L2

There are nontrivial solutions of the incompressible Euler equations which are compactly supported in space and time. If they were to model the motion of a real fluid, we would see it suddenly start moving after staying at rest for a while, without any action by an external force. There are C1 isometric embeddings of a fixed flat rectangle in arbitrarily small balls of the three dimensional space. You should therefore be able to put a fairly large piece of paper in a pocket of your jacket without folding it or crumpling it. I will discuss the corresponding mathematical theorems, point out some surprising relations and give evidences that, maybe, they are not merely a mathematical game.

Recently Frenkel and Hernandez introduced a kind of "Langlands duality" for characters of semisimple Lie algebras. We will discuss a representation-theoretic interpretation of their duality using quantum analogues of exceptional isogenies. Time permitting we will also discuss a branching rule and relations to Littelmann paths.

(Note that the talk will be in L2 and not the usual SR1)

Many interesting features of algebraic varieties are encoded in the spaces of rational curves that they contain. For instance, a smooth cubic surface in complex projective three-dimensional space contains exactly 27 lines; exploiting the configuration of these lines it is possible to find a (rational) parameterization of the points of the cubic by the points in the complex projective plane.

After a general overview, we focus on the Fermat quintic threefold X, namely the hypersurface in four-dimensional projective space with equation x^5+y^5+z^5+u^5+v^5=0. The space of lines on X is well-known. I will explain how to use a mix of algebraic geometry, number theory and computer-assisted calculations to study the space of conics on X.

This talk is based on joint work with R. Heath-Brown.

The isoperimetric inequality is a fundamental tool in many geometric and analytical issues, beside being the starting point for a great variety of other important inequalities.

We shall present some recent results dealing with the quantitative version of this inequality, an old question raised by Bonnesen at the beginning of last century. Applications of the sharp quantitative isoperimetric inequality to other classic inequalities and to eigenvalue problems will be also discussed.

Anomalous ( non local) diffusion processes appear in many subjects: phase transition, fracture dynamics, game theory I will describe some of the issues involved, and in particular, existence and regularity for some non local versions of the p Laplacian, of non variational nature, that appear in non local tug of war.

I will speak about a geometric method, based on the classical trace map, for investigating word maps on groups PSL(2, q) and SL(2, q). In two different papers (with F. Grunewald, B. Kunyavskii, and Sh. Garion, F. Grunewald, respectively) this approach was applied to the following problems.

1. Description of Engel-like sequences of words in two variables which characterize finite

solvable groups. The original problem was reformulated in the language of verbal dynamical

systems on SL(2). This allowed us to explain the mechanism of the proofs for known

sequences and to obtain a method for producing more sequences of the same nature.

2. Investigation of the surjectivity of the word map defined by the n-th Engel word

[[[X, Y ], Y ], . . . , Y ] on the groups PSL(2, q) and SL(2, q). Proven was that for SL(2, q), this

map is surjective onto the subset SL(2, q) $\setminus$ {−id} $\subset$ SL(2, q) provided that q $\ge q_0(n)$ is

sufficiently large. If $n\le 4$ then the map was proven to be surjective for all PSL(2, q).

We discuss homological finiteness Bredon types FPm with respect to the class of finite subgroups and seperately with respect to the class of virtually cyclic subgroups. We will concentrate to the case of solubles groups and if the time allows to the case of generalized R. Thompson groups of type F. The results announced are joint work with Brita Nucinkis

(Southampton) and Conchita Martinez Perez (Zaragoza) and will appear in papers in Bulletin of LMS and Israel Journal of Mathematics.

The new model is that the universal small scale structure of high Reynolds number turbulence is determined by the dynamics of thin evolving shear layers, with thickness of the order of the Taylor micro scale,within which there are the familiar elongated vortices .Local quasi-linear dynamics shows how the shear layers act as barriers to external eddies and a filter for the transfer of energy to their interiors. The model is consistent with direct numerical simulations by Ishihara and Kaneda analysed in terms of conditional statistics relative to the layers and also with recent 4D measurements of lab turbulence by Wirth and Nickels. The model explains how the transport of energy into the layers leads to the observed inertial range spectrum and to the generation of intense structures, on the scale of the Kolmogorov micro-scale.

But the modelling also explains the important discrepancies between data and the Kolmogorov-Richardson cascade concept ,eg larger amplitudes of the smallest scale motions and of the higher moments ,and why the latter are generally less isotropic than lower order moments, eg in thermal convection. Ref JCRHunt , I Eames, P Davidson,J.Westerweel, J Fernando, S Voropayev, M Braza J Hyd Env Res 2010