If $A$ is a set of $n$ positive integers, how small can the set
$\{ x/(x,y) : x,y \in A \}$ be? Here, as usual, $(x,y)$ denotes the highest common factor of
$x$ and $y$. This elegant question was raised by Granville and Roesler, who
also reformulated it in the following way: given a set $A$ of $n$ points in
the integer grid ${\bf Z}^d$, how small can $(A-A)^+$, the projection of the difference
set of $A$ onto the positive orthant, be?
Freiman and Lev gave an example to show that (in any dimension) the size can
be as small as $n^{2/3}$ (up to a constant factor). Granville and Roesler
proved that in two dimensions this bound is correct, i.e. that the size is
always at least $n^{2/3}$, and they asked if this holds in any dimension.
After some background material, the talk will focus on recent developments.
Joint work with B\'ela Bollob\'as.
When modelling the motion of a sperm cell in the female reproductive tract, the Reynolds number is found to be very small, thus allowing the nonlinear Navier-Stokes equations to simplify to the linear Stokes equations stating that pressure, viscous and body forces balance each other at any instant in time. A wide range of analytical techniques can be applied to investigate the Stokes flow past a moving body. In this talk, we introduce various Stokes flow singularities and illustrate how they can provide a handy starting point (ansatz) when trying to determine the form of the flow field around certain bodies, from simple translating spheres to beating sperm tails.
The infamous inverse Galois problem asks whether or not every finite group can be realised as a Galois group over Q. We will see some techniques that have been developed to attack it, and will soon end up in the realms of class field theory, étale fundamental groups and modular representations. We will give some concrete examples and outline the so called 'rigidity method'.
Morgan and Culler proved in the 1980’s that a minimal action of a free group on a tree is
completely determined by its length function. This theorem has been of fundamental importance in the
study of automorphisms of free groups. In particular, it gives rise to a compactification of Culler-Vogtmann's
Outer Space. We prove a 2-dimensional analogue of this theorem for right-angled Artin groups acting on
CAT(0) rectangle complexes. (Joint work with M. Margolis)
I will describe the recent construction of new supersymmetric compactifications of the heterotic string which yield just the spectrum of the MSSM at low energies. The starting point is the standard embedding solution on a Calabi-Yau manifold with Euler number -6 with various choices of Wilson lines, i.e., various choices of discrete holonomy for the gauge bundle. Although they yield three net generations of standard model matter, such models necessarily have a larger gauge group than the standard model, as well as exotic matter content. Families of stable bundles can be obtained by deformation of these simple solutions, the deformation playing the dual role of partially breaking the gauge group and reducing the matter content, and in this way we construct more realistic models. The moduli space breaks up into various branches depending on the initial choice of Wilson lines, and on eight of these branches we find models with exactly the standard model gauge group, three generations of quarks and leptons, two Higgs doublets, and no other massless charged states. I will also comment on why these are possibly the unique models of this type.
The Eilenberg-Ganea conjecture is the statement that every group of cohomological dimension two admits a two-dimensional classifying space. This problem is unsolved after 50 years. I shall discuss the background to this question and negative answers to some other related questions. This includes recent joint work with Martin Fluch.
In a market with one safe and one risky asset, an investor with a long
horizon and constant relative risk aversion trades with constant
investment opportunities and proportional transaction costs. We derive
the optimal investment policy, its welfare, and the resulting trading
volume, explicitly as functions of the market and preference parameters,
and of the implied liquidity premium, which is identified as the
solution of a scalar equation. For small transaction costs, all these
quantities admit asymptotic expansions of arbitrary order. The results
exploit the equivalence of the transaction cost market to another
frictionless market, with a shadow risky asset, in which investment
opportunities are stochastic. The shadow price is also derived
explicitly. (Joint work with Paolo Guasoni, Johannes Muhle-Karbe, and
Walter Schachermayer)
The seminar will take place in Lecture Theatre A, Department of Computer Science.
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Contextuality and non-locality are features of quantum mechanics which stand in sharp contrast to the realistic picture underlying classical physics. We shall describe a unified geometric perspective on these notions in terms of *obstructions to the existence of global sections*. This allows general results and structural notions to be uncovered, with quantum mechanics appearing as a special case. The natural language to use here is that of sheaves and presheaves; and cohomological obstructions can be defined which witness contextuality in a number of salient examples.
