Past

In 1851, Carl Jacobi made the experimental observation that all integers are sums of seven non-negative cubes, with precisely 17 exceptions, the largest of which is 454. Building on previous work by Maillet, Landau, Dickson, Linnik, Watson, Bombieri, Ramaré, Elkies and many others, we complete the proof of Jacobi's observation.

We consider the impact of spatial heterogeneities on the dynamics of 
localized patterns in systems of partial differential equations (in one 
spatial dimension). We will mostly focus on the most simple possible 
heterogeneity: a small jump-like defect that appears in models in which 
some parameters change in value as the spatial variable x crosses 
through a critical value -- which can be due to natural inhomogeneities, 
as is typically the case in ecological models, or can be imposed on the 
model for engineering purposes, as in Josephson junctions. Even such a 
small, simplified heterogeneity may have a crucial impact on the 
dynamics of the PDE. We will especially consider the effect of the 
heterogeneity on the existence of defect solutions, which boils down to 
finding heteroclinic (or homoclinic) orbits in an n-dimensional 
dynamical system in `time' x, for which the vector field for x > 0 
differs slightly from that for x < 0 (under the assumption that there is 
such an orbit in the homogeneous problem). Both the dimension of the 
problem and the nature of the linearized system near the limit points 
have a remarkably rich impact on the defect solutions. We complement the 
general approach by considering two explicit examples: a heterogeneous 
extended Fisher–Kolmogorov equation (n = 4) and a heterogeneous 
generalized FitzHugh–Nagumo system (n = 6).

Oxford Mathematics Public Lectures

MC Escher - Artist, Mathematician, Man 

Roger Penrose and Jon Chapman

This lecture has now sold out

The symbiosis between mathematics and art is personified by the relationship between Roger Penrose and the great Dutch graphic artist MC Escher. In this lecture Roger will give a personal perspective on Escher's work and his own relationship with the artist while Jon Chapman will demonstrate the mathematical imagination inherent in the work. 

The lecture will be preceded by a showing of the BBC 4 documentary on Escher presented by Sir Roger Penrose. Private Escher prints and artefacts will be on display outside the lecture theatre.

5pm

Lecture Theatre 1

Mathematical Institute

Andrew Wiles Building

Radcliffe Observatory Quarter

Woodstock Road

OX2 6GG

Roger Penrose is Emeritus Rouse Ball Professor at the Mathematical Institute in Oxford

Jon Chapman is Statutory Professor of Mathematics and Its Applications at the Mathematical Institute in Oxford

This is an exciting time to study quantum algorithms. As the technological challenges of building a quantum computer continue to be met there is still much to learn about the power of quantum computing. Understanding which problems a quantum computer could solve faster than a classical device and which problems remain hard is particularly relevant to cryptography. We would like to design schemes that are secure against an adversary with a quantum computer. I'll give an overview of the quantum computing that is accessible to a general audience and use a recently declassified project called "soliloquy" as a case study for the development (and breaking) of post-quantum cryptography.

Many people talk about properties that you would expect of a group. When they say this they are considering random groups, I will define what it means to pick a random group in one of many models and will give some properties that these groups will have with overwhelming probability. I will look at the proof of some of these results although the talk will mainly avoid proving things rigorously.

I will explain how to relate the center of a cyclotomic quiver Hecke algebras to the cohomology of Nakajima quiver varieties using a current algebra action. This is a joint work with M. Varagnolo and E. Vasserot.
 

It has become possible in recent years to provide unconditional lower bounds on the time needed to perform a number of basic computational operations. I will briefly discuss some of the main techniques involved and show how one in particular, the information transfer method, can be exploited to give  time lower bounds for computation on streaming data.

I will then go on to present a simple looking mathematical conjecture with a probabilistic combinatorics flavour that derives from this work.  The conjecture is related to the classic "coin weighing with a spring scale" puzzle but has so far resisted our best efforts at resolution.

