Tue, 23 Feb 2016

12:00 - 13:15
L4

The amplituhedron for tree-level scattering amplitudes in N=4 sYM

Dr Livia Ferro
(LMU-Muenchen and Max Planck Institut fuer Physik)
Abstract

In this talk I will present some recent work on the amplituhedron formulation of scattering amplitudes. Very recently it has been conjectured that amplitudes in planar N=4 sYM are nothing else but the volume of a completely new mathematical object, called amplituhedron, which generalises the positive Grassmannian. After a review of the main ingredients which will be used, I will discuss some of the questions which remain open in this framework. I will then describe a new direction which promises to solve these issues and compute the volume of the amplituhedron at tree level.

 

Mon, 22 Feb 2016
16:30
C1

Congruence and non-congruence level structures on elliptic curves: a hands-on tour of the modular tower

Alexander Betts
(Oxford University)
Abstract
Classically, one puts an algebraic structure on certain "congruence" quotients of the upper half plane by interpreting them as spaces parametrising elliptic curves with certain level structures on their torsion subgroups. However, the non-congruence quotients don't admit such a straightforward description.
 
We will sketch the classical theory of congruence modular curves and level structures, and then discuss a preprint by W. Chen which extends the above notions to non-congruence modular curves by considering so-called Teichmueller level structures on the fundamental groups of punctured elliptic curves.
Mon, 22 Feb 2016

16:00 - 17:00
L4

The hydrodynamic limit of the parabolic Ginzburg-Landau equation

Matthias Kurzke
(University of Nottingham)
Abstract

The Ginzburg-Landau functional serves as a model for the formation of vortices in many physical contexts. The natural gradient flow, the parabolic Ginzburg-Landau equation, converges in the limit of small vortex size and finite number of vortices to a system of ODEs. Passing to the limit of many vortices in this ODE, one can derive a mean field PDE, similar to the passage from point vortex systems to the 2D Euler equations. In the talk, I will present quantitative estimates that allow us to directly connect the parabolic GL equation to the limiting mean field PDE. In contrast to recent work by Serfaty, our work is restricted to a fairly low number of vortices, but can handle vortex sheet initial data in bounded domains. This is joint work with Daniel Spirn (University of Minnesota).

Mon, 22 Feb 2016

15:45 - 16:45
L5

Renormalisation in Regularity Structures

Lorenzo Zambotti
(Universite of Paris 6)
Abstract

In this talk we want to present a detailed study of the algebraic objects appearing in the theory of regularity structures. In particular we aim at introducing a class of co-algebras on labelled forests and trees and show that these allow to describe in an unified setting the structure group and the renormalisation group. Based on joint work with Yvain Bruned and Martin Hairer

          

Mon, 22 Feb 2016

14:15 - 15:15
L5

Rough differential equations and random dynamical systems

Sebastian Riedel
(TU Berlin University)
Abstract

We aim to study the long time behaviour of the solution to a rough differential equation (in the sense of Lyons) driven by a random rough path. To do so, we use the theory of random dynamical systems. In a first step, we show that rough differential equations naturally induce random dynamical systems, provided the driving rough path has stationary increments. If the equation satisfies a strong form of stability, we can show that the solution admits an invariant measure.

This is joint work with I. Bailleul (Rennes) and M. Scheutzow (Berlin).    

Mon, 22 Feb 2016
14:15
L4

The Gromoll filtration, Toda brackets and positive scalar curvature

OAC-manifolds meeting: Diarmuid Crowley
(Aberdeen)
Abstract
An exotic (n+1)-sphere has disc of origin D^k if k is the smallest integer such that some clutching diffeomorphism of the n-disc which builds the exotic sphere can be written as an (n-k)-parameter family of diffeomorphisms of the k-disc.
 
In this talk I will present a new method for constructing exotic spheres with small disc of origin via Toda brackets.  
 
This method gives exotic spheres in all dimensions 8j+1 and 8j+2 with disc of origin 6 and with Dirac operators of non-zero index (such spheres are often called "Hitchin spheres").
 
I will also briefly discuss implications of our results for the space of positive scalar curvature metrics on spin manifolds of dimension 6 and higher, and in particular the relationship of this project to the work of Botvinnik, Ebert and Randal-Williams.
 
