide compactifications, depending on a stability parameter epsilon, for moduli spaces of maps from nonsingular algebraic curves to a large class of GIT quotients. These compactifications enjoy good properties and in particular they carry virtual fundamental classes. As the parameter epsilon varies, the resulting invariants are related by wall-crossing formulas. I will present some of these formulas in genus zero, and will explain why they can be viewed as generalizations (in several directions) of Givental's toric mirror theorems. I will also describe extensions of wall-crossing to higher genus, and (time permitting) to orbifold GIT targets as well.
The talk is based on joint works with Bumsig Kim, and partly also with Daewoong Cheong and with Davesh Maulik.

Tue, 22 Oct 2013

15:45 - 16:45
L4

Noncommutative algebraic geometry of isolated hypersurface singularities II

Toby Dyckerhoff
(Oxford)
Abstract

The concept of a matrix factorization was originally introduced by Eisenbud to study syzygies over local rings of singular hypersurfaces. More recently, interactions with mathematical physics, where matrix factorizations appear in quantum field theory, have provided various new insights. I will explain how matrix factorizations can be studied in the context of noncommutative algebraic geometry based on differential graded categories. We will see the relevance of the noncommutative analogue of de Rham cohomology in terms of classical singularity theory. Finally, I will outline how the Kapustin-Li formula for the noncommutative Serre duality pairing (originally computed via path integral methods) can be mathematically explained using a combination of homological perturbation theory and local duality.
Partly based on joint work with Daniel Murfet.

Tue, 22 Oct 2013

14:00 - 15:00
L4

Noncommutative algebraic geometry of isolated hypersurface singularities I

Toby Dyckerhoff
(Oxford)
Abstract

The concept of a matrix factorization was originally introduced by Eisenbud to study syzygies over local rings of singular hypersurfaces. More recently, interactions with mathematical physics, where matrix factorizations appear in quantum field theory, have provided various new insights. I will explain how matrix factorizations can be studied in the context of noncommutative algebraic geometry based on differential graded categories. We will see the relevance of the noncommutative analogue of de Rham cohomology in terms of classical singularity theory. Finally, I will outline how the Kapustin-Li formula for the noncommutative Serre duality pairing (originally computed via path integral methods) can be mathematically explained using a combination of homological perturbation theory and local duality.
Partly based on joint work with Daniel Murfet.

Tue, 15 Oct 2013

15:45 - 16:45
L4

Noncommutative deformations and birational geometry II

Will Donovan
(Edinburgh)
Abstract

I will speak about recent work with Michael Wemyss (arXiv:1309.0698), applying noncommutative deformation theory to study the birational geometry of 3-folds. In particular, I will explain how every flippable or floppable rational curve in a 3-fold has a naturally associated algebra of noncommutative deformations, even in the singular setting. We investigate the properties of this algebra, and indicate how to calculate it in examples using quiver techniques. This gives new information about the (commutative) geometry of 3-folds, and in particular provides a new tool to differentiate between flops.

As a further application, we show how the noncommutative deformation algebra controls the homological properties of a floppable curve. In this setting, work of Bridgeland and Chen yields a Fourier-Mukai flop-flop functor which acts on the derived category of the 3-fold (assuming any singularities are at worst Gorenstein terminal). We show that this functor can be described as a spherical twist about the universal family over the noncommutative deformation algebra.

In the second part, I will talk about further work in progress, and explain some more technical details, such as the use of noncommutative deformation functors, and the categorical mutations of Iyama and Wemyss. If there is time, I will also give some higher-dimensional examples, and discuss situations involving chains of intersecting floppable curves. In this latter case, deformations, intersections and homological properties are encoded by the path algebra of a quiver, generalizing the algebra of noncommutative deformations.

Tue, 15 Oct 2013

14:00 - 15:00
L4

Noncommutative deformations and birational geometry I

Will Donovan
(Edinburgh)
Abstract

I will speak about recent work with Michael Wemyss (arXiv:1309.0698), applying noncommutative deformation theory to study the birational geometry of 3-folds. In particular, I will explain how every flippable or floppable rational curve in a 3-fold has a naturally associated algebra of noncommutative deformations, even in the singular setting. We investigate the properties of this algebra, and indicate how to calculate it in examples using quiver techniques. This gives new information about the (commutative) geometry of 3-folds, and in particular provides a new tool to differentiate between flops.

As a further application, we show how the noncommutative deformation algebra controls the homological properties of a floppable curve. In this setting, work of Bridgeland and Chen yields a Fourier-Mukai flop-flop functor which acts on the derived category of the 3-fold (assuming any singularities are at worst Gorenstein terminal). We show that this functor can be described as a spherical twist about the universal family over the noncommutative deformation algebra.

In the second part, I will talk about further work in progress, and explain some more technical details, such as the use of noncommutative deformation functors, and the categorical mutations of Iyama and Wemyss. If there is time, I will also give some higher-dimensional examples, and discuss situations involving chains of intersecting floppable curves. In this latter case, deformations, intersections and homological properties are encoded by the path algebra of a quiver, generalizing the algebra of noncommutative deformations.

Tue, 11 Jun 2013

15:45 - 16:45
L1

Bridgeland 40 conference: $E_n$-deformations and quantizations of derived stacks II

Chris Brav
(Oxford)
Abstract

We review the theory of $E_n$-algebras (roughly, algebras with $n$ compatible multiplications) and discuss $E_n$-deformation theory in the sense of Lurie. We then describe, to the best of our ability, the use of $E_n$-deformation theory in the on-going work of Calaque, Pantev, Toen, Vezzosi, and Vaquie about deformation quantization of derived stacks with shifted Poisson structure.

Tue, 11 Jun 2013

14:00 - 15:00
SR1

Bridgeland 40 conference: $E_n$-deformations and quantizations of derived stacks I

Chris Brav
(Oxford)
Abstract

We review the theory of $E_n$-algebras (roughly, algebras with $n$ compatible multiplications) and discuss $E_n$-deformation theory in

the sense of Lurie. We then describe, to the best of our ability, the use of $E_n$-deformation theory in the on-going work of Calaque, Pantev, Toen,

Vezzosi, and Vaquie about deformation quantization of derived stacks with shifted Poisson structure.

Tue, 28 May 2013

15:45 - 16:45
L3

Hamiltonian reduction and t-structures in (quantum) symplectic geometry

Tom Nevins
(Illinois)
Abstract

Many interesting examples of singular symplectic algebraic varieties and their symplectic resolutions are built by Hamiltonian reduction. There is a corresponding construction of "quantum Hamiltonian reduction" which is of substantial interest to representation theorists. It starts from a twisted-equivariant D-module, an analogue of an algebraic vector bundle (or coherent sheaf) on a moment map fiber, and produces an object on the quantum analogue of the symplectic resolution. In order to understand how far apart the quantisation of the singular symplectic variety and its symplectic resolution can be, one wants to know "what gets killed by quantum Hamiltonian reduction?" I will give a precise answer to this question in terms of effective combinatorics. The answer has consequences for exactness of direct images, and thus for t-structures, which I will also explain. The beautiful geometry behind the combinatorics is that of a stratification of a GIT-unstable locus called the "Kirwan-Ness stratification." The lecture will not assume familiarity with D-modules, nor with any previous talks by the speaker or McGerty in this series. The new results are joint work with McGerty.