Past

In 1878, Jordan showed that there is a function f on the set of natural numbers such that, if $G$ is a finite subgroup of $GL(n,C)$, then $G$ has an abelian normal subgroup of index at most $f(n)$. Early bounds were given by Frobenius and Schur, and close to optimal bounds were given by Weisfeiler in unpublished work in 1984 using the classification of finite simple groups; about ten years ago I obtained the optimal bounds. Crucially, these are "absolute" bounds; they do not address the wider question of divisibility of orders.

In 1887, Minkowski established a bound for the order of a Sylow p-subgroup of a finite subgroup of GL(n,Z). Recently, Serre asked me whether I could obtain Minkowski-like results for complex linear groups, and posed a very specific question. The answer turns out to be no, but his suggestion is actually quite close to the truth, and I shall address this question in my seminar. The answer addresses the divisibility issue in general, and it turns out that a central technical theorem on the structure of linear groups from my earlier work which there was framed as a replacement theorem can be reinterpreted as an embedding theorem and so can be used to preserve divisibility.

Interactions between micro-swimmers and their complex flow environments are important in many biological systems, such as sperm cells swimming in cervical mucus or bacteria in biofilm initiation areas. We present a theoretical model describing the dynamics of micro-organisms swimming in a plane Poiseuille flow of a viscoelastic fluid, accounting for hydrodynamic interactions and biological noise. General non-Newtonian effects are investigated, including shear-thinning and normal stress differences that lead to migration of the organisms across the streamlines of the background flow. We show that micro-swimmers are driven towards the centre-line of the channel, even if countered by hydrodynamic interactions with the channel walls that typically lead to boundary accumulation. Furthermore, we demonstrate that the normal stress differences reorient the swimmers at the centre-line in the direction against the flow so that they swim upstream. This suggests a natural sorting mechanism to select swimmers with a given swimming speed larger than the tunable Poiseuille flow velocity. This framework is then extended to study trapping and colony formation of pathogens near surfaces, in corners and crevices. 

Loop computations put the 'quantum' into quantum field theory. Much effort has focused on their structure and properties, with most spectacular progress in maximally supersymmetric gauge theories in the planar limit. These theories are however quite far from reality as described for instance in the standard model of particle physics. In this talk I'll report on ongoing work using BCFW on-shell recursion to obtain loop amplitude integrands in a much more realistic theory, pure Yang-Mills theory, using methods which apply directly to the standard model.

In this lecture we describe the effective Chebotarev Theorem for global function fields and show how this can be used to describe the statistics of a polynomial map f in terms of its monodromy groups. With this tool in hand, we will provide a strategy to remove the remaining heuristic in the quasi-polynomial time algorithm for discrete
logarithm problems over finite fields of small characteristic.

I will describe a sketch of the proof of Grothendieck conjecture on fundamental groups.
 

Part of the series 'What do historians of mathematics do?'

"In this talk I present the history and proof of the four-colour theorem: Can every map be coloured with just four colours so that neighbouring countries are coloured differently?  The proof took 124 years to find, and used 1200 hours of computer time. But what did it involve, and is it really a proof?"

In this talk we discuss a new second order parabolic evolution equation

for hypersurfaces in space-time initial data sets, that generalizes mean

curvature flow (MCF). In particular, the 'null mean curvature' - a

space-time extrinsic curvature quantity - replaces the usual mean

curvature in the evolution equation defining MCF.  This flow is motivated

by the study of black holes and mass/energy inequalities in general

relativity. We present a theory of weak solutions using the level-set

method and  outline a natural application of the flow as a parabolic

approach to finding outermost marginally outer trapped surfaces (MOTS),

which play the role of quasi-local black hole boundaries in general

relativity. This is joint work with Kristen Moore.

We will prove an upper bound for the volume of a minimal
hypersurface in a closed Riemannian manifold conformally equivalent to
a manifold with $Ric > -(n-1)$.  In the second part of the talk we will
construct a sweepout of a closed 3-manifold with positive Ricci
curvature by 1-cycles of controlled length and prove an upper bound
for the length of a stationary geodesic net. These are joint works
with Parker Glynn-Adey (Toronto) and Xin Zhou (MIT).

The classical Yamabe problem asks to find in a given conformal class a metric of constant scalar curvature. In fully nonlinear analogues, the scalar curvature is replaced by certain functions of the eigenvalue of the Schouten curvature tensor. I will report on quantitative Liouville theorems and fine blow-up analysis for these problems. Joint work with Yanyan Li.
 

