Past

In this talk we present a work done with M. Di Giacinto (Università di Cassino - Italy) and Salvatore Federico (Scuola Normale - Pisa - Italy). The subject of the work is a continuous time stochastic model of optimal allocation for a defined contribution pension fund with a minimum guarantee. We adopt the point of view of a fund manager maximizing the expected utility from the fund wealth over an infinite horizon.

The level of wealth is constrained to stay above a "solvency level".

The model is naturally formulated as an optimal control problem of a stochastic delay equation with state constraints and is treated by the dynamic programming approach.

We first present the study in the simplified case of no delay where a satisfactory theory can be built proving the existence of regular feedback control strategies and then go to the more general case showing some first results on the value function and on its properties.

I will prove that certain pairs of ordered structures are dependent. There are basically two cases depending on whether the smaller structure is dense or discrete. I will discuss the proofs of two quite general theorems which construe the dividing line between these cases. Among examples are dense pairs of o-minimal structures in the first case, and tame pairs of o-minimal structures in the latter. This is joint work with P. Hieronymi.

Partial differential equations with a nonlinear pointwise constraint defined through a manifold occur in a variety of applications: The magnetization of a ferromagnet can be described by a unit length vector field and the orientation of the rod-like molecules that constitute a liquid crystal is often modeled by a vector field that attains its values in the real projective plane thus respecting the head-to-tail symmetry of the molecules. Other applications arise in geometric

modeling, quantum mechanics, and general relativity. Simple examples reveal that it is impossible to satisfy pointwise constraints exactly by lowest order finite elements. For two model problems we discuss the practical realization of the constraint, the efficient solution of the resulting nonlinear systems of equations, and weak accumulation of approximations at exact solutions.

Fibrations are a valuable tool in the study of the geometry of higher dimensional algebraic varieties. By expressing a higher dimensional variety as a fibration by lower dimensional varieties, we can deduce much about its properties. Whilst the theory of elliptic fibrations is very well developed, fibrations by higher dimensional varieties, especially K3 surfaces, are only just beginning to be studied. In this talk I study a special case of the K3-fibration, where the general fibres admit a <2>-polarisation and the base of the fibration is a nonsingular curve.

We study the Neumann and regularity boundary value problems for a divergence form elliptic equation in the plane. We assume the gradient

of the coefficient matrix satisfies a Carleson measure condition and consider data in L^p, 1

The Navier-Stokes equation with a non-linear viscous term will be considered, p is the exponent of non-linearity.

An existence theorem is proved for the case when the convection term is not subordinate to the viscous

term, in particular for the previously open case p

There has been much recent progress in extending Griffith's criterion for

crack growth into mathematical models for quasi-static crack evolution

that are well-posed, in the sense that there exist solutions that can be

numerically approximated. However, mathematical progress in dynamic

fracture (crack growth consistent with Griffith's criterion, together with

elastodynamics) has been meager. We describe some recent results on a

phase-field model of dynamic fracture, as well as some models based on a

"sharp interface" instead of a phase-field.

Some possible strategies for showing existence for these last models will

also be described.

Y. Brenier, R. Natalini and M. Puel have considered a ``relaxation" of the Euler equations in R². After an approriate scaling, they have obtained the following hyperbolic version of the Navier-Stokes equations, which is similar to the hyperbolic version of the heat equation introduced by Cattaneo, $$\varepsilon u_{tt}^\varepsilon + u_t^\varepsilon -\Delta u^\varepsilon +P (u^\varepsilon \nabla u^\varepsilon) \, = \, Pf~, \quad (u^\varepsilon(.,0), u_t^\varepsilon(.,0)) \, = \, (u_0(.),u_1(.))~, \quad (1) $$ where $P$ is the classical Leray projector and $\varepsilon$ is a small, positive number. Under adequate hypotheses on the forcing term $f$, we prove global existence and uniqueness of a mild solution $(u^\varepsilon,u_t^\varepsilon) \in C^0([0, +\infty), H^{1}({\bf R}^2) \times L^2({\bf R}^2))$ of (1), for large initial data $(u_0,u_1)$ in $H^{1}({\bf R}2) \times L^2({\bf R}2)$, provided that $\varepsilon>0$ is small enough, thus improving the global existence results of Brenier, Natalini and Puel (actually, we can work in less regular Hilbert spaces). The proof uses appropriate Strichartz estimates, combined with energy estimates. We also show that $(u^\varepsilon,u_t^\varepsilon)$ converges to $(v,v_t)$ on finite intervals of time $[t_0,t_1]$, $0 <+ \infty$, when $\varepsilon$ goes to $0$, where $v$ is the solution of the corresponding Navier-Stokes equations $$ v_t -\Delta v +P (v\nabla v) \, = \, Pf~, \quad v(.,0) \, = \, u_0~. \quad (2) $$ We also consider Equation (1) in the three-dimensional case. Here we expect global existence results for small data. Under appropriate assumptions on the forcing term, we prove global existence and uniqueness of a mild solution $(u^\varepsilon,u_t^\varepsilon) \in C^0([0, +\infty), H^{1+\delta}({\bf R}^3) \times H^{\delta}({\bf R}^3))$ of (1), for initial data $(u_0,u_1)$ in $H^{1 +\delta}({\bf R}^3) \times H^{\delta}({\bf R}^3)$ (where $\delta >0 $ is a small positive number), provided that $\varepsilon > 0$ is small enough and that $u_0$ and $f$ satisfy a smallness condition. (Joint work with Marius Paicu)

Baumslag-Solitar groups are very simple groups which are not Hopfian (they are isomorphic to proper quotients). I will discuss these groups, as well as their obvious generalizations, with emphasis on their automorphisms and their generating sets

We will examine the typical structure of random polytopes by projecting the three fundamental regular polytopes: the simplex, cross-polytope, and hypercube. Along the way we will explore the implications of their structure for information acquisition and optimization. Examples of these implications include: that an N-vector with k non-zeros can be recovered computationally efficiently from only n random projections with n=2e k log(N/n), or that for a surprisingly large set of optimization problems the feasible set is actually a point. These implications are driving a new signal processing paradigm, Compressed Sensing, which has already lead to substantive improvements in various imaging modalities. This work is joint with David L. Donoho.

We will consider a (sub) critical Galton-Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We shall specify the law of this allelic partition in terms of the distribution of the number of clone-children and the number of mutant-children of a typical individual. Some limit theorems related to the distribution of the allelic partition will be also presented.