|
|
|
Titles and abstractsGérard Besson: A finiteness theorem for Gromov hyperbolic groups We shall prove that given $H>0$ and $\delta\geq 0$ there is a finite number (up to isomorphisms) of marked groups $(\Gamma , \Sigma)$ which are $\delta$-hyperbolic, torsion-free, non cyclic such that their entropy satisfies ${\rm Entropie} (\Gamma , \Sigma) \le H$. Here $\Gamma$ is finitely generated and endowed with a (symmetric) finite generating set $\Sigma$. The goal of the talk is to explain all the words of this abstract and give a flavour of the proof.
Andrea Bisterzo: Symmetry of solutions to semilinear PDEs on (isoparametric) Riemannian domains
This talk aims to introduce an appropriate notion of symmetry for Riemannian domains (and for functions) and to show some symmetry results for solutions to semilinear PDEs on such domains. In particular, we will see that stable solutions to the Dirichlet problem associated with the equation \Delta u =f(u) are symmetric.
From the beginning we will consider the setting of weighted Riemannian manifolds, giving an interpretation of symmetric domains that is, in some sense, compatible with the presence of the weight.
Reto Buzano: Noncompact self-shrinkers for mean curvature flow with arbitrary genus In his lecture notes on mean curvature flow, Tom Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. Here, we employ min-max techniques to give a rigorous existence proof for these surfaces. Conjecturally, the self-shrinkers that we obtain have precisely one (asymptotically conical) end. We confirm this for large genus via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity. Finally, we present some numerical evidence for a further new family of noncompact self-shrinkers with odd genus and two asymptotically conical ends. This is joint work with Huy Nguyen and Mario Schulz.
Davide Dameno: Riemannian four-manifolds and twistor spaces: some rigidity results
It is well-known that four-dimensional Riemannian manifolds carry many peculiar properties, which give rise to the existence of unique canonical metrics. In order to find conditions for the existence of such metrics, in 1978 Atiyah, Hitchin and Singer adapted Penrose’s construction of twistor spaces to the Riemannian context, paving the way for the study of many other characterizations of curvature properties for Riemannian four-manifolds. After giving an overview of the Riemannian and Hermitian structures of twistor spaces in the four-dimensional case, we present some new rigidity results for Riemannian four-manifolds whose twistor spaces satisfy specific vanishing curvature conditions. This is based on joint work with Professors Giovanni Catino and Paolo Mastrolia.
Baptiste Devyver: Lp cohomology and Hodge decomposition on ALE manifolds
Much work has been devoted in differential geometry to the understanding of the relationship between the topology (for instance Betti numbers) and the geometry of the manifold (for instance through curvature assumptions). A prominent example is the Bochner method for compact manifolds, which gives vanishing of the first Betti number under the positivity of the Ricci curvature. Betti numbers have been the subject of intense research, both in the compact and in the non-compact settings. In the compact setting, one crucial analytical tool is the Hodge-De Rham decomposition, which allows to interpret the kth Betti numbers as the dimension of the space of harmonic k-forms. In the non-compact setting, several types of Betti numbers can be defined, and in particular one must specify the integrability of the form. Arguably the most popular choice is the reduced L2 Betti numbers, which are closely related to a general L2 Hodge-De Rham decomposition which holds for any complete, non-compact manifold. However, other choices are interesting: one could look at reduced Lp Betti numbers as well, for an exponent p different from 2. And, as a related question, one can ask whether there is an associated Lp Hodge-De Rham decomposition. For an integrability exponent p different from 2, the analytic side of the theory becomes very subtle, in fact it is not a trivial task to even prove the Lp Hodge-De Rham decomposition for the Euclidean space itself! It turns out that curvature plays a role in this story: loosely speaking, if the curvature is "positive" then Lp Hodge-De Rham decompositions can be proved, and Lp Betti numbers vanish. In this talk, we will investigate these questions for one of the simplest class of non globally postively curved manifolds, namely Asymptotically Locally Euclidean (ALE) manifolds.
This is a joint work with K. Kröncke (KTH Stockholm). Anne Parreau: Boundaries of higher Teichmuller spaces and geodesic currents
Higher Teichmuller spaces are special components of spaces of conjugacy classes of representations of a surface group in a semisimple Lie group $G$, that consist of faithfull discrete representations. Focusing on the case where $G=SL_3(R)$, which correspond to the space of convex real projective structures on a surface, I will explain the link between length boundaries, non-archimedean representations, and geodesic currents on the surface, and present some applications.
Michele Rimoldi: Quantitative index bounds for translators via topology Translators for the mean curvature flow give rise to a special class of eternal solutions, that besides having their own intrinsic interest, play a key role in the analysis of singularities of the flow. Recently there has been a great effort in trying to classify them under certain geometric conditions and in constructing new classes of examples. After reviewing some of the main results in literature we will present recent progresses on quantitative estimates on the stability index of these objects via their topology. This is a joint project with D. Impera.
Alberto Roncoroni: On the stable Bernstein problem In a recent paper Chodosh and Li proved that the only complete, immersed, orientable, stable minimal hypersurfaces in R^4 are the hyperplanes. This is known as the stable Bernstein problem in R^4. In this talk I will discuss an alternative proof of the stable Bernstein problem in R^4 obtained in collaboration with G. Catino and P. Mastrolia.
|
| Online user: 2 | Privacy |
|
