Now that the geometrization conjecture has been proven, and the virtual Haken conjecture has been proven, what is left in 3-manifold topology ? One remaining topic is the computational complexity of geometric topology problems. How difficult is it to distinguish the unknot ? Or 3-manifolds from each other ? The right approach to these questions is not just to consider quantitative complexity, i.e., how much work they take for a computer ; but also qualitative complexity, whether there are efficient algorithms with one or another kind of help. I will discuss various results on this theme, such as that knottedness and unknottedness are both in NP ; and I will discuss high-dimensional questions for context.
The theory of motives, introduced in the sixties, studies the common properties of the different cohomology theories (de Rham, Betti, etale, crystalline, etc) of algebraic varieties. In the same vein, the theory of noncommutative motives, introduced much more recently, studies the common properties of the different invariants (K-theory, cyclic homology, topological Hochschild homology, etc) of "noncommutative algebraic varieties". The bridge from the former theory to the latter consists of the passage from an algebraic variety to its derived category. The aim of this talk, prepared for a broad audience, is to give an overview of the theory of noncommutative motives and to describe some of its manyfold applications to adjacent areas of mathematics.
Le modèle d’Ising, introduit au début du vingtième siècle, décrit la transition de phase des matériaux ferro/paramagnétiques. Depuis son introduction, ce modèle a non seulement eu un impact retentissant sur la compréhension physique des phénomènes de changements d’états, mais il a également été à l’origine de profondes théories en mathématiques. Dans cet exposé, nous présenterons certaines de ces théories ainsi que le concept d’universalité en physique statistique en prenant l’exemple du modèle d’Ising sur les réseaux planaires.
We discuss a project, started by Kummer and Darboux, to describe all surfaces that contain at least 2 circles through every point by relating it to the problem of finding polynomial solutions of quadratic forms.
This talk will describe the main features of symplectic topology, why it is interesting, some known results and some open questions. It will be aimed at a general audience who have some mathematical knowledge, but know nothing about symplectic geometry and topology.