Past

A Kähler group is a group which can be realised as the fundamental group of a close Kähler manifold. We will prove that for a Kähler group $G$ we have that $G$ is residually free if and only if $G$ is a full subdirect product of a free abelian group and finitely many closed hyperbolic surface groups. We will then address Delzant-Gromov's question of which subgroups of direct products of surface groups are Kähler: We explain how to construct subgroups of direct products of surface groups which have even first Betti number but are not Kähler. All relevant notions will be explained in the talk.

The Algebraic Eraser is a cryptosystem (more precisely, a class of key
agreement schemes) introduced by Anshel, Anshel, Goldfeld and Lemieux
about 10 years ago. There is a concrete instantiation of the Algebraic
Eraser called the Colored Burau Key Agreement Protocol (CBKAP), which
uses a blend of techniques from permutation groups, matrix groups and
braid groups. SecureRF, the company owning the trademark to the
Algebraic Eraser, is marketing this system for lightweight
environments such as RFID tags and other Internet of Things
applications; they have proposed making this scheme the basis for an
ISO RFID standard.

This talk gives an introduction to the Algebraic Eraser, a brief
history of the attacks on this scheme using ideas from group-theoretic
cryptography, and describes the countermeasures that have been
proposed. I would not recommend the scheme for the proposed
applications: the talk ends with a brief sketch of a recent convincing
cryptanalysis of this scheme due to Ben-Zvi, Blackburn and Tsaban
(which appeared at CRYPTO this summer), and significant attacks
on the protocol in the proposed ISO standard due to Blackburn and
Robshaw (which appeared at ACNS earlier this year).

The talk will be a survey of recent work on the connections between group operator systems and non-signalling correlations. I will show how tensor products in the operator system category can be interpreted as physical theories, and will exhibit a refinement of non-signalling corrections giving rise to quantum attributes of graphs, including quantum versions of the chromatic and the fractional chromatic number. Relations between the parameters will be examined and further directions will be discussed.

The study of separating invariants is a new trend in Invariant Theory and a return to its roots: invariants as a classification tool. For a finite group acting linearly on a vector space, a separating set is simply a set of invariants whose elements separate the orbits o the action. Such a set need not generate the ring of invariants. In this talk, we give lower bounds on the size of separating sets based on the geometry of the action. These results are obtained via the study of the local cohomology with support at an arrangement of linear subspaces naturally arising from the action.

(Joint with Jack Jeffries)

The threshold for existence of a giant component in a random graph with given vertex degrees was found by Molloy and Reed (1995), and several authors have since studied the size of the largest and other components in various cases. The critical window was found by Hatami and Molloy (2012), and has a width that depends on whether the asymptotic degree distribution has a finite third moment or not. I will describe some new results (joint work with Remco van der Hofstad and Malwina Luczak) on the barely supercritical case, where this difference between finite and infinite third moment also is seen.

Multi-index methods are a generalization of multilevel methods in high dimensional problem and are based on taking mixed first-order differences along all dimensions. With these methods, we can accurately and efficiently compute a quadrature or construct an interpolation where the integrand requires some form of high dimensional discretization. Multi-index methods are related to Sparse Grid methods and the Combination Technique and have been applied to multiple sampling methods, i.e., Monte Carlo, Stochastic Collocation and, more recently, Quasi Monte Carlo.

In this talk, we describe and analyse the Multi-Index Monte Carlo (MIMC) and Multi-Index Stochastic Collocation (MISC) methods for computing statistics of the solution of a PDE with random data. Provided sufficient mixed regularity, MIMC and MISC achieve better complexity than their corresponding multilevel methods. We propose optimization procedures to select the most effective mixed differences to include in these multi-index methods. We also observe that in the optimal case, the convergence rate of MIMC and MISC is only dictated by the convergence of the deterministic solver applied to a one-dimensional spatial problem. We finally show the effectiveness of MIMC and MISC in some computational tests, including PDEs with random coefficients and Stochastic Particle Systems.
 

Given a complex vector space $V$, consider the ring $R_{a,b}(V)$ of polynomial functions on the space of configurations of $a$ vectors and $b$ covectors which are invariant under the natural action of $SL(V)$. Rings of this type play a central role in representation theory, and their study dates back to Hilbert. Over the last three decades, different bases of these spaces with remarkable properties were found. To explicitly construct, as well as to compare, some of these bases remains a challenging problem, already open when $V$ is 3-dimensional. 
In this talk, I report on recent developments in the 3-dimensional setting of this theory.

I will discuss some of the relationships between ODE IVPs, usually solved by marching, and ODE BVPs, usually solved by global discretizations.

