Past

We also state some connections to some open problems.

Hollow vortices are vortices whose interior is at rest. They posses vortex sheets on their boundaries and can be viewed as a desingularization of point vortices. We give a brief history of point vortices. We then obtain exact solutions for hollow vortices in linear and nonlinear strain and examine the properties of streets of hollow vortices. The former can be viewed as a canonical example of a hollow vortex in an arbitrary flow, and its stability properties depend. In the latter case, we reexamine the hollow vortex street of Baker, Saffman and Sheffield and examine its stability to arbitrary disturbances, and then investigate the double hollow vortex street. Implications and extensions of this work are discussed.

The geometric Langlands correspondence relates rank n integrable connections 
on a complex Riemann surface $X$ to regular holonomic D-modules on 
$Bun_n(X)$, the moduli stack of rank n vector bundles on $X$.  If we replace 
$X$ by a smooth irreducible curve over a finite field of characteristic p 
then there is a version of the geometric Langlands correspondence involving 
$l$-adic perverse sheaves for $l\neq p$.  In this lecture we consider the 
case $l=p$, proposing a $p$-adic version of the geometric Langlands 
correspondence relating convergent $F$-isocrystals on $X$ to arithmetic 
$D$-modules on $Bun_n(X)$.

The geometric Langlands correspondence relates rank n integrable connections on a complex Riemann surface $X$ to regular holonomic D-modules on  $Bun_n(X)$, the moduli stack of rank n vector bundles on $X$.  If we replace $X$ by a smooth irreducible curve over a finite field of characteristic p then there is a version of the geometric Langlands correspondence involving $l$-adic perverse sheaves for $l\neq p$.  In this lecture we consider the case $l=p$, proposing a $p$-adic version of the geometric Langlands correspondence relating convergent $F$-isocrystals on $X$ to arithmetic $D$-modules on $Bun_n(X)$.

We present recent numerical techniques for the treatment of integral formulations of Helmholtz boundary value problems in the case of high frequencies. The combination of $H^2$-matrices with further developments of the adaptive cross approximation allows to solve such problems with logarithmic-linear complexity independent of the frequency. An advantage of this new approach over existing techniques such as fast multipole methods is its stability over the whole range of frequencies, whereas other methods are efficient either for low or high frequencies.

We find and describe four futures markets where the bid-ask spread is bid down to the fixed price tick size practically all the time, and which match coun- terparties using a pro-rata rule. These four markets’ offered depths at the quotes on average exceed mean market order size by two orders of magnitude, and their order cancellation rates (the probability of any given offered lot being cancelled) are significantly over 96 per cent. We develop a simple theoretical model to explain these facts, where strategic complementarities in the choice of limit order size cause traders to risk overtrading by submitting over-sized limit orders, most of which they expect to cancel.

Joint work with Jonathan Field.

We shall dedicate the first half of the talk to introduce

classical Higgs bundles and describe the fibres of the corresponding

Hitchin fibration in terms of spectral data. Then, we shall define

principal Higgs bundles and look at some examples. Finally, we

consider the particular case of $SL(2,R)$, $U(p,p)$ and $Sp(2p,2p)$ Higgs

bundles and study their spectral data. Time permitting, we shall look

at different applications of our new methods.

In this talk, we make quantitative comparisons between two widely-used liquid crystal modelling approaches - the continuum Landau-de Gennes theory and mesoscopic mean-field theories, such as the Maier-Saupe and Onsager theories. We use maximum principle arguments for elliptic partial differential equations to compute explicit bounds for the norm of static equilibria within the Landau-de Gennes framework. These bounds yield an explicit prescription of the temperature regime within which the LdG and the mean-field predictions are consistent, for both spatially homogeneous and inhomogeneous systems. We find that the Landau-de Gennes theory can make physically unrealistic predictions in the low-temperature regime. In my joint work with John Ball, we formulate a new theory that interpolates between mean-field and continuum approaches and remedies the deficiencies of the Landau-de Gennes theory in the low-temperature regime. In particular, we define a new thermotropic potential that blows up whenever the mean-field constraints are violated. The main novelty of this work is the incorporation of spatial inhomogeneities (outside the scope of mean-field theory) along with retention of mean-field level information.