This is joint work with Adam Brandenburger
http://iopscience.iop.org/1367-2630/13/11/113036/
http://arxiv.org/abs/1102.0264
and Shane Mansfield and Rui Soares Barbosa
http://arxiv.org/abs/1111.3620
This work builds on the foundation laid by Benney & Timson (1980), who
examined the flow near a contact line and showed that, if the contact
angle is 180 degrees, the usual contact-line singularity does not arise.
Their local analysis, however, does not allow one to determine the
velocity of the contact line and their expression for the shape of the
free boundary involves undetermined constants - for which they have been
severely criticised by Ngan & Dussan V. (1984). As a result, the ideas
of Benny & Timson (1980) have been largely forgotten.
The present work shows that the criticism of Ngan & Dussan V. (1984)
was, in fact, unjust. We consider a two-dimensional steady Couette flow
with a free boundary, for which the local analysis of Benney & Timson
(1980) can be complemented by an analysis of the global flow (provided
the slope of the free boundary is small, so the lubrication
approximation can be used). We show that the undetermined constants in
the solution of Benney & Timson (1980) can all be fixed by matching
their local solution to the global one. The latter also determines the
contact line's velocity, which we compute among other characteristics of
the global flow.
We show that the steepest-descent and Newton's methods for unconstrained nonconvex optimization
under standard assumptions may both require a number of iterations and function evaluations
arbitrarily close to the steepest-descent's global worst-case complexity bound. This implies that
the latter upper bound is essentially tight for steepest descent and that Newton's method may be as
slow as the steepest-descent method in the worst case. Then the cubic regularization of Newton's
method (Griewank (1981), Nesterov & Polyak (2006)) is considered and extended to large-scale
problems, while preserving the same order of its improved worst-case complexity (by comparison to
that of steepest-descent); this improved worst-case bound is also shown to be tight. We further
show that the cubic regularization approach is, in fact, optimal from a worst-case complexity point
of view amongst a wide class of second-order methods. The worst-case problem-evaluation complexity
of constrained optimization will also be discussed. This is joint work with Nick Gould (Rutherford
Appleton Laboratory, UK) and Philippe Toint (University of Namur, Belgium).
I will give an introduction to the variational characterisation of the Ricci flow that was first introduced by G. Perelman in his paper on "The entropy formula for the Ricci flow and its geometric applications" http://arxiv.org/abs/math.DG/0211159. The first in a series of three papers on the geometrisation conjecture. The discussion will be restricted to sections 1 through 5 beginning first with the gradient flow formalism. Techniques from the Calculus of Variations will be emphasised, notably in proving the monotonicity of particular functionals. An overview of the local noncollapsing theorem (Perelman’s first breakthrough result) will be presented with refinements from Topping [Comm. Anal. Geom. 13 (2005), no. 5, 1039–1055.]. Some remarks will also be made on connections to implicit structures seen in the physics literature, for instance of those seen in D. Friedan [Ann. Physics 163 (1985), no. 2, 318–419].
Decision making in the presence of uncertainty is a mathematically delicate topic. In this talk, we consider coherent sublinear expectations on a measurable space, without assuming the existence of a dominating probability measure. By considering discrete-time `martingale' processes, we show that the classical results of martingale convergence and the up/downcrossing inqualities hold in a `quasi-sure' sense. We also give conditions, for a general filtration, under which an `aggregation' property holds, generalising an approach of Soner, Touzi and Zhang (2011). From this, we extend various results on the representation of conditional sublinear expectations to general filtrations under uncertainty.
In the talk I will mention two regularity results: the SBV regularity for strictly hyperbolic, genuinely nonlinear 1D systems of conservation laws and the characterization of intrinsic Lipschitz codimension 1 graphs in the Heisenberg groups. In both the contexts suitable scalar, 1D balance laws arise with very low regularity. I will in particular highlight the role of characteristics.
This seminar will be based on joint works with G. Alberti, S. Bianchini, F. Bigolin and F. Serra Cassano, and the main previous literature.
I will first present Priestley style topological dualities for
several categories of distributive meet-semilattices
and implicative semilattices developed by G. Bezhanishvili and myself.
Using these dualities I will introduce a topological duality for Hilbert
algebras,
the algebras that correspond to the implicative reduct of intuitionistic logic.
In this talk I'll describe recent work with Henry Wilton (UCL) in which
we prove that there does not exist an algorithm that can determine which
finitely presented groups have a non-trivial finite quotient.