Past

We discuss general and structured matrix polynomials which may be singular and may have eigenvalues at infinity. We discuss several real industrial applications ranging from acoustic field computations to optimal control problems.

We discuss linearization and first order formulations and their relationship to the corresponding techniques used in the treatment of systems of higher order differential equations.

In order to deal with structure preservation, we derive condensed/canonical forms that allow (partial) deflation of critical eigenvalues and the singular structure of the matrix polynomial. The remaining reduced order staircase form leads to new types of linearizations which determine the finite eigenvalues and corresponding eigenvectors.

Based on these new linearization techniques we discuss new structure preserving eigenvalue methods and present several real world numerical examples.

As all analysts know, solving a problem which contains some kind of nonlinearity is generally far from being obvious. This is of course the case when we deal with nonlinear PDEs, but also when we have linear systems with some nonlinear compatibility conditions. One example is constituted by the systems of first order linear partial differential equations. If the coefficients are regular, there is a classical way to solve this systems. However, if the coefficients are only of class L^p, the classical methods cannot be applied. We will show how this problem can be solved by using a method of regularization which allows to preserve the nonlinear compatibility conditions. Then we will present some possible applications to the theory of elasticity. In the end some open problems related to similar aspects will be discussed. Example of such a problem: the rigidity of deformations of class H¹.

A group is called boundedly generated if it is the product of a finite sequence of its cyclic subgroups. Bounded generation is a property possessed by finitely generated abelian groups and by some other linear groups.

Apparently it was not known before whether all boundedly generated groups are linear. Another question about such groups has also been open for a while: If a torsion-free group $G$ has a finite sequence of generators $a_1,\dotsc,a_n$ such that every element of $G$ can be written in a unique way as $a_1^{k_1}\dotsm a_n^{k_n}$, where $k_i\in\mathbb Z$, is it true then that $G$ is virtually polycyclic? (Vasiliy Bludov, Kourovka Notebook, 1995.)

Counterexamples to resolve these two questions have been constructed using small-cancellation method of combinatorial group theory. In particular boundedly generated simple groups have been constructed.

Let d be a fixed natural number. There is a theorem, due to Chvátal, Rodl,

Szemerédi and Trotter (CRST), saying that the Ramsey number of any graph G

with maximum degree d and n vertices is at most c(d)n, that is it grows

linearly with the size of n. The original proof of this theorem uses the

regularity lemma and the resulting dependence of c on d is of tower-type.

This bound has been improved over the years to the stage where we are now

grappling with proving the correct dependency, believed to be an

exponential in d. Our first main result is a proof that this is indeed the

case if we assume additionally that G is bipartite, that is, for a

bipartite graph G with n vertices and maximum degree d, we have r(G)

In the case of Einstein's equations coupled to a non-linear scalar field with a suitable exponential potential, there are solutions for which the expansion is accelerated and of power law type. In the talk I will discuss the future global non-linear stability of such models. The results generalize those of Mark Heinzle and Alan Rendall obtained using different methods.

One of the intrinsic methods in elasticity is to consider the Cauchy-Green tensor as the primary unknown, instead of the deformation realizing this tensor, as in the classical approach.

Then one can ask whether it is possible to recover the deformation from its Cauchy-Green tensor. From a differential geometry viewpoint, this amounts to finding an isometric immersion of a Riemannian manifold into the Euclidian space of the same dimension, say d. It is well known that this is possible, at least locally, if and only if the Riemann curvature tensor vanishes. However, the classical results assume at least a C2 regularity for the Cauchy-Green tensor (a.k.a. the metric tensor). From an elasticity theory viewpoint, weaker regularity assumptions on the data would be suitable.

We generalize this classical result under the hypothesis that the Cauchy-Green tensor is only of class W^{1,p} for some p>d.

The proof is based on a general result of PDE concerning the solvability and stability of a system of first order partial differential equations with L^p coefficients.

If $G$ is a group and $g$ an element of the derived subgroup $[G,G]$, the commutator length of $g$ is the least positive integer $n$ such that $g$ can be written as a product of $n$ commutators. The commutator width of $G$ is the maximum of the commutator lengths of elements of $[G,G]$. Until 1991, to my knowledge, it has not been known whether there exist simple groups of commutator width greater than $1$. The same question for finite simple groups still remains unsolved. In 1992, Jean Barge and Étienne Ghys showed that the commutator width of certain simple groups of diffeomorphisms is infinite. However, those groups are not finitely generated. Finitely generated infinite simple groups of infinite commutator width can be constructed using "small cancellations." Additionally, finitely generated infinite boundedly simple groups of arbitrarily large (but necessarily finite) commutator width can be constructed in a similar way.

I will present two new results in the context of the title. Both are joint work with B. Veto.