The extension of standard stochastic models (SDEs, SPDEs) to general fractional noises is known to be a tricky issue, which cannot be studied within the classical martingale setting. We will see how the recently-introduced theory of regularity structures allows us to overcome these difficulties, in the case of a heat equation model with non-linear perturbation driven by a space-time fractional Brownian motion.

The analysis relies in particular on the exhibition of an explicit process at the core of the dynamics, the so-called K-rough path, the definition of which shows strong similarities with that of a classical rough path.

Heterotic vacua, defined with a holomorphic bundle and connection satisfying hermitian Yang-Mills, realise four-dimensional chiral gauge theories. We exploit the rich interplay between four-dimensional physics, supersymmetry and  geometry to construct a natural Kaehler metric for the moduli space, with a shockingly simple Kaehler potential. Along the way, we discover a natural geometric structure for the heterotic moduli.
 

What is the point of giving a talk?  What is the point of going to a talk?  In this presentation, which is intended to have a lot of audience participation, I would like to explore how one should prepare talks for different audiences and different occasions, and what one should try to get out of going to a talk.

This is a survey (with some proofs) of chapter 2 of the notes http://renyi.mta.hu/~szamuely/beilintronew.pdf of T. Szamuely and G. Zabradi on Beilinson's approach to the p-adic Hodge decomposition theorem.

Science is giving us unprecedented insight into the big questions that have challenged humanity. Where did we come from? What is the ultimate destiny of the universe? What are the building blocks of the physical world? What is consciousness?

‘What We Cannot Know’ asks us to rein in this unbridled enthusiasm for the power of science. Are there limits to what we can discover about our physical universe? Are some regions of the future beyond the predictive powers of science and mathematics? Are there ideas so complex that they are beyond the conception of our finite human brains? Can brains even investigate themselves or does the analysis enter an infinite loop from which it is impossible to rescue itself? 

To coincide with the launch of his new book of the same title, Marcus du Sautoy will be answering (or not answering) those questions. He will also be signing copies of the book before and after the lecture.

To book please email @email

This seminar is run jointly with OMI.

Throughout the Western world, defined benefit pension plans are disappearing, replaced by defined contribution (DC) plans. Retail investors are thus faced with managing investments over a thirty year accumulation period followed by a twenty year decumulation phase. Holders of DC plans are thus truly long term investors. We consider dynamic mean variance asset allocation strategies for long term investors. We derive the "embedding result" which converts the mean variance objective into a form suitable for dynamic programming using an intuitive approach. We then discuss a semi-Lagrangian technique for numerical solution of the optimal control problem via a Hamilton-Jacob-Bellman PDE. Parameters for the inflation adjusted return of a stock index and a risk free bond are determined by examining 89 years of US data. Extensive synthetic market tests, and resampled backtests of historical data, indicate that the multi-period mean variance strategy achieves approximately the same expected terminal wealth as a constant weight strategy, while reducing the probability of shortfall by a factor of two to three.

My main purpose in this talk is try and convey a sense of my enthusiasm for mathematical modelling generally and how I've come to use it in a range of transport applications. For concreteness, I am going to talk in particular about work I have been doing on EPSRC grant EP/K000438/1 (PI: Jillian Anable, Aberdeen) where we are using the UK government's Department for Transport MOT data to estimate mileage totals and study how they are broken down across the population in various different ways. Embedded inside this practical problem is a whole set of miniature mathematical puzzles and challenges which are quite particular to the problem area itself, and one wider question which is rather deeper and more general: whether it is possible (and how) to convert usage data that is low-resolution in time but high-resolution in individuals to knowledge that is high-resolution in time but only expressed at a population level.

In many applications we need to find or estimate the $p \ge 1$ largest elements of a matrix, along with their locations. This is required for recommender systems used by Amazon and Netflix, link prediction in graphs, and in finding the most important links in a complex network, for example. 

Our algorithm uses only matrix vector products and is based upon a power method for mixed subordinate norms. We have obtained theoretical results on the convergence of this algorithm via a comparison with rook pivoting for the LU  decomposition. We have also improved the practicality of the algorithm by producing a blocked version iterating on $n \times t$ matrices, as opposed to vectors, where $t$ is a tunable parameter. For $p > 1$ we show how deflation can be used to improve the convergence of the algorithm. 

Finally, numerical experiments on both randomly generated matrices and real-life datasets (the latter for $A^TA$ and $e^A$) show how our algorithms can reliably estimate the largest elements of a matrix whilst obtaining considerable speedups when compared to forming the matrix explicitly: over 1000x in some cases.