Past

A topological space $(X,\tau)$ is submaximal if $\tau$ is the maximal element of $[{\tau}_{s}]$. Submaximality was first defined and characterized by Bourbaki. Since then, some mathematicians presented several characterizations of submaximal spaces.

In this paper, we will attempt to develop the concept of submaximality and offer some new results. Furthermore, some results concerning $\alpha$-scattered space will be obtained.

A factorability structure on a group G is a specification of normal forms

of group elements as words over a fixed generating set. There is a chain

complex computing the (co)homology of G. In contrast to the well-known bar

resolution, there are much less generators in each dimension of the chain

complex. Although it is often difficult to understand the differential,

there are examples where the differential is particularly simple, allowing

computations by hand. This leads to the cohomology ring of hv-groups,

which I define at the end of the talk in terms of so called "horizontal"

and "vertical" generators.

Word maps on groups were studied extensively in the past few years, in connection to various conjectures on profinite groups, finite groups, finite simple groups, etc. I will provide background, as well as very recent works (joint with Larsen, Larsen-Tiep,

Liebeck-O'Brien-Tiep) on word maps with relations to representations (e.g. Gowers' method and character ratios), geometry and probability.

Recent applications, e.g. to subgroup growth and representation varieties, will also be described.

I will conclude with a list of problems and conjectures which are still very much open.  The talk should be accessible to a wide audience.

Counting the number of curves of degree $d$ with $n$ nodes (and no further singularities) going through $(d^2+3d)/2 - n$ points in general position in the projective plane is a problem which was already considered more than 150 years ago. More recently, people conjectured that for sufficiently large $d$ this number should be given by a polynomial of degree $2n$ in $d$. More generally, the Göttsche conjecture states that the number of $n$-nodal curves in a general $n$-dimensional linear subsystem of a sufficiently ample line bundle $L$ on a nonsingular projective surface $S$ is given by a universal polynomial of degree $n$ in the 4 topological numbers $L^2, L.K_S, (K_S)^2$ and $c_2(S)$. In a joint work with Vivek Shende and Richard Thomas, we give a short (compared to existing) proof of this conjecture.

In 1961 Hajos conjectured that if a graph contains no subdivsion of a clique of order t then its chromatic number is less than t. In 1981, Erdos and Fajtlowicz showed that the conjecture is almost always false. We show it is almost always true. This is joint work with Keevash, Mohar, and McDiarmid.

We study exponential Levy models with change-point which is a random variable, independent from initial Levy processes. On canonical space with initially enlarged filtration we describe all equivalent martingale measures for change-

point model and we give the conditions for the existence of f-minimal equivalent martingale measure. Using the connection between utility maximisation and f-divergence minimisation, we obtain a general formula for optimal strategy in change-point case for initially enlarged filtration and also for progressively enlarged filtration when the utility is exponential. We illustrate our results considering the Black-Scholes model with change-point.

Key words and phrases: f-divergence, exponential Levy models, change-point, optimal portfolio

MSC 2010 subject classifications: 60G46, 60G48, 60G51, 91B70

Fluids that are not adequately described by a linear constitutive relation are usually referred to as   "non-Newtonian fluids". In the last 15 years we have seen a significant progress in the mathematical theory of generalized Newtonian fluids, which is an important subclass of non-Newtonian fluids. We present some recent results in the existence theory and in the error analysis for approximate solutions. We will also indicate how these techniques can be generalized to more general constitutive relations.

Over the last several decades, many mesh generation methods and a plethora of adaptive methods for solving differential equations have been developed.  In this talk, we take a general approach for describing the mesh generation problem, which can be considered as being in some sense equivalent to determining a coordinate transformation between physical space and a computational space.  Our description provides some new theoretical insights into precisely what is accomplished from mesh equidistribution (which is a standard adaptivity tool used in practice) and mesh alignment.  We show how variational mesh generation algorithms, which have historically been the most common and important ones, can generally be compared using these mesh generation principles.  Lastly, we relate these to a variety of moving mesh methods for solving time-dependent PDEs.

This is joint work with Weizhang Huang, Kansas University

We consider the problem of cavitation in nonlinear elasticity, or the formation of macroscopic cavities in elastic materials from microscopic defects, when subjected to large tension at the boundary.

The main goal is to determine the optimal locations where the body prefers the cavities to open, the preferred number of cavities, their optimal sizes, and their optimal shapes. To this aim it is necessary to analyze the elastic energy of an incompressible deformation creating multiple cavities, in a way that accounts for the interaction between the cavitation singularities. Based on the quantitative version of the isoperimetric inequality, as well as on new explicit constructions of incompressible deformations creating cavities of different shapes and sizes, we provide energy estimates showing that, for certain loading conditions, there are only the following possibilities:

• only one cavity is created, and if the loading is isotropic then it is created at the centre
• multiple cavities are created, they are spherical, and the singularities are well separated
• there are multiple cavities, but they act as a single spherical cavity, they are considerably distorted, and the distance between the cavitation singularities must be of the same order as the size of the initial defects contained in the domain.

In the latter case, the formation of thin structures between the cavities is observed, reminiscent of the initiation of ductile fracture by void coalesence.

This is joint work with Sylvia Serfaty (LJLL, Univ. Paris VI).

Postponed until May

I'll describe an approach to perturbative quantum field theory
which is philosophically similar to the deformation quantization approach
to quantum mechanics. The algebraic objects which appear in our approach --
factorization algebras -- also play an important role in some recent work
in topology (by Francis, Lurie and others).  This is joint work with Owen
Gwilliam.