Graphs and digraphs behave quite differently, and many classical results for graphs are often trivially false when extended to general digraphs. Therefore it is usually necessary to restrict to a smaller family of digraphs to obtain meaningful results. One such very natural family is Eulerian digraphs, in which the in-degree equals out-degree at every vertex.

In this talk, we discuss several natural parameters for Eulerian digraphs and study their connections. In particular, we show that for any Eulerian digraph G with n vertices and m arcs, the minimum feedback arc set (the smallest set of arcs whose removal makes G acyclic) has size at least $m^2/2n^2+m/2n$, and this bound is tight. Using this result, we show how to find subgraphs of high minimum degrees, and also long cycles in Eulerian digraphs. These results were motivated by a conjecture of Bollob\'as and Scott.

Joint work with Ma, Shapira, Sudakov and Yuster

Stability plays an important role in engineering, for it limits the load carrying capacity of all kinds of structures. Many failure mechanisms in advanced engineering materials are stability-related, such as localized deformation zones occurring in fiber-reinforced composites and cellular materials, used in aerospace and packaging applications. Moreover, modern biomedical applications, such as vascular stents, orthodontic wire etc., are based on shape memory alloys (SMA’s) that exploit the displacive phase transformations in these solids, which are macroscopic manifestations of lattice-level instabilities.

The presentation starts with the introduction of the concepts of stability and bifurcation for conservative elastic systems with a particular emphasis on solids with periodic microstructures. The concept of Bloch wave analysis is introduced, which allows one to find the lowest load instability mode of an infinite, perfect structure, based solely on unit cell considerations. The relation between instability at the microscopic level and macroscopic properties of the solid is studied for several types of applications involving different scales: composites (fiber-reinforced), cellular solids (hexagonal honeycomb) and finally SMA's, where temperature- or stress-induced instabilities at the atomic level have macroscopic manifestations visible to the naked eye.

This talk will attempt to say something about the p-adic zeta function, a p-adic analytic object which encodes information about Galois cohomology of Tate twists in its special values. We first explain the construction of the p-adic zeta function, via p-adic Fourier theory. Then, after saying something about Coleman integration, we will explain the interpretation of special values of the p-adic zeta function as limiting values of p-adic polylogarithms, in analogy with the Archimedean case. Finally, we will explore the consequences for the de Rham and etale fundamental groupoids of the projective line minus three points.

In this talk, I will focus on an infinite class of 3d N=4 gauge theories

which can be constructed from a certain set of ordered pairs of integer

partitions. These theories can be elegantly realised on brane intervals in

string theory.  I will give an elementary review on such brane constructions

and introduce to the audience a symmetry, known as mirror symmetry, which

exchanges two different phases (namely the Higgs and Coulomb phases) of such

theories.  Using mirror symmetry as a tool, I will discuss a certain

geometrical aspect of the vacuum moduli spaces of such theories in the

Coulomb phase. It turns out that there are certain infinite subclasses of

such spaces which are special and rather simple to study; they are complete intersections. I will mention some details and many interesting features of these spaces.

 Consider a fully-connected social network of people, companies,
or countries, modeled as an undirected complete graph with real numbers on
its edges. Positive edges link friends; negative edges link enemies.
I'll discuss two simple models of how the edge weights of such networks
might evolve over time, as they seek a balanced state in which "the enemy of
my enemy is my friend." The mathematical techniques involve elementary
ideas from linear algebra, random graphs, statistical physics, and
differential equations. Some motivating examples from international
relations and social psychology will also be discussed. This is joint work
with Seth Marvel, Jon Kleinberg, and Bobby Kleinberg. 

Data assimilation in highly nonlinear and high dimensional systems is a hard

problem. We do have efficient data-assimilation methods for high-dimensional

weakly nonlinear systems, exploited in e.g. numerical weather forecasting.

And we have good methods for low-dimensional (

In this talk, we elaborate on the notions of no-free-lunch that have proved essential in the theory of financial mathematics---most notably, arbitrage of the first kind. Focus will be given in most recent developments. The precise connections with existence of deflators, numeraires and pricing measures are explained, as well as the consequences that these notions have in the existence of bubbles and the valuation of illiquid assets in the market.

ARKeX is a geophysical exploration company that conducts airborne gravity gradiometer surveys for the oil industry. By measuring the variations in the gravity field it is possible to infer valuable information about the sub-surface geology and help find prospective areas.