My research focuses on numerical techniques that help provide scalable computation within simulations of two-phase fluid flow problems. The efficient solution of the linear systems which arise is key to obtaining practical computation. I will motivate and discuss new methods which seek to generalise effective techniques for a single phase to the more challenging setting of two-phase flow where the governing equations have discontinuous coefficients.

The $\phi^4$ model in statistical physics describes the
continous phase transition in the liquid-vapour system, transition to
the superfluid phase in helium, etc. Experimentally measured values in
this model are critical exponents and universal amplitude ratios.
These values can also be calculated in the framework of the
renormalization group approach. It turns out that the obtained series
are divergent asymptotic series, but it is possible to perform Borel
resummation of such a series. To make this procedure more accurate we
need as much terms of the expansion as possible.
The results of the recent six loop analitical calculations of the
anomalous dimensions, beta function and critical exponents of the
$O(N)$ symmetric $\phi^4$ model will be presented. Different technical
aspects of these calculations (IBP method, R* operation and parametric
integration in Feynman representation) will be discussed. The

numerical estimations of critical exponents obtained with Borel
resummation procedure are compared with experimental values and
results of Monte-Carlo simulations.

In this talk I will begin by reviewing classical geometric properties of constant mean curvature surfaces, H>0, in R^3. I will then talk about several more recent results for surfaces embedded in R^3 with constant mean curvature, such as curvature and radius estimates. Finally I will show applications of such estimates including a characterisation of the round sphere as the only simply-connected surface embedded in R^3 with constant mean curvature and area estimates for compact surfaces embedded in a flat torus with constant mean curvature and finite genus. This is joint work with Meeks.

 In this introduction for topologists, we explain the role that extensions of L-infinity algebras by taking homotopy fibers plays in physics. This first appeared with the work of physicists D'Auria and Fre in 1982, but is beautifully captured by the "brane bouquet" of Fiorenza, Sati and Schreiber which shows how physical objects such as "strings", "D-branes" and "M-branes" can be classified by taking successive homotopy fibers of an especially simple L-infinity algebra called the "supertranslation algebra". We then conclude by describing our joint work with Schreiber where we build the brane bouquet out of the homotopy theory of an even simpler L-infinity algebra called the superpoint.

The Regularity Structures introduced by Martin Hairer allow us to describe the solution of a singular SPDEs by a Taylor expansion with new monomials.  We present the two Hopf Algebras used in this theory for defining the structure group and the renormalisation group. We will point out the importance of recursive formulae with twisted antipodes.

Since Donaldson-Thomas proposed a programme for studying gauge theory in higher dimensions, there has
been significant interest in understanding special Yang-Mills connections in Ricci-flat 7-manifolds with holonomy
G_2 called G_2-instantons.  However, still relatively little is known about these connections, so we begin the
systematic study of G_2-instantons in the SU(2)^2-invariant setting.  We provide existence, non-existence and
classification results, and exhibit explicit sequences of G_2-instantons where “bubbling" and "removable
singularity" phenomena occur in the limit.  This is joint work with Goncalo Oliveira (Duke).

Signature of a path provides a top down summary of the path as a driving signal. There have been substantial recent progress in reconstructing paths from its signature, (Lyons-Xu 2016, Geng 2016). In this talk, we focus on obtaining certain quantitative features of paths from their signatures. Hambly-Lyons' showed that the normalized limit of signature gives the length of a C^3 path. The result was recently extended by Lyons-Xu to C^1 paths. The extension of this result to bounded variation paths remains open. We will discuss this open problem.

I will review the recently exposed connection between N=2 superconformal field theories in four dimensions and vertex operator algebras (VOAs). I will outline some general features of the VOAs that arise in this manner and describe the manner in which they reflect four-dimensional operations such as gauging and Higgsing. Time permitting, I will also touch on the modular properties of characters of these VOAs.

There are many opportunities within Oxford to communicate your excitement about mathematics and your own research to a wider audience, whether adults or school students.  In this session we'll hear about some of those opportunities, and have some training on how to write a press release, so that you are well placed to share your next research paper with the public.

Featuring 
Rebecca Cotton-Barratt, Schools Liaison Officer and Admissions Coordinator in the Mathematical Institute
Mareli Grady, Schools Liaison Officer in the Statistics Department and Mathemagicians Coordinator in the Mathematical Institute
Stuart Gillespie, Media Relations Officer for the University of Oxford

We will introduce some ideas from geometric representation theory, including basic D-modules, (a realisation of) infinity categories, and the Ran space, and then go on to define chiral and factorisation algebras.
This talk has no prerequisites.