This is part of joint work with Thomas Schick and Wolfgang Steimle.
Sat, 20 Feb 2016

16:00 - 17:00

TBA

Piotr Mucha
(Warsaw)
Fri, 19 Feb 2016

16:00 - 17:00
L1

North meets South Colloquium

Patrick Farrell + Yufei Zhao
(University of Oxford)
Abstract

Computing distinct solutions of differential equations -- Patrick Farrell

Abstract: TBA

Triangles and equations -- Yufei Zhao

Abstract: I will explain how tools in graph theory can be useful for understanding certain problems in additive combinatorics, in particular the existence of arithmetic progressions in sets of integers. 

Fri, 19 Feb 2016

11:00 - 12:00
C1

\zeta(3) in graviton-graviton scattering and the moduli space of CY manifolds

Philip Candelas
(Oxford)
Abstract

I will discuss how \zeta(3) occurs in quantum corrections to the Einstein action, and how this causes \zeta(3) to be seen in the moduli space of CY manifolds. I will also draw attention to the fact that the dependence of the moduli space on \zeta(3) has a p-adic analogue.

Thu, 18 Feb 2016

16:00 - 17:00
C5

Equivariant Topological Quantum Field Theory

Thomas Wasserman
(Oxford)
Abstract

Topological Quantum Field Theories are functors from a category of bordisms of manifolds to (usually) some categorification of the notion of vector spaces. In this talk we will first discuss why mathematicians are interested in these in general and an overview of the relevant notions. After this we will have a closer look at the example of functors from the bordism category of 1-, 2- and 3-dimensional manifolds equipped with principal G-bundles, for G a finite group, to nice categorifications of vector spaces.

Thu, 18 Feb 2016
16:00
L5

Joint Number Theory/Logic Seminar: On a modular Fermat equation

Jonathan Pila
(Oxford University)
Abstract
`I will describe some diophantine problems and results motivated
by the analogy between powers of the modular curve and powers of the
multiplicative group in the context of the Zilber-Pink conjecture.
Thu, 18 Feb 2016

16:00 - 17:00
L5

(Joint Number Theory and Logic) On a modular Fermat equation

Jonathan Pila
(University of Oxford)
Abstract

I will describe some diophantine problems and results motivated by the analogy between powers of the modular curve and powers of the multiplicative group in the context of the Zilber-Pink conjecture.

Thu, 18 Feb 2016

16:00 - 17:30
L4

A pathwise dynamic programming approach to nonlinear option pricing

Christian Bender
(Department of Mathematics Saarland university)
Abstract

In this talk, we present a pathwise method to construct confidence 
intervals on the value of some discrete time stochastic dynamic 
programming equations, which arise, e.g., in nonlinear option pricing 
problems such as credit value adjustment and pricing under model 
uncertainty. Our method generalizes the primal-dual approach, which is 
popular and well-studied for Bermudan option pricing problems. In a 
nutshell, the idea is to derive a maximization problem and a 
minimization problem such that the value processes of both problems 
coincide with the solution of the dynamic program and such that 
optimizers can be represented in terms of the solution of the dynamic 
program. Applying an approximate solution to the dynamic program, which 
can be precomputed by any algorithm, then leads to `close-to-optimal' 
controls for these optimization problems and to `tight' lower and upper 
bounds for the value of the dynamic program, provided that the algorithm 
for constructing the approximate solution was `successful'. We 
illustrate the method numerically in the context of credit value 
adjustment and pricing under uncertain volatility.
The talk is based on joint work with C. Gärtner, N. Schweizer, and J. 
Zhuo.

Thu, 18 Feb 2016

16:00 - 17:00
L3

Interactions of noise and discontinuities: transitions and qualitative changes

Rachel Kuske
(University of British Colombia)
Abstract

While there have been recent advances for analyzing the complex deterministic
behavior of systems with discontinuous dynamics, there are many open questions about
understanding and predicting noise-driven and noise-sensitive phenomena in the
non-smooth context.  Stochastic effects can often change the picture dramatically,
particularly if multiple time scales are present.  We demonstrate novel approaches
for exploring and explaining surprising phenomena driven by the interplay of
nonlinearities, delays, randomness, in specific applications with piecewise smooth
dynamics - nonlinear models of balance,  relay control, and impacting dynamics.
Effective techniques typically depend on the combination of mathematical techniques,
multiple scales techniques, and phenomenological intuition from seemingly unrelated
canonical models of biophysics, mechanics, and chemical dynamics.  The appropriate
strategy is not always immediately obvious from the area of application or model
type. This gap may follow from the limited attention that stochastic models with
discontinuous dynamics have received in the past, or it may be the reason for this
limited attention.  Combining the geometrical perspective with asymptotic approaches
in physical and phase space appears to be a critical part of developing effective
approaches.