A new type of gravity gradiometer instrument is being developed to have higher resolution than the current technology. The basic operating principles are fairly simple - essentially measuring the relative displacement of two proof masses in response to a change in the gravity field. The challenge is to be able to see typical signals from geological features in the presence of large amounts of motional noise due to the aircraft. Fortunately, by making a gradient measurement, a lot of this noise is cancelled by the instrument itself. However, due to engineering tolerances, the instrument is not perfect and residual interference remains in the measurement.

Accelerometers and gyroscopes record the motional disturbances and can be used to mathematically model how the noise appears in the instrument and remove it during a software processing stage. To achieve this, we have employed methods taken from the field of system identification to produce models having typically 12 inputs and a single output. Generally, the models contain linear transfer functions that are optimised during a training stage where controlled accelerations are applied to the instrument in the absence of any anomalous gravity signal. After training, the models can be used to predict and remove the noise from data sets that contain signals of interest.

High levels of accuracy are required in the noise correction schemes to achieve the levels of data quality required for airborne exploration. We are therefore investigating ways to improve on our existing methods, or find alternative techniques. In particular, we believe non-linear and non-stationary models show benefits for this situation.

Nematic liquid crystals (NLCs) are materials that flow like liquids, but have some crystalline features. Their molecules are typically long and thin, and tend to align locally, which imparts some elastic character to the NLC. Moreover at interfaces between the NLC and some other material (such as a rigid silicon substrate, or air) the molecules tend to have a preferred direction (so-called "surface anchoring"). This preferred behaviour at interfaces, coupled with the internal "elasticity", can give rise to complex instabilities in spreading free surface films. This talk will discuss modelling approaches to describe such flows. The models presented are capable of capturing many of the key features observed experimentally, including arrested spreading (with or without instability). Both 2D and 3D spreading scenarios will be considered, and simple ways to model nontrivial surface anchoring patterns, and "defects" within the flows will also be discussed.

Although much work has been done on using RBFs for reconstructing arbitrary surfaces, using RBFs to solve PDEs on arbitrary manifolds is only now being considered and is the subject of this talk. We will review current methods and introduce a new technique that is loosely inspired by the Closest Point Method. This new technique, the Orthogonal Gradients Method (OGr), benefits from the advantages of using RBFs: the simplicity, the high accuracy but also the meshfree character, which gives the flexibility to represent the most complex geometries in any dimension.

In this talk I will survey the results on the existence of solutions of the semigeostrophic system, a fully nonlinear reduction of the Navier-Stokes equation that constitute a valid model when the effect of rotation dominate the atmospheric flow. I will give an account of the theory developed since the pioneering work of Brenier in the early 90's, to more recent results obtained in a joint work with Mike Cullen and David Gilbert.

Expander graphs are sparse finite graphs with strong connectivity properties, on account of which they are much sought after in the construction of networks and in coding theory. Surprisingly, the first examples of large expander graphs came not from combinatorics, but from the representation theory of semisimple Lie groups. In this introductory talk, I will outline some of the history of the emergence of such examples from group theory, and give several applications of expander graphs to group theoretic problems.

See http://www.essex.ac.uk/maths/people/fremlin/chap53.pdf (538S)

The reaction-diffusion master equation (RDME) is a popular model in systems biology. In the RDME, diffusion is modeled as discrete jumps between voxels in the computational domain. However, it has been demonstrated that a more fine-grained model is required to resolve all the dynamics of some highly diffusion-limited systems.

In Greenʼs Function Reaction Dynamics (GFRD), a method based on the Smoluchowski model, diffusion is modeled continuously in space.

This will be more accurate at fine scales, but also less efficient than a discrete-space model.

We have developed a hybrid method, combining the RDME and the GFRD method, making it possible to do the more expensive fine-grained simulations only for the species and in the parts of space where it is required in order to resolve all the dynamics, and more coarse-grained simulations where possible. We have applied this method to a model of a MAPK-pathway, and managed to reduce the number of molecules simulated with GFRD by orders of magnitude and without an appreciable loss of accuracy.

Derived moduli stacks extend moduli stacks to give families over simplicial or dg rings. Lurie's representability theorem gives criteria for a functor to be representable by a derived geometric stack, and I will introduce a variant of it. This establishes representability for problems such as moduli of sheaves and moduli of polarised schemes.

