Fri, 26 Jan 2024
12:00
L3

Geometric action for extended Bondi-Metzner-Sachs group in four dimensions

Romain Ruzziconi
(Oxford)
Abstract

This will be an informal discussion seminar based on https://arxiv.org/abs/2211.07592:

The constrained Hamiltonian analysis of geometric actions is worked out before applying the construction to the extended Bondi-Metzner-Sachs group in four dimensions. For any Hamiltonian associated with an extended BMS4 generator, this action provides a field theory in two plus one spacetime dimensions whose Poisson bracket algebra of Noether charges realizes the extended BMS4 Lie algebra. The Poisson structure of the model includes the classical version of the operator product expansions that have appeared in the context of celestial holography. Furthermore, the model reproduces the evolution equations of non-radiative asymptotically flat spacetimes at null infinity.

Mon, 26 Feb 2024
14:15
L4

Hessian geometry of $G_2$-moduli spaces

Thibault Langlais
(Oxford)
Abstract

The moduli space of torsion-free $G_2$-structures on a compact $7$-manifold $M$ is a smooth manifold, locally diffeomorphic to an open subset of $H^3(M)$. It is endowed with a natural metric which arises as the Hessian of a potential, the properties of which are still poorly understood. In this talk, we will review what is known of the geometry of $G_2$-moduli spaces and present new formulae for the fourth derivative of the potential and the curvatures of the associated metric. We explain some interesting consequences for the simplest examples of $G_2$-manifolds, when the universal cover of $M$ is $\mathbb{R}^7$ or $\mathbb{R}^3 \times K3$. If time permits, we also make some comments on the general case.

Mon, 29 Jan 2024
14:15
L4

Floer cohomology for symplectic ${\mathbb C}^*$-manifolds

Alexander Ritter
(Oxford)
Abstract

In this joint work with Filip Zivanovic, we construct symplectic cohomology for a class of symplectic manifolds that admit ${\mathbb C}^*$-actions and which project equivariantly and properly to a convex symplectic manifold. The motivation for studying these is a large class of examples known as Conical Symplectic Resolutions, which includes quiver varieties, resolutions of Slodowy varieties, and hypertoric varieties. These spaces are highly non-exact at infinity, so along the way we develop foundational results to be able to apply Floer theory. Motivated by joint work with Mark McLean on the Cohomological McKay Correspondence, our goal is to describe the ordinary cohomology of the resolution in terms of a Morse-Bott spectral sequence for positive symplectic cohomology. These spectral sequences turn out to be quite computable in many examples. We obtain a filtration on ordinary cohomology by cup-product ideals, and interestingly the filtration can be dependent on the choice of circle action.

Fri, 19 Jan 2024
12:00
L3

Topological Recursion: Introduction, Overview and Applications

Alex Hock
(Oxford)
Abstract
I will give a talk about the topological recursion (TR) of Eynard and Orantin, which generates from some initial data (the so-called the spectral curve) a family of symmetric multi-differentials on a Riemann surface. Symplectic transformations of the spectral curve play an important role and are conjectured to leave the free energies $F_g$ invariant. TR has nowadays a lot of applications ranging random matrix theory, integrable systems, intersection theory on the moduli space of complex curves $\mathcal{M}_{g,n}$, topological string theory over knot theory to free probability theory. I will highlight specific examples, such as the Airy curve (also sometimes called the Kontsevich-Witten curve) which enumerates $\psi$-class intersection numbers on $\mathcal{M}_{g,n}$, the Mirzakhani curve for computing Weil–Petersson volumes, the spectral curve of the hermitian 1-matrix model, and the topological vertex curve which derives the $B$-model correlators in topological string theory. Should time allow, I will also discuss the quantum spectral curve as a quantisation of the classical spectral curve annihilating a wave function constructed from the family of multi-differentials. 
 
 
Fri, 08 Dec 2023
12:00
L3

A Positive Way to Scatter Strings and Particles

Hadleigh Frost
(Oxford)
Abstract

We present a new formulation of string and particle amplitudes that emerges from simple one-dimensional models. The key is a new way to parametrize the positive part of Teichmüller space. It also builds on the results of Mirzakhani for computing Weil-Petterson volumes. The formulation works at all orders in the perturbation series, including non-planar contributions. The relationship between strings and particles is made manifest as a "tropical limit". The results are well adapted to studying the scattering of large numbers of particles or amplitudes at high loop order. The talk will in part cover results from arXiv:2309.15913, 2311.09284.