Thu, 18 Feb 2016

14:00 - 15:00
L5

Ten things you should know about quadrature

Professor Nick Trefethen
(Oxford)
Abstract

Quadrature is the term for the numerical evaluation of integrals.  It's a beautiful subject because it's so accessible, yet full of conceptual surprises and challenges.  This talk will review ten of these, with plenty of history and numerical demonstrations.  Some are old if not well known, some are new, and two are subjects of my current research.

Thu, 18 Feb 2016
12:00
L6

Time-Periodic Einstein-Klein-Gordon Bifurcations Of Kerr

Yakov Shlapentokh-Rothman
(Princeton University)
Abstract

For a positive measure set of Klein-Gordon masses mu^2 > 0, we construct one-parameter families of solutions to the Einstein-Klein-Gordon equations bifurcating off the Kerr solution such that the underlying family of spacetimes are each an asymptotically flat, stationary, axisymmetric, black hole spacetime, and such that the corresponding scalar fields are non-zero and time-periodic. An immediate corollary is that for these Klein-Gordon masses, the Kerr family is not asymptotically stable as a solution to the Einstein-Klein-Gordon equations. This is joint work with Otis Chodosh.

 
Wed, 17 Feb 2016
15:00
L4

The evolution of discrete logarithm in GF(p^n)

Razvan Barbulescu
(CNRS Paris)
Abstract
The security of pairings-based cryptography relies on the difficulty of two problems: computing discrete logarithms over elliptic curves and, respectively, finite fields GF(p^n) when n is a small integer larger than 1. The real-life difficulty of the latter problem was tested in 2006 by a record in a field GF(p^3) and in 2014 and 2015 by new records in GF(p^2), GF(p^3) and GF(p^4). We will present the new methods of polynomial selection which allowed to obtain these records. Then we discuss the difficulty of DLP in GF(p^6) and GF(p^12) when p has a special form (SNFS) for which two theoretical algorithms were presented recently.
Wed, 17 Feb 2016

11:00 - 11:30
N3.12

The Riemann zeta function, quantum chaos and random matrices

Simon Myerson
(Oxford)
Abstract
The Riemann zeta function is linked to quantum chaology by some totally neat results and utterly wacky conjectures concerning random matrices. Join me to see the horrifying extent of these unexpected connections!
Tue, 16 Feb 2016

15:45 - 16:45
L4

The K3 category of a cubic fourfold

Daniel Huybrechts
(Bonn)
Abstract

Smooth cubic fourfolds are linked to K3 surfaces via their Hodge structures, due to work of Hassett, and via Kuznetsov's K3 category A. The relation between these two viewpoints has recently been elucidated by Addington and Thomas. 
We study both of these aspects further and extend them to twisted K3 surfaces, which in particular allows us to determine the group of autoequivalences of A for the general cubic fourfold. Furthermore, we prove finiteness results for cubics with equivalent K3 categories and study periods of cubics in terms of generalized K3 surfaces.

Tue, 16 Feb 2016

15:00 - 16:00
L5

Hrushovski's construction

Felix Weitkamper
(Oxford University)
Abstract
I will give a general overview of the versatile method behind Hrushovski's construction and then sketch the proof that the original strongly minimal set considered by him does not interpret an infinite group using a group configuration.
 
Tue, 16 Feb 2016
14:30
L5

How accurate must solves be in interior point methods?

Tyrone Rees
(Rutherford Appleton Laboratory)
Abstract

At the heart of the interior point method in optimization is a linear system solve, but how accurate must this solve be?  The behaviour of such methods is well-understood when a direct solver is used, but the scale of problems being tackled today means that users increasingly turn to iterative methods to approximate its solution.  Current suggestions of the accuracy required can be seen to be too stringent, leading to inefficiency.

In this talk I will give conditions on the accuracy of the solution in order to guarantee the inexact interior point method converges at the same rate as if there was an exact solve.  These conditions can be shown numerically to be tight, in that performance degrades rapidly if a weaker condition is used.  Finally, I will describe how the norms that appear in these condition are related to the natural norms that are minimized in several popular Krylov subspace methods. This, in turn, could help in the development of new preconditioners in this important field.