Past

Neural field models describe the coarse-grained activity of populations of

interacting neurons. Because of the laminar structure of real cortical

tissue they are often studied in two spatial dimensions, where they are well

known to generate rich patterns of spatiotemporal activity. Such patterns

have been interpreted in a variety of contexts ranging from the

understanding of visual hallucinations to the generation of

electroencephalographic signals. Typical patterns include localised

solutions in the form of travelling spots, as well as intricate labyrinthine

structures. These patterns are naturally defined by the interface between

low and high states of neural activity. Here we derive the equations of

motion for such interfaces and show, for a Heaviside firing rate, that the

normal velocity of an interface is given in terms of a non-local Biot-Savart

type interaction over the boundaries of the high activity regions. This

exact, but dimensionally reduced, system of equations is solved numerically

and shown to be in excellent agreement with the full nonlinear integral

equation defining the neural field. We develop a linear stability analysis

for the interface dynamics that allows us to understand the mechanisms of

pattern formation that arise from instabilities of spots, rings, stripes and

fronts. We further show how to analyse neural field models with

linear adaptation currents, and determine the conditions for the dynamic

instability of spots that can give rise to breathers and travelling waves.

We end with a discussion of amplitude equations for analysing behaviour in

the vicinity of a bifurcation point (for smooth firing rates). The condition

for a drift instability is derived and a center manifold reduction is used

to describe a slowly moving spot in the vicinity of this bifurcation. This

analysis is extended to cover the case of two slowly moving spots, and

establishes that these will reflect from each other in a head-on collision.

Given a root system, the choice of a root basis divides the set of roots into the positive and the negative ones, it also yields an ordering on the set of positive roots. The set of positive roots with respect to this ordering is called a root poset. The root posets have attracted a lot of interest in the last years. The set of antichains (with a suitable ordering) in a root poset turns out to be a lattice, it is called lattice of (generalized) non-crossing partitions. As Ingalls and Thomas have shown, this lattice is isomorphic to the lattice of thick subcategories of a hereditary abelian category of Dynkin type. The isomorphism can be used in order to provide conceptual proofs of several intriguing counting results for non-crossing partitions.

In this talk, we investigate in a unified way the structural properties of a large class of convex regularizers for linear inverse problems. We consider regularizations with convex positively 1-homogenous functionals (so-called gauges) which are piecewise smooth. Singularies of such functionals are crucial to force the solution to the regularization to belong to an union of linear space of low dimension. These spaces (the so-called "models") allows one to encode many priors on the data to be recovered, conforming to some notion of simplicity/low complexity. This family of priors encompasses many special instances routinely used in regularized inverse problems such as L^1, L^1-L^2 (group sparsity), nuclear norm, or the L^infty norm. The piecewise-regular requirement is flexible enough to cope with analysis-type priors that include a pre-composition with a linear operator, such as for instance the total variation and polyhedral gauges. This notion is also stable under summation of regularizers, thus enabling to handle mixed regularizations.

The main set of contributions of this talk is dedicated to assessing the theoretical recovery performance of this class of regularizers. We provide sufficient conditions that allow to provably controlling the deviation of the recovered solution from the true underlying object, as a function of the noise level. More precisely we establish two main results. The first one ensures that the solution to the inverse problem is unique and lives on the same low dimensional sub-space as the true vector to recover, with the proviso that the minimal signal to noise ratio is large enough. This extends previous results well-known for the L^1 norm [1], analysis L^1 semi-norm [2], and the nuclear norm [3] to the general class of piecewise smooth gauges. In the second result, we establish L^2 stability by showing that the L^2 distance between the recovered and true vectors is within a factor of the noise level, thus extending results that hold for coercive convex positively 1-homogenous functionals [4].

This is a joint work with S. Vaiter, C. Deledalle, M. Golbabaee and J. Fadili. For more details, see [5].

Bibliography:
[1] J.J. Fuchs, On sparse representations in arbitrary redundant bases. IEEE Transactions on Information Theory, 50(6):1341-1344, 2004.
[2] S. Vaiter, G. Peyré, C. Dossal, J. Fadili, Robust Sparse Analysis Regularization, to appear in IEEE Transactions on Information Theory, 2013.
[3] F. Bach, Consistency of trace norm minimization, Journal of Machine Learning Research, 9, 1019-1048, 2008.
[4] M. Grasmair, Linear convergence rates for Tikhonov regularization with positively homogeneous functionals. Inverse Problems, 27(7):075014, 2011.
[5] S. Vaiter, M. Golbabaee, J. Fadili, G. Peyré, Model Selection with Piecewise Regular Gauges, Preprint hal-00842603, 2013

Penrose’s Weyl Curvature Hypothesis, which dates from the late 70s, is a hypothesis, motivated by observation, about the nature of the Big Bang as a singularity of the space-time manifold. His Conformal Cyclic Cosmology is a remarkable suggestion, made a few years ago and still being explored, about the nature of the universe, in the light of the current consensus among cosmologists that there is a positive cosmological constant.  I shall review both sets of ideas within the framework of general relativity, and emphasise how the second set solves a problem posed by the first.

It is an open question whether a group with a finite classifying space is hyperbolic or contains a Baumslag Solitar Subgroup. An idea of Gromov was to use aperiodic tilings of the plane to try and disprove this conjecture. I will be looking at some of the attempts to find a counterexample.

I shall outline a procedure for efficiently approximating arbitrary elements of certain topological groups by words in a finite set. The method is suprisingly general and is based upon the assumption that close to the identity, group elements may be easily expressible as commutators. Time permitting, I shall discuss some applications to uniform diameter bounds for finite groups and to quantum computation.

