Tue, 24 May 2022

15:30 - 16:30
L3

Moment Polyptychs and the Equivariant Quantisation of Hypertoric Varieties

Ben Brown
(Edinburgh)
Abstract

We develop a method to investigate the geometric quantisation of a hypertoric variety from an equivariant viewpoint, in analogy with the equivariant Verlinde for Higgs bundles. We do this by first using the residual circle action on a hypertoric variety to construct its symplectic cut, resulting in a compact cut space which is needed for localisation. We introduce the notion of a moment polyptych associated to a hypertoric variety and prove that the necessary isotropy data can be read off from it. Finally, the equivariant Hirzebruch-Riemann-Roch formula is applied to the cut spaces and expresses the dimension of the equivariant quantisation space as a finite sum over the fixed-points. This is joint work with Johan Martens.

Tue, 08 Mar 2022
12:00
L5

Classical physics and scattering amplitudes on curved backgrounds

Andrea Christofoli
(Edinburgh)
Abstract

A particle physics approach to describing black hole interactions is opening new avenues for understanding gravitational-wave observations. We will start by reviewing this paradigm change, showing how to compute observables in general relativity from amplitudes on flat spacetime. We will then present a generalization of this framework for amplitudes on curved backgrounds. Evaluating the required one-to-one amplitudes already shows remarkable structures. We will discuss them in detail, including eikonal behaviours and unexpected KLT-like factorization properties for amplitudes on stationary backgrounds. We will then conclude by discussing applications of these amplitudes to strong field observables such as the impulse on a curved background and memory effects

 

 

 

Mon, 07 Feb 2022
14:15
L5

Nonabelian Hodge theory and the decomposition theorem for 2-CY categories

Ben Davison
(Edinburgh)
Further Information

The talk will be both online (Teams) and in person (L5)

Abstract

Examples of 2CY categories include the category of coherent sheaves on a K3 surface, the category of Higgs bundles, and the category of modules over preprojective algebras or fundamental group algebras of compact Riemann surfaces.  Let p:M->N be the morphism from the stack of semistable objects in a 2CY category to the coarse moduli space.  I'll explain, using cohomological DT theory, formality in 2CY categories, and structure theorems for good moduli stacks, how to prove a version of the BBDG decomposition theorem for the exceptional direct image of the constant sheaf along p, even though none of the usual conditions for the decomposition theorem apply: p isn't projective or representable, M isn't smooth, the constant mixed Hodge module complex Q_M isn't pure...  As an application, I'll explain how this allows us to extend nonabelian Hodge theory to Betti/Dolbeault stacks.

Tue, 09 Feb 2021
12:00

The stability of Kaluza-Klein spacetimes

Zoe Wyatt
(Edinburgh)
Abstract

Spacetimes with compact directions play an important role in supergravity and string theory. The simplest such example is the Kaluza-Klein spacetime, where the compact space is a flat torus. An interesting question to ask is whether this spacetime, when viewed as an initial value problem, is stable to small perturbations of initial data. In this talk I will discuss the global, non-linear stability of the Kaluza-Klein spacetime to toroidal-independent perturbations and the particular nonlinear structure appearing in the associated PDE system.

Mon, 26 Oct 2020

14:15 - 15:15
Virtual

Coproducts in the cohomological DT theory of 3-Calabi-Yau completions

Ben Davison
(Edinburgh)
Abstract
Given a suitably friendly category D we can take the 3-Calabi Yau completion of D and obtain a 3-Calabi-Yau category E. The archetypal example has D as the category of coherent sheaves on a smooth quasiprojective surface, then E is the category of coherent sheaves on the total space of the canonical bundle - a quasiprojective 3CY variety. The moduli stack of semistable objects in the 3CY completion E supports a vanishing cycle-type sheaf, the hypercohomology of which is the basic object in the study of the DT theory of E. Something extra happens when our input category is itself 2CY: examples include the category of local systems on a Riemann surface, the category of coherent sheaves on a K3/Abelian surface, the category of Higgs bundles on a smooth complete curve, or the category of representations of a preprojective algebra. In these cases, the DT cohomology of E carries a cocommutative coproduct. I'll also explain how this interacts with older algebraic structures in cohomological DT theory to provide a geometric construction of both well-known and new quantum groups.
Mon, 07 Dec 2020

11:00 - 12:00
Virtual

Two perspectives on the stack of principal bundles on an elliptic curve and its slices

Dougal Davis
(Edinburgh)
Abstract

Let G be a reductive group, E an elliptic curve, and Bun_G the moduli stack of principal G-bundles on E. In this talk, I will attempt to explain why Bun_G is a very interesting object from the perspectives of both singularity theory on the one hand, and shifted symplectic geometry and representation theory on the other. In the first part of the talk, I will explain how to construct slices of Bun_G through points corresponding to unstable bundles, and how these are linked to certain singular algebraic surfaces and their deformations in the case of a "subregular" bundle. In the second (probably much shorter) part, I will discuss the shifted symplectic geometry of Bun_G and its slices. If time permits, I will sketch how (conjectural) quantisations of these structures should be related to some well known algebras of an "elliptic" flavour, such as Sklyanin and Feigin-Odesskii algebras, and elliptic quantum groups.

Mon, 22 Jun 2020
14:15
Virtual

Geometry of genus 4 curves in P^3 and wall-crossing

Fatemeh Rezaee
(Edinburgh)
Abstract

In this talk, I will explain a new wall-crossing phenomenon on P^3 that induces non-Q-factorial singularities and thus cannot be understood as an operation in the MMP of the moduli space, unlike the case for many surfaces.  If time permits, I will explain how the wall-crossing could help to understand the geometry of the associated Hilbert scheme and PT moduli space.

Mon, 28 Oct 2019
14:15
L4

The Hitchin connection in (almost) arbitrary characteristic.

Johan Martens
(Edinburgh)
Further Information

The Hitchin connection is a flat projective connection on bundles of non-abelian theta-functions over the moduli space of curves, originally introduced by Hitchin in a Kahler context.  We will describe a purely algebra-geometric construction of this connection that also works in (most)positive characteristics.  A key ingredient is an alternative to the Narasimhan-Atiyah-Bott Kahler form on the moduli space of bundles on a curve.  We will comment on the connection with some related topics, such as the Grothendieck-Katz p-curvature conjecture.  This is joint work with Baier, Bolognesi and Pauly.

 

Tue, 29 May 2018
15:45
L4

Frobenius splittings of toric varieties

Milena Hering
(Edinburgh)
Abstract



Varieties admitting Frobenius splittings exhibit very nice properties.
For example, many nice properties of toric varieties can be deduced from
the fact that they are Frobenius split. Varieties admitting a diagonal
splitting exhibit even nicer properties. In this talk I will give an
overview over the consequences of the existence of such splittings and
then discuss criteria for toric varieties to be diagonally split.

Mon, 14 May 2018

14:15 - 15:15
L4

Families of Hyperkaehler varieties via families of stability conditions

Arend Bayer
(Edinburgh)
Abstract

Stability conditions on derived categories of algebraic varieties and their wall-crossings have given algebraic geometers a number of new tools to study the geometry of moduli spaces of stable sheaves. In work in progress with Macri, Lahoz, Nuer, Perry and Stellari, we are extending this toolkit to a the "relative" setting, i.e. for a family of varieties. Our construction comes with relative moduli spaces of stable objects; this gives additional ways of constructing new families of varieties from a given family, thereby potentially relating different moduli spaces of varieties.

 

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