We investigate rational homology cobordisms of 3-manifolds with non-zero first Betti number. This is motivated by the natural generalization of the slice-ribbon conjecture to multicomponent links.
Sala Seminari (Dip. Matematica)
Abstract: Topological Quantum Field Theories (TQFTs for short) were
axiomatized by Atiyah in 1988 (). Following his exposition, we will
define the concept of TQFT and give some examples. In particular, we will
describe the (1+1)-TQFTs and their algebraic counterparts, the Frobenius
algebras. Time permitting we will give some more examples in higher
 M. Atiyah, Topological Quantum Field Theory, Publications
Mathématiques de l'IHÉS 68 (68): 175–186
The splice of two links is an operation defined by Eisenbund and Neumann which generalizes several other operations on links, such as the connected sum, the cabling or the disjoint union. The precise definition will be given in the talk but the rough idea goes as follows: the splice of the links L’ and L'' along the components K' and K'' is the link (L' \ K') U (L''\ K'') obtained by identifying the exterior of K' with the exterior of K''.
A celebrated result by Poincaré states that a compact Riemann surface of positive genus has a conformal metric of constant curvature, unique up to rescaling. Clearly, the case of genus 0 is not so exciting: there is a unique complex structure and a unique metric of curvature 1 up to Möbius transformations.
In the spirit of famous papers by Pila & Bombieri and Pila & Wilkie, I will explain how to bound the number
of rational points, with respect to their height, in various kinds of sets, such as algebraic varieties of a given degree,
transcendental sets definable in some o-minimal (or not) structure over the real field, and, after joint work
with R. Cluckers and F. Loeser, also definable sets in non-archimedean contexts.
The talk will consist in a survey of results and I will go into some of the proofs only in order to give the main ideas relating
We show hos the gradient of SBD functions can control, on "most" of the domain, their global behaviour (up to a rigid motion), provided the jump set is not too large.
This is a joint work with Sergio Conti and Gilles Francfort.
We consider a coupled system of Liouville equations on compact surfaces motivated by the study of non-abelian Chern-Simons vortices, and also describing holomorphic curves in projective spaces. We will see how geometric inequalities à la Moser-Trudinger allow to study the variational features of the problem in terms of concentration of the components of the system at finitely-many points of the surface. Existence results can then be derived via min-max theory. These are joint works with L.Battaglia, A.Jevnikar, S.Kallel, C.Ndiaye and
The Bernstein problem, namely the problem of classifying all entire minimal hypergraphs in Euclidean spaces has played a crucial role in the development of Analysis throughout the whole course of the twentieth century. In this talk, I will discuss its natural extension to asymptotically flat manifolds, where it is motivated by the study of the large-scale structure of initial data sets for the Einstein field equation. I will first present the basic non-existence result and its relation to the asymptotic Plateau problem and then mention the application of similar techniques to the study of 1) large CMC spheres and isoperimetric domains (C.-Chodosh-Eichmair), 2) marginally outer-trapped surfaces (C.) and 3) the zero set of static potentials (Galloway-Miao).