Van den Dries has proved the decidability of the ring of algebraic integers, the integral closure of the ring of integers in
the algebraic closure of the rationals.  A well-established analogy leads to ask the same question for the ring of complex polynomials.
This turns out to go the other way, interpreting the rational field.    An interesting structure on the
limit of Jacobians of all complex curves is encountered along the way. 

What can fashionable ideas, blind faith, or pure fantasy have to do with the scientific quest to understand the universe? Surely, scientists are immune to trends, dogmatic beliefs, or flights of fancy? In fact, Roger Penrose argues that researchers working at the extreme frontiers of mathematics and physics are just as susceptible to these forces as anyone else. In this lecture, based on his new book, Roger will argue that fashion, faith, and fantasy, while sometimes productive and even essential, may be leading today's researchers astray, most notably in three of science's most important areas - string theory, quantum mechanics, and cosmology. Yet Roger will also describe how fashion, faith, and fantasy have, ironically, also been invaluable in shaping his own work.

Roger will be signing copies of his book after the lecture.

This lecture is now SOLD OUT. Any questions, please email: @email

Let $F$ be a binary form of degree $d \geq 3$ with integer coefficients and non-zero discriminant. In this talk we give an asymptotic formula for the quantity $R_F(Z)$, the number of integers in the interval $[-Z,Z]$ representable by the binary form $F$.

This is joint work with C.L. Stewart.

We introduce a novel stochastic volatility model where the squared volatility of the asset return follows a Jacobi process. It contains the Heston model as a limit case. We show that the the joint distribution of any finite sequence of log returns admits a Gram--Charlier A expansion in closed-form. We use this to derive closed-form series representations for option prices whose payoff is a function of the underlying asset price trajectory at finitely many time points. This includes European call, put, and digital options, forward start options, and forward start options on the underlying return. We derive sharp analytical and numerical bounds on the series truncation errors. We illustrate the performance by numerical examples, which show that our approach offers a viable alternative to Fourier transform techniques. This is joint work with Damien Ackerer and Damir Filipovic.

Graham Benham

The Fluid Mechanics of Low-Head Hydropower Illuminated by Particle Image Velocimetry

We study a new type of hydropower which is cost-effective in rivers and tides where there are small pressure drops. The concept goes as follows: The cost of water turbines scales with the flow rate they deal with.  Therefore, in order to render this hydropower desirable, we make use of the Venturi principle, a natural fluid mechanical gear system which involves splitting the flow into two streams. The turbine deals with a small fraction of the flow at slow speed and high pressure, whilst the majority avoids the turbine, going at high speed and low pressure. Now the turbine feels an amplified pressure drop, thus maintaining its power output, whilst becoming much cheaper. But it turns out that the efficiency of the whole system depends strongly on the way in which these streams mix back together again.

Here we discuss some new experimental results and compare them to a simplified mathematical model for the mixing of these streams. The experimental results were achieved using particle image velocimetry (PIV), which is a type of flow visualisation. Using a laser sheet and a high speed camera, we are able to capture flow velocity fields at high resolution. Pressure measurements were also taken. The mathematical model is derived from the Navier Stokes equations using boundary layer theory alongside a flow-averaging method and reduces the problem to solving a set of ODE’s for the bulk components of the flow.

Nabil Fadai

Asymptotic Analysis of a Multiphase Drying Model Motivated by Coffee Bean Roasting

Recent modelling of coffee bean roasting suggests that in the early stages of roasting, within each coffee bean, there are two emergent regions: a dried outer region and a saturated interior region. The two regions are separated by a transition layer (or drying front). In this talk, we consider the asymptotic analysis of a multiphase model of this roasting process which was recently put forth and studied numerically, in order to gain a better understanding of its salient features. The model consists of a PDE system governing the thermal, moisture, and gas pressure profiles throughout the interior of the bean. Obtaining asymptotic expansions for these quantities in relevant limits of the physical parameters, we are able to determine the qualitative behaviour of the outer and interior regions, as well as the dynamics of the drying front. Although a number of simplifications and scaling are used, we take care not to discard aspects of the model which are fundamental to the roasting process. Indeed, we find that for all of the asymptotic limits considered, our approximate solutions faithfully reproduce the qualitative features evident from numerical simulations of the full model. From these asymptotic results we have a better qualitative understanding of the drying front (which is hard to resolve precisely in numerical simulations), and hence of the various mechanisms at play as heating, evaporation, and pressure changes result in a roasted bean. This qualitative understanding of solutions to the multiphase model is essential if one is to create more involved models that incorporate chemical reactions and solid mechanics effects.