Le séminaire quimpériodique se tient sur deux jours jeudi 7 et vendredi 8 novembre. Renseignements et Inscription ici

Everything has been said about Stein fillings of lens spaces when they are endowed with their standard (tight) contact structure. Nevertheless, lens spaces support many more tight structures, that are all classified, but for which a complete list of their Stein fillings is still missing. I will present a series of methods and techniques which can be used to investigate the topology of these fillings, giving constraints, for example, on their Euler characteristic and fundamental group.

On va s'intéresser au théorème de Perron-Frobenius qui énonce notamment qu'une matrice positive fortement irréductible a une valeur propre dominante. Après quelques explications sur ce théorème et quelques exemples d'applications à de (nombreux !) domaines des mathématiques, on abordera une preuve due à Garrett Birkhoff. Cette preuve exhibe le vecteur propre de la valeur propre dominante comme point fixe d'une application contractante pour la métrique de Hilbert, que l'on introduira à l'occasion. L'étude de cette métrique fait appel à des propriétés de géométrie élémentaire. On parlera éventuellement plus généralement de métriques projectives et de la pertinence du choix de la métrique de Hilbert dans la preuve de Birkhoff. Enfin, on parlera éventuellement de la situation en dimension infinie.

Oriented knots are said to be concordant if they cobound an embedded cylinder in the interval times the 3–sphere. This defines an equivalence relation under which the set of knots becomes an abelian group with the connected sum operation. The importance of this group lies in its strong connection with the study of 4-manifolds. Indeed, many questions pertaining to 4–manifolds with small topology (like the 4–sphere) can be addressed in terms of concordance. A powerful tool for studying the algebraic structure of this group comes from satellite operations or the process of tying a given knot P along another knot K to produce a third knot P(K). In the talk I will describe how to use SO(3) gauge theory to provide a general criterion sufficient for the image of a satellite operation to generate an infinite rank subgroup of the smooth concordance group. This is joint work with Matt Hedden.

To each coprime pair of natural numbers is associated a rational homology 4-ball B_{p,q}; these are of interest in algebraic geometry and in constructions of smooth 4-manifolds. Evans and Smith have completely determined which of these may be embedded symplectically in CP^2; the answer coincides with an algebraic geometric result of Hacking and Prokhorov, and is described in terms of solutions to the Markov diophantine equation.

Using double branched covers, we exhibit an infinite family of such balls which embed smoothly but not symplectically in CP^2. We also describe an obstruction using Donaldson’s diagonalisation which may be used to show that no two of our examples may be embedded disjointly.

We will give a survey on the main problems concerning rigidity, stability and formation of black holes and some of the recent results.