With the advent of powerful computers and the internet, our ability to collect and store large amounts of data has improved tremendously over the past decades. As a result, methods for extracting useful information from these large datasets have gained in importance. In many cases the data can be conveniently represented as a network, where the nodes are entities of interest and the edges encode the relationships between them. Community detection aims to identify sets of nodes that are more densely connected internally than to the rest of the network. Many popular methods for partitioning a network into communities rely on heuristically optimising a quality function. This approach can run into problems for large networks, as the quality function often becomes near degenerate with many near optimal partitions that can potentially be quite different from each other. In this talk I will show that this near degeneracy, rather than being a severe problem, can potentially allow us to extract additional information

based on joint work with Charles Fefferman (Princeton) and Kevin Luli (Yale).

Vinogradov's three prime theorem resolves the weak Goldbach conjecture for sufficiently large integers. We discuss some of the ideas behind the proof, and discuss some of the obstacles to completing a proof of the odd goldbach conjecture.

For a fixed background manifold $M$ and parameter-space $X$, the associated configuration space is the space of $n$-point subsets of $M$ with parameters drawn from $X$ attached to each point of the subset, topologised in a natural way so that points cannot collide. One can either remember or forget the ordering of the n points in the configuration, so there are ordered and unordered versions of each configuration space.

It is a classical result that the sequence of unordered configuration spaces, as $n$ increases, is homologically stable: for each $k$ the degree-$k$ homology is eventually independent of $n$. However, a simple counterexample shows that this result fails for ordered configuration spaces. So one could ask whether it's possible to remember part of the ordering information and still have homological stability.

The goal of this talk is to explain the ideas behind a positive answer to this question, using 'oriented configuration spaces', in which configurations are equipped with an ordering - up to even permutations - of their points. I will also explain how this case differs from the unordered case: for example the 'rate' at which the homology stabilises is strictly slower for oriented configurations.

If time permits, I will also say something about homological stability with twisted coefficients.

We find the order of growth of the typical number of components of zero sets of smooth random functions of several real variables. This might be thought as a statistical version of the (first half of) 16th Hilbert problem. The primary examples are various ensembles of Gaussian real-valued polynomials (algebraic or trigonometric) of large degree, and smooth Gaussian functions on the Euclidean space with translation-invariant distribution.

Joint work with Fedor Nazarov.

It is well known that electrical resistance arguments provide (usually) the best method for determining whether a graph is transient or recurrent. In this talk I will discuss a similar characterization of 'sub-diffusive behaviour' -- this occurs in spaces with many obstacles or traps.

The characterization is in terms of the energy of functions in annuli.

A simple formula is given for the $n$-field tree-level MHV gravitational

amplitude, based on soft limit factors. It expresses the full $S_n$ symmetry

naturally, as a determinant of elements of a symmetric ($n \times n$) matrix.

Shapiro's Conjecture says that two classical exponential polynomials over the complexes can have infinitely many common zeros only for algebraic reasons. I will explain the history of this, the connection to Schanuel's Conjecture, and sketch a proof for the complexes using Schanuel, as well as an unconditional proof for Zilber's fields.

In a variety of problems in engineering and applied science, the goal is to design or control a network of dynamical agents so as to achieve some desired asymptotic behaviour. Examples include consensus and rendez-vous problems in robotics, synchronization of generator angles in power grids or coordination of oscillations in bacterial populations. A pressing challenge in all of these problems is to derive appropriate analytical tools to prove convergence towards the target behaviour. Such tools are not only invaluable to guarantee the desired performance, but can also provide important guidelines for the design of decentralized control strategies to steer the collective behaviour of the network of interest in a desired manner. During this talk, a methodology for analysis and design of convergence in networks will be presented which is based on the use of a classical, yet not fully exploited, tool for convergence analysis: contraction theory. As opposed to classical methods for stability analysis, the idea is to look at convergence between trajectories of a system of interest rather that at their asymptotic convergence towards some solution of interest. After introducing the problem, a methodology will be derived based on the use of matrix measures induced by non-Euclidean norms that will be exploited to design strategies to control the collective behaviour of networks of dynamical agents. Representative examples will be used to illustrate the theoretical results.

We study exponential sums of Weyl type taken over roots of quadratic congruences. We are particularly interested in the situation where the range of summation is small compared to the discriminant of the polynomial. We are then able to give a number of arithmetic applications.

This is work which is joint with W. Duke and H. Iwaniec.