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 Early bounds were given by Frobenius and Schur, and close to optimal bounds were given by Weisfeiler in unpublished work in 1984 using the classification of finite simple groups; about ten years ago I obtained the optimal bounds.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.

Quasimorphisms (QM) of groups to the reals are well studied and are linked to stable commutator length (scl) via Bavard Duality- Theorem. The notion of QM can be generalized to yield maps  between groups such that each QM from one group pulls back to a QM in the other.

We will give both a short overview of features of scl and investigate these generalized QMs with large scale properties of the commutator group. 

One of the peculiar features of quantum mechanics is
entanglement. It is known that entanglement is monogamous in the sense
that a quantum system can only be strongly entangled to one other
system. In this talk, I will show how this so-called monogamy of
entanglement can be captured and quantified by a "game". We show that,
in this particular game, the monogamy completely "cancels out" the
advantage of entanglement.
As an application of our analysis, we show that - in theory - the
standard BB84 quantum-key-distribution scheme is one-sided
device-independent, meaning that one of the parties, say Bob, does not
need to trust his quantum measurement device: security is guaranteed
even if his device is completely malicious.
The talk will be fully self-contained; no prior knowledge on quantum
mechanics/cryptography is necessary.

In 1924, James W. Alexander constructed a 2-sphere in R3 that is not ambiently homeomorphic to the standard 2-sphere, which demonstrated the failure of the Schoenflies theorem in higher dimensions. I will describe the construction of the Alexander horned sphere and the Antoine necklace and describe some of their properties.

Ever since the compilers of Euclid's Elements gave the "definitions" that "a point is that which has no part" and "a line is breadthless length", philosophers and mathematicians have worried that the basic concepts of geometry are too abstract and too idealized.  In the 20th century writers such as Husserl, Lesniewski, Whitehead, Tarski, Blumenthal, and von Neumann have proposed "pointless" approaches.  A problem more recent authors have emphasized it that there are difficulties in having a rich theory of a part-whole relationship without atoms and providing both size and geometric dimension as part of the theory.  A possible solution is proposed using the Boolean algebra of measurable sets modulo null sets along with relations derived from the group of rigid motions in Euclidean n-space. 

Low-rank compression of matrices and tensors is a huge and growing business.  Closely related is low-rank compression of multivariate functions, a technique used in Chebfun2 and Chebfun3.  Not all functions can be compressed, so the question becomes, which ones?  Here we focus on two kinds of functions for which compression is effective: those with some alignment with the coordinate axes, and those dominated by small regions of localized complexity.

For any given graph H, we may define a natural corresponding functional ||.||_H. We then say that H is norming if ||.||_H is a semi-norm. A similar notion ||.||_{r(H)} is defined by || f ||_{r(H)}:=|| | f | ||_H and H is said to be weakly norming if ||.||_{r(H)} is a norm. Classical results show that weakly norming graphs are necessarily bipartite. In the other direction, Hatami showed that even cycles, complete bipartite graphs, and hypercubes are all weakly norming. Using results from the theory of finite reflection groups, we demonstrate that any graph which is edge-transitive under the action of a certain natural family of automorphisms is weakly norming. This result includes all previous examples of weakly norming graphs and adds many more. We also include several applications of our results. In particular, we define and compare a number of generalisations of Gowers' octahedral norms and we prove some new instances of Sidorenko's conjecture. Joint work with David Conlon.

We establish necessary and sufficient conditions for linear convergence of operator splitting methods for a general class of convex optimization problems where the associated fixed-point operator is averaged. We also provide a tight bound on the achievable convergence rate. Most existing results establishing linear convergence in such methods require restrictive assumptions regarding strong convexity and smoothness of the constituent functions in the optimization problem. However, there are several examples in the literature showing that linear convergence is possible even when these properties do not hold. We provide a unifying analysis method for establishing linear convergence based on linear regularity and show that many existing results are special cases of our approach.

Black holes are one of the few available laboratories for testing theoretical ideas in fundamental physics. Since Hawking's result that they radiate a thermal spectrum, black holes have been regarded as thermodynamic objects with associated temperature, entropy, etc. While this is an extremely beautiful picture it has also lead to numerous puzzles. In this talk I will describe the two-loop correction to scalar correlation functions due to \phi^{^4} interactions and explain why this might have implications for our current view of semi-classical black holes.
 

Given an integer $d$ such that $2 \leq d \leq 50$, we want to
answer the question: When is the sum of
$d$ consecutive cubes a perfect power? In other words, we want to find all
integer solutions to the equation
$(x+1)^3 + (x+2)^3 + \cdots + (x+d)^3 = y^p$. In this talk, we present some
of the techniques used to tackle such diophantine problems.

We explore a specific system in which geometry and loading conspire to generate fine-scale wrinkling. This system -- a twisted ribbon held with small tension -- was examined experimentally by Chopin and Kudrolli 
[Phys Rev Lett 111, 174302, 2013].

There is a regime where the ribbon wrinkles near its center. A recent paper by Chopin, D\'{e}mery, and Davidovitch models this regime using a von-K\'{a}rm\'{a}n-like 
variational framework [J Elasticity 119, 137-189, 2015]. Our contribution is to give upper and lower bounds for the minimum energy as the thickness tends to zero. Since the bounds differ by a thickness-independent prefactor, we have determined how the minimum energy scales with thickness. Along the way we find estimates on Sobolev norms of the minimizers, which provide some information on the character of the wrinkling. This is a joint work with  Robert V. Kohn in Courant Institute, NYU.