Given a hypersurface, the Bernstein-Sato polynomial gives deep information about its singularities.  It is defined by a D-module (the algebraic formalism of differential equations) closely related to analytic continuation of the gamma function. On the other hand, given a hypersurface (in a Calabi-Yau variety) one can also consider the Hamiltonian flow by divergence-free vector fields, which also defines a D-module considered by Etingof and myself. I will explain how, in the case of quasihomogeneous hypersurfaces with isolated singularities, the two actually coincide. As a consequence I affirmatively answer a folklore question (to which M. Saito recently found a counterexample in the non-quasihomogeneous case): if c$ is a root of the b-function, is the D-module D f^c / D f^{c+1} nonzero? We also compute this D-module, and for c=-1 its length is one more than the genus (conjecturally in the non-quasihomogenous case), matching an analogous D-module in characteristic p. This is joint work with Bitoun.
 

An edge (vertex) coloured graph is rainbow-connected if there is a rainbow path between any two vertices, i.e. a path all of whose edges (internal vertices) carry distinct colours. Rainbow edge (vertex) connectivity of a graph G is the smallest number of colours needed for a rainbow edge (vertex) colouring of G. We propose a very simple approach to studying rainbow connectivity in graphs. Using this idea, we give a unified proof of several new and known results, focusing on random regular graphs. This is joint work with Michael Krivelevich and Benny Sudakov.

 If $R = F_q[t]$ is the polynomial ring over a finite field
then the group $SL_2(R)$ is not finitely generated. The group $SL_3(R)$ is
finitely generated but not finitely presented, while $SL_4(R)$ is
finitely presented. These examples are facets of a larger picture that
I will talk about.

TBA

This will be an overview of Prof Stroffolini's research and precursor to the eight-hour mini-course Prof Stroffolini will be giving later in October.

One way of understanding groups is by investigating their actions on special spaces, such as Hilbert and Banach spaces, non-positively curved spaces etc. Classical properties like Kazhdan property (T) and the Haagerup property are formulated in terms of such actions and turn out to be relevant in a wide range of areas, from the conjectures of Baum-Connes and Novikov to constructions of expanders. In this talk I shall overview various generalisations of property (T) and Haagerup to Banach spaces, especially in connection with classes of groups acting on non-positively curved spaces.

The 3d/3d correspondence is about the correspondence between 3d N=2 supersymmetric gauge theories and the 3d complex Chern-Simons theory on a 3-manifold.

In this talk I will describe codimension 2 and 4 supersymmetric defects in this correspondence, by a combination of various existing techniques, such as state-integral models, cluster algebras, holographic dual, and 5d SYM.

Studies of thermal convection in planetary interiors have largely focused on convection above the critical Rayleigh number. However, convection in planetary mantles and crusts can also occur under subcritical conditions. Subcritical convection exhibits phenomena which do not exist above the critical Rayleigh number. One such phenomenon is spatial localization characterized by the formation of stable, spatially isolated convective cells. Spatial localization occurs in a broad range of viscosity laws including temperature-dependent viscosity and power-law viscosity and may explain formation of some surface features observed on rocky and icy bodies in the Solar System.

If you do not know quasicircles, you will understand what they are.
If you hate quasicircles, you will change your mind.
If you already love quasicircles, they will astonish you once more.

Recently, a series of unprecedented leaks by Edward Snowden had made it possible for the first time to get a glimpse into the actual capabilities and limitations of the techniques used by the NSA and GCHQ to eavesdrop to computers and other communication devices. In this talk, I will survey some of the things we have learned, and discuss possible countermeasures against these capabilities.

Seminar series `Symmetries and Correspondences'

We give a classification of open Klein topological conformal field theories in terms of Calabi-Yau $A_\infty$-categories endowed with an involution. Given an open Klein topological conformal field theory, there is a universal open-closed extension whose closed part is the involutive version of the Hochschild chains associated to the open part.

Already Serre's "Cohomologie Galoisienne" contains an exercise regarding the following condition on a field F: For every finite field extension E of F and every n, the index of the n-th powers (E*)^n in the multiplicative group E* is finite. Model theorists recently got interested in this condition, as it is satisfied by every superrosy field and also by every strongly2 dependent field, and occurs in a conjecture of Shelah-Hasson on NIP fields. I will explain how it relates to the better known condition that F is bounded (i.e. F has only finitely many extensions of degree n, for any n - in other words, the absolute Galois group of F is a small profinite group) and why it is not preserved under elementary equivalence. Joint work with Franziska Jahnke.

