Past

It is well known that one can attach Galois representations to certain modular forms, it is natural to ask how one might generalise this to produce more Galois representations. One such approach, due to Gross, defines objects called algebraic modular forms on certain types of reductive groups and then conjectures the existence of Galois representations attached to them. In this talk I will outline how for a particular choice of reductive group the conjectured Galois representations exist and are the classical modular Galois representations, thus providing some evidence that this is a good generalisation to consider.

The description of the behaviour of local minima or evolution problems for families of energies cannot in general be deduced from their Gamma-limit, which is a concept designed to treat static global minimum problems. Nevertheless this can be taken as a starting point. Various issues that have been addressed are:

Find criteria that ensure the convergence of local minimizers and critical points. In case this does not occur then modify the Gamma-limit in order to match this requirement. We note that in this way we `correct' some limit theories, finding (or `validating') other ones present in the literature;

Modify the concept of local minimizer, so that it may be more `compatible' with the process of Gamma-limit;

Treat evolution problems for energies with many local minima obtained by a time-discrete scheme introducing the notion of `minimizing movements along a sequence of functionals'. In this case the minimizing movement of the Gamma-limit can always be obtained by a choice of the space- and time-scale, but more interesting behaviors can be obtained at a critical ratio between them. In many cases a `critical scale' can be computed and an effective motion, from which all other minimizing movements are obtained by scaling.

Relate minimizing movements to general variational evolution results, in particular recent theories of quasistatic motion and gradient flow in metric spaces.

I will illustrate some of these points.

''Regression analysis aims to use observational data from multiple observations to develop a functional relationship relating explanatory variables to response variables, which is important for much of modern statistics, and econometrics, and also the field of machine learning. In this paper, we consider the special case where the explanatory variable is a stream of information, and the response is also potentially a stream. We provide an approach based on identifying carefully chosen features of the stream which allows linear regression to be used to characterise the functional relationship between explanatory variables and the conditional distribution of the response; the methods used to develop and justify this approach, such as the signature of a stream and the shue product of tensors, are standard tools in the theory of rough paths and seem appropriate in this context of regression as well and provide a surprisingly unified and non-parametric approach.''

The performance of the shares of a closed end bond fund is based on the returns of an underlying portfolio of bonds. The returns on closed end bond funds are typically higher than those of comparable open ended bond funds and this result is attributed to the use of leverage by closed end bond funds. This talk develops a simple model to assess the impact of leverage on the expected return and riskiness of a closed end bond fund. We illustrate the model with some examples

Let $A$ be a finite-dimensional $K$-algebra over an algebraically closed field $K$. Denote by $\Omega_A$ the syzygy operator on the category $\mod A$ of finite-dimensional right $A$-modules, which assigns to a module $M$ in $\mod A$ the kernel $\Omega_A(M)$ of a minimal projective cover $P_A(M) \to M$ of $M$ in $\mod A$. A module $M$ in $\mod A$ is said to be periodic if $\Omega_A^n(M) \cong M$ for some $n \geq 1$. Then $A$ is said to be a periodic algebra if $A$ is periodic in the module category $\mod A^e$ of the enveloping algebra $A^e = A^{\op} \otimes_K A$. The periodic algebras $A$ are self-injective and their module categories $\mod A$ are periodic (all modules in $\mod A$ without projective direct summands are periodic). The periodicity of an algebra $A$ is related with periodicity of its Hochschild cohomology algebra $HH^{*}(A)$ and is invariant under equivalences of the derived categories $D^b(\mod A)$ of bounded complexes over $\mod A$. One of the exciting open problems in the representation theory of self-injective algebras is to determine the Morita equivalence classes of periodic algebras.

We will present the current stage of the solution of this problem and exhibit prominent classes of periodic algebras.

We investigate symmetric quotient algebras of symmetric algebras,

with an emphasis on finite group algebras over a complete discrete

valuation ring R with residue field of positive characteristic p. Using elementary methods, we show that if an

ordinary irreducible character of a finite group gives

rise to a symmetric quotient over R which is not a matrix algebra,

then the decomposition numbers of the row labelled by the character are

all divisible by p. In a different direction, we show that if is P is a finite

p-group with a cyclic normal subgroup of index p, then every ordinary irreducible character of P gives rise to a

symmetric quotient of RP. This is joint work with Shigeo Koshitani and Markus Linckelmann.

The majority of epidemic models fall into two categories: 1) deterministic models represented by differential equations and 2) stochastic models which can be evaluated by simulation. In this presentation I will discuss the precise connection between these models. Until recently, exact correspondence was only established in situations exhibiting large degrees of symmetry or for infinite populations.

I will consider SIR dynamics on finite static contact networks. I will give an overview of two provably exact deterministic representations of the underlying stochastic model for tree-like networks. These are the message passing description of Karrer and Newman and my pair-based moment closure representation. I will discuss relationship between the two representations and the relative merits of both.

We explain how Khovanov-Lauda-Rouquier algebras in finite type A are affine cellular in the sense of Koenig and Xi. In particular this reproves finiteness of their global dimension. This is joint work with Alexander Kleshchev and Joseph Loubert.

This talk is about some recent joint work with Sarah Witherspoon. The representations of some finite dimensional Hopf algebras have curious behaviour: Nonprojective modules may have projective tensor powers, and the variety of a tensor product of modules may not be contained in the intersection of their varieties. I shall describe a family of examples of such Hopf algebras and their modules, and the classification of left, right, and two-sided ideals in their stable module categories.

To a finite connected acyclic quiver Q there is associated a path algebra kQ, for an algebraically closed field k, a Coxeter group W and a preprojective algebra. We discuss a bijection between elements of the Coxeter group W and the cofinite quotient closed subcategories of mod kQ, obtained by using the preprojective algebra. This is taken from a paper with Oppermann and Thomas. We also include a related result by Mizuno in the case when Q is Dynkin.

In the talk we will discuss Quillen's construction of a determinant line bundle associated to a family of Cauchy-Riemann operators. I will first of all try to convince you why this is a cool thing and mention some of the many different applications. The bulk of the talk will be focused on constructing the line bundle, its hermitian metric and calculating the curvature. Hopefully a talk accessible to many.

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.