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

I'm currently working on a biography of Charles Hutton (1737–1823): pit lad, FRS, and professor of Mathematics. No-one much has heard of him today, but to his contemporaries he was "one of the greatest mathematicians in Europe". I'll give an outline of his remarkable story and say something about why he's worth my time.

In this talk we will give an overview of the recent research on global quantizations on spaces of different types: compact and nilpotent Lie groups, general locally compact groups, compact manifolds with boundary.

Motivated by work of Borel and Serre on arithmetic groups, Bestvina and Feighn defined a bordification of Outer space; this is an enlargement of Outer space which is highly-connected at infinity and on which the action of $Out(F_n)$ extends, with compact quotient. They conclude that $Out(F_n)$ satisfies a type of duality between homology and cohomology.  We show that Bestvina and Feighn’s  bordification can be realized as a deformation retract of Outer space instead of an extension, answering some questions left open by Bestvina and Feighn and considerably simplifying their proof that the bordification is highly connected at infinity.

We give a noncommutative analogue of Castelnuovo's classic theorem that (-1) lines on a smooth surface can be contracted, and show how this may be used to construct an explicit birational map between a noncommutative P^2 and a noncommutative quadric surface. This has applications to the classification of noncommutative projective surfaces, one of the major open problems in noncommutative algebraic geometry. We will not assume a background in noncommutative ring theory.  The talk is based on joint work with Rogalski and Staffor

The talk will focus on continuous time random walk with unbounded i.i.d. random conductances on the grid $\mathbb{Z}^d$  In the first place, in a joint work with Kumagai and Mathieu, we obtain Gaussian heat kernel bounds and also local CLT for bounded from above and not bounded from below conductances. The proof is given at first in a general framework, then it is specified in the case of plynomial lower tail conductances. It is essentially based on percolation and spectral analysis arguments, and Harnack inequalities. Then we will discuss the same questions for the same model with i.i.d. random conductances, bounded from below and with finite expectation.

Recently, there has been some progress in examining mirror symmetry beyond Calabi-Yau threefolds. I will discuss how this is related to flux vacua of type II supergravity on eight-dimensional manifolds equipped with SU(4)-structure. It will be shown that the natural framework to describe such vacua is generalized complex geometry. Two classes of type IIB solutions will be given, one of which is complex, the other symplectic, and I will describe in what sense these are mirror to one another.  

From the finite Fourier transform to topological quantum field theory -- Bruce Bartlett

Abstract: In 1979, Auslander and Tolimieri wrote the influential "Is computing with the finite Fourier transform pure or applied mathematics?".  It was a homage to the indivisibility of our two subjects, by demonstrating the interwoven nature of the finite Fourier transform, Gauss sums, and the finite Heisenberg group.  My talk is intended as a new chapter in this story. I will explain how all these topics come together yet again in 3-dimensional topological quantum field theory, namely Chern-Simons theory with gauge group U(1).

Defects in liquid crystals: mathematical approaches -- Giacomo Canevari

Abstract: Liquid crystals are matter in an intermediate state between liquids and crystalline solids.  They are composed by molecules which can flow, but retain some form of ordering.  For instance, in the so-called nematic phase the molecules tend to align along some locally preferred directions.  However, the ordering is not perfect, and defects are commonly observed.

The mathematical theory of defects in liquid crystals combines tools from different fields, ranging from topology - which provides a convenient language to describe the main properties of defects -to calculus of variations and partial differential equations.  I will compare a few mathematical approaches to defects in nematic liquid crystals, and discuss how they relate to each other via asymptotic analysis.

In this talk, I will present two different aspects of the ice flow modelling, including a theoretical part and an applied part. In the theoretical part, I will derive some "mechanical error estimators'', i.e. estimators that can measure the mechanical error between the most accurate ice flow model (Glen-Stokes) and some approximations based on shallowness assumption. To do so, I will follow residual techniques used to obtain a posteriori estimators of the numerical error in finite element methods for non-linear elliptic problems. In the applied part, I will present some simulations of the ice flow generated by the Rhone Glacier, Switzerland, during the last glacial maximum (~ 22 000 years ago), analyse the trajectories taken by erratic boulders of different origins, and compare these results to geomorphological observations. In particular, I will show that erratic boulders, whose origin is known, constitute valuable data to infer information about paleo-climate, which is the most uncertain input of any paleo ice sheet model. 

A holy grail of nano-technology is to create truly complex, multi-component structures by self assembly.
 

Most self-assembly has focused on the creation of `structural complexity'. In my talk, I will discuss `Addressable Complexity': the creation of structures that contain hundreds or thousands of
distinct building blocks that all have to find their place in a 3D structure.

In the first part of the talk I will review Bob Fernholz' theory of functionally generated portfolios. In the second part I will discuss questions related to the existence of short-term arbitrage opportunities.
This is joint work with Bob Fernholz and Ioannis Karatzas

This is the first talk of the workshop organised by F. Brown, M. Kim and D. Rössler on Beilinson's approach to p-adic Hodge theory. 

In this talk, we shall give the definition and recall various properties of the cotangent complex, which was originally defined by L. Illusie in his monograph "Complexe cotangent et déformations" (Springer LNM 239, 1971).