*** Note unusual day and time ***

I will describe a categorical approach to analytic geometry using the theory of quasi-abelian closed symmetric monoidal categories which works both for Archimedean and non-Archimdedean base fields. In particular I will show how the weak G-topologies of (dagger) affinoid subdomains can be characterized by homological method. I will end by briefly saying how to generalize these results for characterizing open embeddings of Stein spaces. This project is a collaboration with Oren Ben-Bassat and Kobi Kremnizer.

We will discuss our approach to global analytic geometry, based on overconvergent power series and functors of functions. We will explain how slight modifications of it allow us to develop a derived version of global analytic geometry. We will finish by discussing applications to the cohomological study of arithmetic varieties.

I will talk in down to earth terms about several main features of this theory.

We describe how p-adic height pairings allow us to find integral points on hyperelliptic curves, in the spirit of Kim's nonabelian Chabauty program. In particular, we discuss how to carry out this ``quadratic Chabauty'' method over quadratic number fields (joint work with Amnon Besser and Steffen Mueller) and present related ideas to find rational points on bielliptic genus 2 curves (joint work with Netan Dogra).

There has been a great deal of attention paid to "Big Data" over the last few years.  However, often as not, the problem with the analysis of data is not as much the size as the complexity of the data.  Even very small data sets can exhibit substantial complexity.  There is therefore a need for methods for representing complex data sets, beyond the usual linear or even polynomial models.  The mathematical notion of shape, encoded in a metric, provides a very useful way to represent complex data sets.  On the other hand, Topology is the mathematical sub discipline which concerns itself with studying shape, in all dimensions.  In recent years, methods from topology have been adapted to the study of data sets, i.e. finite metric spaces.  In this talk, we will discuss what has been
done in this direction and what the future might hold, with numerous examples.

As the fundamental unit of life, the biological cell is a natural focus for computational simulations of growing cell population and tissues. However, models developed at the cellular scale can also be integrated into more complex multiscale models in order to examine complex biological and physical process that scan scales from the molecule to the organ.

This seminar will present a selection of the cellular scale agent-based modelling that has taken place at the University of Sheffield (where one software agent represents one biological cell) and how such models can be used to examine collective behaviour in cellular systems. Finally some of the issues in extending to multiscale models and the theoretical and computational methodologies being developed in Sheffield and by the wider community in this area will be presented.

G’s Growers supply salad and vegetable crops throughout the UK and Europe; primarily as a direct supplier to supermarkets. We are currently working on a project to improve the availability of Iceberg Lettuce throughout the year as this has historically been a very volatile crop. It is also by far the highest volume crop that we produce with typical weekly sales in the summer season being about 3m heads per week.

In order to continue to grow our business we must maintain continuous supply to the supermarkets. Our current method for achieving this is to grow more crop than we will actually harvest. We then aim to use the wholesale markets to sell the extra crop that is grown rather than ploughing it back in and then we reduce availability to these markets when the availability is tight.

We currently use a relatively simple computer Heat Unit model to help predict availability however we know that this is not the full picture. In order to try to help improve our position we have started the IceCAM project (Iceberg Crop Adaptive Model) which has 3 aims.

1. Forecast crop availability spikes and troughs and use this to have better planting programmes from the start of the season.
2. Identify the growth stages of Iceberg to measure more accurately whether crop is ahead or behind expectation when it is physically examined in the field.
3. The final utopian aim would be to match the market so that in times of general shortage when price are high we have sufficient crop to meet all of our supermarket customer requirements and still have spare to sell onto the markets to benefit from the higher prices. Equally when there is a general surplus we would only look to have sufficient to supply the primary customer base.

We believe that statistical mathematics can help us to solve these problems!!

Accurate simulation of coastal and hydraulic structures is challenging due to a range of complex processes such as turbulent air-water flow and breaking waves. Many engineering studies are based on scale models in laboratory flumes, which are often expensive and insufficient for fully exploring these complex processes. To extend the physical laboratory facility, the US Army Engineer Research and Development Center has developed a computational flume capability for this class of problems. I will discuss the turbulent air-water flow model equations, which govern the computational flume, and the order-independent, unstructured finite element discretization on which our implementation is based. Results from our air-water verification and validation test set, which is being developed along with the computational flume, demonstrate the ability of the computational flume to predict the target phenomena, but the test results and our experience developing the computational flume suggest that significant improvements in accuracy, efficiency, and robustness may be obtained by incorporating recent improvements in numerical methods.