1. In earlier work a limit theorem with $t^{2/3}$ scaling was established for a class of self repelling random walks on $\mathbb Z$ with long memory, where the self-interaction was defined in terms of the local time spent on unoriented edges. For combinatorial reasons this proof was not extendable to the natural case when the self-repellence is defined in trems of local time on sites. Now we prove a similar result for a *continuous time* random walk on $\mathbb Z$, with self-repellence defined in terms of local time on sites.

2. Defining the self-repelling mechanism in terms of the local time on *oriented edges* results in totally different asymptotic behaviour than the unoriented cases. We prove limit theorems for this random walk with long memory.

I will take as my starting point a problem which is classical in

population genetics: we wish to understand the distribution of numbers

of individuals in a population who carry different alleles of a

certain gene. We imagine a sample of size n from a population in

which individuals are subject to neutral mutation at a certain

constant rate. Every mutation gives rise to a completely new type.

The genealogy of the sample is modelled by a coalescent process and we

imagine the mutations as a Poisson process of marks along the

coalescent tree. The allelic partition is obtained by tracing back to

the most recent mutation for each individual and grouping together

individuals whose most recent mutations are the same. The number of

blocks of each of the different possible sizes in this partition is

called the allele frequency spectrum. Recently, there has been much

interest in this problem when the underlying coalescent process is a

so-called Lambda-coalescent (even when this is not a biologically

``reasonable'' model) because the allelic partition is a nice example

of an exchangeable random partition. In this talk, I will describe

the asymptotics (as n tends to infinity) of the allele frequency

spectrum when the coalescent process is a particular Lambda-coalescent

which was introduced by Bolthausen and Sznitman. It turns out that

the frequency spectrum scales in a rather unusual way, and that we

need somewhat unusual tools in order to tackle it.

This is joint work with Anne-Laure Basdevant (Toulouse III).

Abstract: We will discuss string $AdS_4$ domain wall solutions with stabilized moduli. These solutions are interesting, since they potentially induce decay processes between different vacua within the string landscape. Moreover, we discuss how black hole physics provide another tool of seeing through the vacuum landscape.

A lattice in the plane is a discrete subgroup in R^2 isomorphic to Z^2 ; it is unimodular if the area of the quotient is 1. The space of unimodular lattices is a venerable object in mathematics related to topology, dynamics and number theory. In this talk, I'd like to present a guided tour of this space, focusing on its topological aspect. I will describe in particular the periodic orbits of the modular flow, giving rise to beautiful "modular knots". I will show some animations

Bardakov and Tolstykh have recently shown that Richard Thompson's group

$F$ interprets the Arithmetic $(\mathbb Z,+,\times)$ with parameters. We

consider a class of infinite groups of piecewise affine permutations of

an interval which contains all the three groups of Thompson and some

classical families of finitely presented infinite simple groups. We have

interpreted the Arithmetic in all the groups of this class. In particular

we have obtained that the elementary theories of all these groups are

undecidable. Additionally, we have interpreted the Arithmetic in $F$ and

some of its generalizations without parameters.

This is a joint work with Tuna Altınel.

The Mutual Fund Theorem (MFT) is considered in a general semimartingale financial market S with a finite time horizon T, where agents maximize expected utility of terminal wealth. The main results are:

(i) Let N be the wealth process of the numéraire portfolio (i.e. the optimal portfolio for the log utility). If any path-independent option with maturity T written on the numéraire portfolio can be replicated by trading only in N and the risk-free asset, then the (MFT) holds true for general utility functions, and the numéraire portfolio may serve as mutual fund. This generalizes Merton’s classical result on Black-Merton-Scholes markets.

Conversely, under a supplementary weak completeness assumption, we show that the validity of the (MFT) for general utility functions implies the replicability property for options on the numéraire portfolio described above.

(ii) If for a given class of utility functions (i.e. investors) the

(MFT) holds true in all complete Brownian financial markets S, then all investors use the same utility function U, which must be of HARA type.

This is a result in the spirit of the classical work by Cass and Stiglitz.

Multiscale ensembles of reaction networks with well separated constants are introduced and typical properties of such systems are studied. For any given ordering of reaction rate constants the explicit approximation of steady state, relaxation spectrum and related eigenvectors (``modes") is presented. The obtained multiscale approximations are computationally cheap and robust. Some of results obtained are rather surprising and unexpected. First of all is the zero-one asymptotic of eigenvectors (asymptotically exact lumping; but these asymptotic lumps could intersect). Our main mathematical tools are auxiliary discrete dynamical systems on finite sets and specially developed algorithms of ``cycles surgery"

for reaction graphs. Roughly speaking, the dynamics of linear multiscale networks transforms into the dynamics on finite sets of reagent names.

A.N. Gorban and O. Radulescu, Dynamic and static limitation in multiscale reaction networks, Advances in Chemical Engineering, 2008 (in press), arXiv e-print: http://arxiv.org/abs/physics/0703278

A.N. Gorban and O. Radulescu, Dynamical robustness of biological networks with hierarchical distribution of time scales, IET Syst.

Biol., 2007, 1, (4), pp. 238-246.