I will speak about recent work with Michael Wemyss (arXiv:1309.0698), applying noncommutative deformation theory to study the birational geometry of 3-folds. In particular, I will explain how every flippable or floppable rational curve in a 3-fold has a naturally associated algebra of noncommutative deformations, even in the singular setting. We investigate the properties of this algebra, and indicate how to calculate it in examples using quiver techniques. This gives new information about the (commutative) geometry of 3-folds, and in particular provides a new tool to differentiate between flops.

As a further application, we show how the noncommutative deformation algebra controls the homological properties of a floppable curve. In this setting, work of Bridgeland and Chen yields a Fourier-Mukai flop-flop functor which acts on the derived category of the 3-fold (assuming any singularities are at worst Gorenstein terminal). We show that this functor can be described as a spherical twist about the universal family over the noncommutative deformation algebra.

In the second part, I will talk about further work in progress, and explain some more technical details, such as the use of noncommutative deformation functors, and the categorical mutations of Iyama and Wemyss. If there is time, I will also give some higher-dimensional examples, and discuss situations involving chains of intersecting floppable curves. In this latter case, deformations, intersections and homological properties are encoded by the path algebra of a quiver, generalizing the algebra of noncommutative deformations.

We present a multilevel preconditioner for the mixed finite element discretization of the biharmonic equation of first kind. While for the interior degrees of freedom a standard multigrid methods can be applied, a different approach is required on the boundary. The construction of the preconditioner is based on a BPX type multilevel representation in fractional Sobolev spaces. Numerical examples illustrate the obtained theoretical results.

An independent set in an $r$-uniform hypergraph is a subset of the vertices that contains no edges. A container for the independent set is a superset of it. It turns out to be desirable for many applications to find a small collection of containers, none of which is large, but which between them contain every independent set. ("Large" and "small" have reasonable meanings which will be explained.)

Applications include giving bounds on the list chromatic number of hypergraphs (including improving known bounds for graphs), counting the solutions to equations in Abelian groups, counting Sidon sets, establishing extremal properties of random graphs, etc.

The work is joint with David Saxton.

Linear wave scattering problems (e.g. for acoustic, electromagnetic and elastic waves) are ubiquitous in science and engineering applications. However, conventional numerical methods for such problems (e.g. FEM or BEM with piecewise polynomial basis functions) are prohibitively expensive when the wavelength of scattered wave is small compared to typical lengthscales of the scatterer (the so-called "high frequency" regime). This is because the solution possesses rapid oscillations which are expensive to capture using conventional approximation spaces. In this talk I will outline some of my recent work in the development of "hybrid numerical-asymptotic" methods, which incur significantly reduced computational cost. These methods use approximation spaces containing oscillatory basis functions, carefully chosen to capture the high frequency asymptotic behaviour. In particular I will discuss some of the interesting challenges arising from non convex, penetrable and three-dimensional scatterers.

I will speak about recent work with Michael Wemyss (arXiv:1309.0698), applying noncommutative deformation theory to study the birational geometry of 3-folds. In particular, I will explain how every flippable or floppable rational curve in a 3-fold has a naturally associated algebra of noncommutative deformations, even in the singular setting. We investigate the properties of this algebra, and indicate how to calculate it in examples using quiver techniques. This gives new information about the (commutative) geometry of 3-folds, and in particular provides a new tool to differentiate between flops.

As a further application, we show how the noncommutative deformation algebra controls the homological properties of a floppable curve. In this setting, work of Bridgeland and Chen yields a Fourier-Mukai flop-flop functor which acts on the derived category of the 3-fold (assuming any singularities are at worst Gorenstein terminal). We show that this functor can be described as a spherical twist about the universal family over the noncommutative deformation algebra.

In the second part, I will talk about further work in progress, and explain some more technical details, such as the use of noncommutative deformation functors, and the categorical mutations of Iyama and Wemyss. If there is time, I will also give some higher-dimensional examples, and discuss situations involving chains of intersecting floppable curves. In this latter case, deformations, intersections and homological properties are encoded by the path algebra of a quiver, generalizing the algebra of noncommutative deformations.

Affinoid enveloping algebras arise as certain p-adic completions of ordinary enveloping algebras, and are closely related to Iwasawa algebras. I will explain how to use Beilinson-Bernstein localisation to compute their (non-commutative) Krull dimension. This is recent joint work with Ian Grojnowski.

We will discuss some geometric methods to study Diophantine equations. We focus on the case of elliptic curves and their natural generalisations: Abelian varieties, Calabi-Yau manifolds and hyperelliptic curves. 

(joint work with Aurélien Alfonsi and Arturo Kohatsu-Higa)

We are interested in the Wasserstein distance on the space of continuous sample-paths equipped with the supremum norm between the laws of a uniformly elliptic one-dimensional diffusion process and its continuous-time Euler scheme with N steps. This distance controls the discretization biais for a large class of path-dependent payoffs.

Its convergence rate to 0 is clearly intermediate between -the rate -1/2 of the strong error estimation obtained when coupling the stochastic differential equation and its Euler scheme with the same Brownian motion -and the rate -1 of the weak error estimation obtained when comparing the expectations of the same function of the diffusion and its Euler scheme at the terminal time.

For uniformly elliptic one-dimensional stochastic differential equations, we prove that this rate is not worse than -2/3.

Abstract: In 2010, Erschler, Tóth and Werner introduced the so-called Stuck Walks, which are a class of self-interacting random walks on Z for which there is competition between repulsion at small scale and attraction at large scale. They proved that, for any positive integer L, if the relevant parameter belongs to a certain interval, then such random walks localize on L + 2 sites with positive probability. They also conjectured that it is the almost sure behaviour. We settle this conjecture partially, proving that the walk localizes on L + 2 or L + 3 sites almost surely, under the same assumptions.