Key Words:

Multiphase flow, Navier-Stokes, level set methods, finite element methods, water waves

In 1937 Quine introduced an interesting, rather unusual, set theory called New Foundations - NF for short.  Since then the consistency of NF has been a problem that remains open today.  But there has been considerable progress in our understanding of the problem. In particular NF was shown, by Specker in 1962, to be equiconsistent with a certain theory, TST^+ of simple types. Moreover Randall Holmes, who has been a long-term investigator of the problem, claims to have  solved the problem by showing that TST^+ is indeed consistent.  But the working manuscripts available on his web page that describe his possible proofs are not easy to understand - at least not by me.

Abstract: The simplest solutions of integrable systems are special functions that have been known since the time of Newton, Gauss and Euler. These functions satisfy not only differential equations as functions of their independent variable but also difference equations as functions of their parameter(s).  We show how the inverse scattering transform method, which was invented to solve the Korteweg-de Vries equation, can be extended to its discrete version.

S.Butler and N.Joshi, An inverse scattering transform for the lattice potential KdV equation, Inverse Problems 26 (2010) 115012 (28pp)

We revisit the optimal exit problem by adding a moral hazard problem where a firm owner contracts out with an agent to run a project. We analyse the optimal contracting problem between the owner and the agent in a Brownian framework, when the latter modifies the project cash-flows with an hidden action. The analysis leads to the resolution of a constrained optimal stopping problem that we solve explicitly.

The talk will discuss the mean value theorem and Wooley's breakthrough with his "efficent congruencing" method.

Although not all complex networks are embedded into physical spaces, it is possible to find an abstract Euclidean space in which they are embedded. This Euclidean space naturally arises from the use of the concept of network communicability. In this talk I will introduce the basic concepts of communicability, communicability distance and communicability angles. Both, analytic and computational evidences will be provided that shows that the average communicability angle represents a measure of the spatial efficiency of a network. We will see how this abstract spatial efficiency is related to the real-world efficiency with which networks uses the available physical space for classes of networks embedded into physical spaces. More interesting, we will show how this abstract concept give important insights about properties of networks not embedded in physical spaces.

Law in mathematics and mathematics in law: Probability theory and the fair price in contracts in England and France 1700–1850

From the middle of the eighteenth century, references to mathematicians such as Edmond Halley and Abraham De Moivre begin to appear in judgments in English courts on the law of contract and French mathematicians such as Antoine Deparcieux and Emmanuel-Etienne Duvillard de Durand are mentioned in French treatises on contract law in the first half of the nineteenth century. In books on the then nascent subject of probability at the beginning of the eighteenth century, discussions of legal problems and principally contracts, are especially prominent. Nicolas Bernoulli’s thesis at Basle in 1705 on The Use of the Art of Conjecturing in Law was aptly called a Dissertatio Inauguralis Matematico-Juridica. In England, twenty years later, De Moivre dedicated one of his books on probability to the Lord Chancellor, Lord Macclesfield and expressly referred to its significance for contract law.

The objective of this paper is to highlight this textual interaction between law and mathematics and consider its significance for both disciplines but primarily for law. Probability was an applied science before it became theoretical. Legal problems, particularly those raised by the law of contract, were one of the most frequent applications and as such played an essential role in the development of this subject from its inception. In law, probability was particularly important in contracts. The idea that exchanges must be fair, that what one receives must be the just price for what one gives, has had a significant influence on European contract law since the Middle Ages. Probability theory allowed, for the first time, such an idea to be applied to the sale of interests which began or terminated on the death of certain people. These interests, particularly reversionary interests in land and personal property in English law and rentes viagères in French law were very common in practice at this time. This paper will consider the surprising and very different practical effects of these mathematical texts on English and French contract law especially during their formative period in the late eighteenth and nineteenth centuries.

When formulated appropriately, the broad families of sequential quadratic programming, augmented Lagrangian and interior-point methods all require the solution of symmetric saddle-point linear systems. When regularization is employed, the systems become symmetric and quasi definite. The latter are
indefinite but their rich structure and strong relationships with definite systems enable specialized linear algebra, and make them prime candidates for matrix-free implementations of optimization methods. In this talk, I explore various formulations of the step equations in optimization and corresponding
iterative methods that exploit their structure.