Tue, 31 Oct 2023
14:00
L5

Elliptic representations

Dan Ciubotaru
(Oxford)
Abstract

In representation theory, the characters of induced representations are explicitly known in terms of the character of the inducing representation. This leads to the question of understanding the elliptic representation space, i.e., the space of representations modulo the properly (parabolically) induced characters. I will give an overview of the description of the elliptic space for finite Weyl groups, affine Weyl groups, affine Hecke algebras, and their connection with the geometry of the nilpotent cone of a semisimple complex Lie algebra. These results fit together in the representation theory of semisimple p-adic groups, where they lead to a new description of the elliptic space within the framework of the local Langlands parameterisation.

Thu, 23 Nov 2023

17:00 - 18:00

Imaginaries in products and in the ring of adeles

Jamshid Derakhshan
(Oxford)
Abstract

In this talk I will present joint work with Ehud Hrushovski on imaginaries in the ring of adeles and more generally in products and restricted products of structures (including the generalised products of Feferman-Vaught).

 

We prove a general theorem on weak elimination of imaginaries in products with respect to additional sorts which we deduce from an elimination of imaginaries for atomic and atomless Booleanizations of a theory. This combined with uniform elimination of imaginaries for p-adic numbers in a language with extra sorts as p-adic lattices proved first by Hrushovski-Martin-Rideau and more recently by Hils-Rideau-Kikuchi in a slightly different language, yields weak elimination of imaginaries for the ring of adeles in a language with extra sorts as adelic versions of the p-adic lattices. 

 

The proofs of the general results on products use Boolean valued model theory, stability theory, analysis of definable groups and liaison groups, and descriptive set theory of smooth Borel equivalence relations including Harrington-Kechris-Louveau and Glimm-Efros dichotomy. 

Tue, 23 Jan 2024

14:00 - 14:30
L6

Scalable Gaussian Process Regression with Quadrature-based Features

Paz Fink Shustin
(Oxford)
Abstract

Gaussian processes provide a powerful probabilistic kernel learning framework, which allows high-quality nonparametric learning via methods such as Gaussian process regression. Nevertheless, its learning phase requires unrealistic massive computations for large datasets. In this talk, we present a quadrature-based approach for scaling up Gaussian process regression via a low-rank approximation of the kernel matrix. The low-rank structure is utilized to achieve effective hyperparameter learning, training, and prediction. Our Gauss-Legendre features method is inspired by the well-known random Fourier features approach, which also builds low-rank approximations via numerical integration. However, our method is capable of generating high-quality kernel approximation using a number of features that is poly-logarithmic in the number of training points, while similar guarantees will require an amount that is at the very least linear in the number of training points when using random Fourier features. The utility of our method for learning with low-dimensional datasets is demonstrated using numerical experiments.

Tue, 24 Oct 2023

14:30 - 15:00
VC

Redefining the finite element

India Marsden
(Oxford)
Abstract

The Ciarlet definition of a finite element has been used for many years to describe the requisite parts of a finite element. In that time, finite element theory and implementation have both developed and improved, which has left scope for a redefinition of the concept of a finite element. In this redefinition, we look to encapsulate some of the assumptions that have historically been required to complete Ciarlet’s definition, as well as incorporate more information, in particular relating to the symmetries of finite elements, using concepts from Group Theory. This talk will present the machinery of the proposed new definition, discuss its features and provide some examples of commonly used elements.

Mon, 09 Oct 2023
14:15
L4

How homotopy theory helps to classify algebraic vector bundles

Mura Yakerson
(Oxford)
Abstract

Classically, topological vector bundles are classified by homotopy classes of maps into infinite Grassmannians. This allows us to study topological vector bundles using obstruction theory: we can detect whether a vector bundle has a trivial subbundle by means of cohomological invariants. In the context of algebraic geometry, one can ask whether algebraic vector bundles over smooth affine varieties can be classified in a similar way. Recent advances in motivic homotopy theory give a positive answer, at least over an algebraically closed base field. Moreover, the behaviour of vector bundles over general base fields has surprising connections with the theory of quadratic forms.

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