Localization of the continuous Anderson hamiltonian in 1d
Resumo: We consider the operator obtained by perturbing the Laplacian with a white noise on a segment of size L. This operator arises as scaling limit of simple random matrix models, and plays an important role in the study of the parabolic Anderson model. We show that, as L goes to infinity, the eigenvalues converge to a Poisson point process on R with an explicit intensity, and the eigenfunctions converge to Dirac masses located at iid uniform points. Furthermore, we show that the asymptotic shape of each eigenfunction around its maximum is given by an explicit, deterministic function which does not depend on the corresponding eigenvalue. This is a joint work with Laure Dumaz (Paris-Dauphine).
O Problema de Kakeya para Corpos Finitos
Resumo: Em 1917, Soichi Kakeya propôs o seguinte problema:
Qual é a menor área do plano necessária para rotacionar uma agulha de comprimento unitário continuamente por $360^\circ$?
Essa simples pergunta motivou conjecturas que, ainda hoje, continuam sem resposta definitiva. Neste colóquio, conversaremos um pouco sobre esse problema e sobre alguns problemas relacionados como, por exemplo: a (famosa) Conjectura de Kakeya, possíveis generalizações de Conjuntos de Besicovitch e uma versão da Conjectura de Kakeya para corpos finitos. Para esta última, apresentaremos uma solução que faz uso apenas de ferramentas de Álgebra Linear.
Ricci flow on cohomogeneity one manifolds
Resumo: We study the Ricci flow in the setting of cohomogeneity one manifolds, i.e. a Riemannian manifold M with a group G acting isometrically such that the orbit space M/G is one-dimensional. Since isometries are preserved under the flow, the evolving metrics continue to be invariant. In several past works, this structure has been utilized to gain new information about the Ricci flow. In particular, we showed that in dimension 4 nonnegative sectional curvature is not preserved under the flow. We will describe the challenges in systematically studying Ricci flow on cohomogeneity one manifolds arising from both the degenerate parabolic nature of the Ricci flow PDE and the structure of invariant metrics on a cohomogeneity one manifold. We will also present a strategy to overcome these.
On the Moduli Spaces of Metrics with Nonnegative Sectional Curvature
Resumo: The Kreck-Stolz s invariant is used to distinguish connected components of the moduli space of positive scalar curvature metrics. We use a formula of Kreck and Stolz to calculate the s invariant for metrics on Sn bundles with nonnegative sectional curvature. We then apply it to show that the moduli spaces of metrics with nonnegative sectional curvature on certain 7-manifolds have infinitely many path components. These include the first non-homogeneous examples of this type and certain positively curved Eschenburg and Aloff-Wallach spaces.
Fibrewise bundles over the Gromov-Hausdorff space
Resumo: The Gromov-Hausdorff space is the set Mof compact metric spaces up to isometries, equippedwith the Gromov-Hausdorff metric.In this lecture we will studyAct(G), the spaceofG-actions over compact metric spacesup to isometric conjugacy. We present a metricinAct(G)extending [Arbieto-Morales, DCDS2017] yieldinga fibrewise space structure overM.Some properties of Act(G)asfibrewise space over Mwill be discused.