Sala Seminari (Dip. Matematica)
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).
We consider polynomial‐exponential equations over the complex
the variables run through the rational numbers. Classically, the integer
such equations are considered and there are several finiteness results in the
literature for those solutions. We present a method to reduce the rational
solutions to integer ones and give a description of them using the earlier
results. As a corollary, we get a finiteness result. If time permits,
we will present
I will define two families of representations of the mapping class group of surfaces, namely the Weil and Reshetikhin-Turaev representations. Then I will state some results concernings the (in)finiteness of their image and their decomposition into irreducible factors.
Uniformly finite homology is a coarse homology theory, defined via chains that satisfy a uniform boundedness condition. Byconstruction, uniformly finite homology carries a canonical $\ell^\infty$-semi-norm. We show that, for UDBG spaces, this semi-norm on uniformly finite homology in degree~$0$ with $\mathbb Z$-coefficients allows for a new formulation of Whyte's rigidity result. In contrast, we prove that this semi-norm is trivial on uniformly homology in higher degrees with $\mathbb R$-coefficients.
Nel seminario presenteremo la relazione tra coomologia limitata e
quasimorfismi di un gruppo.
Ci concentreremo poi sui quasimorfisimi mediani e daremo una
caratterizzazione delle azioni su alberi che inducono quasimorfismi mediani non banali.
Bounded cohomology is a functional analytic modification of
regular cohomology, with applications to geometry, topology and (geometric) group theory. After giving a very short introduction to groupoids, I will present our construction of (relative) bounded cohomology for (pairs) of groupoids. This includes a natural setting for bounded cohomology relative to a family of subgroups. Finally, I will discuss a relative version of Gromov's mapping Theorem in this context.
Consider a tower of finite coverings of a closed manifold.
We usually assume that the fundamental group is residually finite and that the tower corresponds to a residual chain of the fundamental group.
We then ask: How does the size of the k-th homology group grow as we go up in the tower?
By size we mean the rank or the cardinality of the torsion subgroup.
We connect the homology growth for aspherical Riemannian manifolds in the above sense to the volume of balls.