Tue, 22 Oct 2019

14:30 - 15:00
L5

An optimal polynomial approximation of Brownian motion

James Foster
(Oxford)
Abstract

In this talk, I will present a strong (or pathwise) approximation of standard Brownian motion by a class of orthogonal polynomials. Most notably, the coefficients obtained from this expansion are independent Gaussian random variables. This will enable us to generate approximate Brownian paths by matching certain polynomial moments. To conclude the talk, I will discuss related works and applications to numerical methods for SDEs.
 

Mon, 18 Nov 2019
14:15
L4

Quantization through Morita equivalence

Francis Bischoff
(Oxford)
Abstract

In this talk I will discuss a new proposal for constructing quantizations of holomorphic Poisson structures, and generalized complex manifolds more generally, which is based on using the A model of an associated symplectic manifold known as a Morita equivalence. This construction will be illustrated through the example of toric Poisson structures.

 

Tue, 26 Nov 2019
15:30
L4

Degenerate Morse theory and quivers

Frances Kirwan
(Oxford)
Abstract


This talk is an update on joint work with Geoff Penington on extending Morse theory to smooth functions on compact manifolds with very mild nondegeneracy assumptions. The only requirement is that the critical locus should have just finitely many connected components. To such a function we associate a quiver with vertices labelled by the connected components of the critical locus. The analogue of the Morse–Witten complex in this situation is a spectral sequence of multicomplexes supported on this quiver which abuts to the homology of the manifold.

Mon, 21 Oct 2019

14:15 - 15:15
L4

The pure cohomology of multiplicative quiver varieties

Kevin McGerty
(Oxford)
Further Information

Multiplicative quiver varieties are a variant of Nakajima's "additive" quiver varieties which were introduced by Crawley-Boevey and Shaw.
They arise naturally in the study of various moduli spaces, in particular in Boalch's work on irregular connections. In this talk we will discuss joint work with Tom Nevins which shows that the tautological classes for these varieties generate the largest possible subalgebra of the cohomology ring, namely the pure part.

 

Tue, 11 Jun 2019
16:00
C5

The momentum amplituhedron

Matteo Parisi
(Oxford)
Abstract

In this paper we define a new object, the momentum amplituhedron, which is the long sought-after positive geometry for tree-level scattering amplitudes in N=4 super Yang-Mills theory in spinor helicity space. Inspired by the construction of the ordinary amplituhedron, we introduce bosonized spinor helicity variables to represent our external kinematical data, and restrict them to a particular positive region. The momentum amplituhedron Mn,k is then the image of the positive Grassmannian via a map determined by such kinematics. The scattering amplitudes are extracted from the canonical form with logarithmic singularities on the boundaries of this geometry.

Tue, 18 Jun 2019

14:00 - 14:30
L3

Improving the scalability of derivative-free optimisation for nonlinear least-squares problems

Lindon Roberts
(Oxford)
Abstract

In existing techniques for model-based derivative-free optimisation, the computational cost of constructing local models and Lagrange polynomials can be high. As a result, these algorithms are not as suitable for large-scale problems as derivative-based methods. In this talk, I will introduce a derivative-free method based on exploration of random subspaces, suitable for nonlinear least-squares problems. This method has a substantially reduced computational cost (in terms of linear algebra), while still making progress using few objective evaluations.

Tue, 28 May 2019

14:30 - 15:00
L5

Optimisation of 1D Piecewise Smooth Functions

Jonathan Grant-Peters
(Oxford)
Abstract

Optimisation in 1D is far simpler than multidimensional optimisation and this is largely due to the notion of a bracket. A bracket is a trio of points such that the middle point is the one with the smallest objective function value (of the three). The existence of a bracket is sufficient to guarantee that a continuous function has a local minimum within the bracket. The most stable 1D optimisation methods, such as Golden Section or Brent's Method, make use of this fact. The mentality behind these methods is to maintain a bracket at all times, all the while finding smaller brackets until the local minimum can be guaranteed to lie within a sufficiently small range. For smooth functions, Brent's method in particular converges quickly with a minimum of function evaluations required. However, when applied to a piece-wise smooth functions, it achieves its realistic worst case convergence rate. In this presentation, I will present a new method which uses ideas from Brent and Golden Section, while being designed to converge quickly for piece-wise smooth functions.

Mon, 03 Jun 2019
12:45
L3

Brackets, involutivity and generalised geometry for 4d, N=1 backgrounds

Anthony Ashmore
(Oxford)
Abstract

Supergravity backgrounds are an essential ingredient in string theory or field theories via AdS/CFT. The simplest example of a 4d, N=1 background is the product of four-dimensional Minkowski space with a seven-dimensional manifold with G_2 holonomy in M-theory. For more complicated backgrounds where we allow non-zero fluxes, the supersymmetry conditions can be rephrased in terms of G-structure data. The geometry of these backgrounds is often complicated and their general features are not well understood.

In this talk, I will define the analogue of G_2 geometry for generic 4d, N=1 backgrounds with flux in both type II and eleven-dimensional supergravity. The geometry is characterised by a G-structure in 'exceptional generalised geometry' that includes G_2 structures and Hitchin's generalised geometry as subcases. Supersymmetry is then equivalent to integrability of the structures, which appears as an involutivity condition and a moment map for diffeomorphisms and gauge transformations. I will show how this works in a few simple examples and discuss how this can be used to understand general properties of supersymmetric backgrounds.

 

Mon, 24 Jun 2019

14:15 - 15:15
L4

Higher Segal spaces and lax A-infinity structure

Elena Gal
(Oxford)
Abstract

The notion of a higher Segal object was introduces by Dyckerhoff and Kapranov as a general framework for studying (higher) associativity inherent
in a wide range of mathematical objects. Most of the examples are related to Hall algebra type constructions, which include quantum groups. We describe a construction that assigns to a simplicial object S a datum H(S)  which is naturally interpreted as a "d-lax A-infinity algebra” precisely when S is a (d+1)-Segal object. This extends the extensively studied d=2 case.

Tue, 07 May 2019

14:30 - 15:00
L5

Fireshape, a look under the hood

Alberto Paganini
(Oxford)
Abstract

Fireshape is a shape optimization library based on finite elements. In this talk I will describe how Florian Wechsung and I developed Fireshape and will share my reflections on lessons learned through the process.

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