Past

For $k \geq 3$ we give new values of $\rho_k$ such that
$$ \| \alpha p^k + \beta \| < p^{-\rho_k} $$
has infinitely many solutions in primes whenever $\alpha$ is irrational and $\beta$ is real. The mean
value results of Bourgain, Demeter, and Guth are useful for $k \geq 6$; for all $k$, the results also
depend on bounding the number of solutions of a congruence of the form

$$ \left\| \frac{sy^k}{q} \right\| < \frac{1}{Z} \ \ (1 \leq y \leq Y < q) $$

where $q$ is a given large natural number.

The limit order book is the at the core of every modern, electronic financial market. In this talk, I will present some results pertaining to their statistical properties, mathematical modelling and numerical simulation. Questions such as ergodicity, dependencies, relation betwen time scales... will be addressed and sometimes answered to. Some on-going research projects, with applications to optimal trading and market making, will be evoked.

Permanent deformations of crystalline materials are known to be carried out by a large
number of atomistic line defects, i.e. dislocations. For specimens on micron scales or above, it
is more computationally tractable to investigate macroscopic material properties based on the
evolution of underlying dislocation densities. However, classical models of dislocation
continua struggle to resolve short-range elastic interactions of dislocations, which are believed
responsible for the formation of various heterogeneous dislocation substructures in crystals. In
this talk, we start with discussion on formulating the collective behaviour of a row of
dislocation dipoles, which would be considered equivalent to a dislocation-free state in
classical continuum models. It is shown that the underlying discrete dislocation dynamics can
be asymptotically captured by a set of evolution equations for dislocation densities along with
a set of equilibrium equations for variables characterising the self-sustained dislocation
substructures residing on a shorter length scale, and the strength of the dislocation
substructures is associated with the solvability conditions of their governing equilibrium
equations. Under the same strategy, a (continuum) flow stress formula for multi-slip systems
is also derived, and the formula resolves more details from the underlying dynamics than the
ubiquitously adopted Taylor-type formulae.

The requirement to compute Jordan blocks for multiple eigenvalues arises in a number of physical problems, for example panel flutter problems in aerodynamical stability, the stability of electrical power systems, and in quantum mechanics. We introduce a general method for computing a 2-dimensional Jordan block in a parameter-dependent matrix eigenvalue problem based on the so called Implicit Determinant Method. This is joint work with Alastair Spence (Bath).

I will discuss the circumstances in which residual finiteness properties of an amalgamated free product $A\ast_c B$ may be deduced from the properties of $A$ and $B$, with particular regard to the pro-p residual properties.

ABSTRACT: A space of countable extent, also called an omega_1-compact space, is one in which every closed discrete subspace is countable.  The axiom used in the following theorem is consistent if it is consistent that there is a supercompact cardinal.

Theorem 1  The LCT axiom implies that every hereditarily normal, omega_1-compact space
is sigma-countably compact,  i.e., the union of countably many countably compact subspaces.

Even for the specialized subclass of monotonically normal spaces, this is only a consistency result:

Theorem 2   If club, then there exists a locally compact, omega_1-compact monotonically
normal space that is not sigma-countably compact.

These two results together are unusual in that most independence results on
monotonically normal spaces depend on whether Souslin's Hypothesis (SH) is true,
and do not involve large cardinal axioms. Here, it is not known whether either
SH or its negation affect either direction in this independence result.

The following unsolved problem is also discussed:

Problem  Is there a ZFC example of a locally compact, omega_1-compact space
of cardinality aleph_1 that is not sigma-countably compact?

The cube attack of Dinur and Shamir and the AIDA attack of Vielhaber have been used successfully on 

reduced round versions of the Trivium stream cipher and a few other ciphers. 

These attacks can be viewed in the framework of higher order differentiation, as introduced by Lai in 

the cryptographic context. We generalise these attacks from the binary case to general finite fields, 

showing that we would need to differentiate several times with respect to each variable in order to have

a reasonable chance of a successful attack.

We also investigate the notion of “fast points” for a binary polynomial function f  

(i.e. vectors such that the derivative of f with respect to this vector has a lower 

than expected degree). These were  introduced by Duan and Lai, motivated by the fact that higher order 

differential attacks are usually more efficient if they use such points. The number of functions which 

admit fast points were computed by Duan et al in a few particular cases; we give explicit formulae for 

all remaining cases and discuss the cryptographic significance of these results.

Diffusion processes are commonly used in image processing. In particular, complex diffusion models have been successfully applied in medical imaging denoising. The interpretation of a complex diffusion equation as a cross-diffusion system motivates the introduction of more general models of this type and their study in the context of image processing. In this talk we will discuss the use of nonlinear cross-diffusion systems to perform image restoration. We will analyse the well-posedness, scale-space properties and
long time behaviour of the models along with their performance to treat image filtering problems. Examples of application will be highlighted.

For a fixed degree sequence D=(d_1,...,d_n), let G(D) be a uniformly chosen (simple) graph on {1,...,n} where the vertex i has degree d_i. The study of G(D) is of special interest in order to model real-world networks that can be described by their degree sequence, such as scale-free networks. While many aspects of G(D) have been extensively studied, most of the obtained results only hold provided that the degree sequence D satisfies some technical conditions. In this talk we will introduce a new approach (based on the switching method) that allows us to study the random graph G(D) imposing no conditions on D. Most notably, this approach provides a new criterion on the existence of a giant component in G(D). Moreover, this method is also useful to determine whether there exists a percolation threshold in G(D). The first part of this talk is joint work with F. Joos, D. Rautenbach and B. Reed, and the second part, with N. Fountoulakis and F. Joos.

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.