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Terça feira

Geometria em Lisboa

Contact three-manifolds with exactly two simple Reeb orbits.
Umberto Hryniewicz, Aachen University.

Resumo

The goal of this talk is to present a complete characterization of Reeb flows on closed 3-manifolds with precisely two periodic orbits. The main step consists in showing that a contact form with exactly two periodic Reeb orbits is non-degenerate. The proof combines the ECH volume formula with a study of the behavior of the ECH index under non-degenerate perturbations of the contact form. As a consequence, the ambient contact 3-manifold is a standard lens space, the contact form is dynamically convex, the Reeb flow admits a rational disk-like global surface of section and the dynamics are described by a pseudorotation of the 2-disk. Moreover, the periods and rotation numbers of the closed orbits satisfy the same relations as (quotients of) irrational ellipsoids, and in the case of $S^3$ the transverse knot-type of the periodic orbits is determined. Joint work with Cristofaro-Gardiner, Hutchings and Liu.


Quarta feira

Matemática, Física & Aprendizagem Automática


, University of Washington.

Resumo

Representation learning has been widely used in many applications. In this talk, I will present our work, which uncovers when and why representation learning provably improves the sample efficiency, from a statistical learning point of view. I will show

  1. the existence of a good representation among all tasks, and
  2. the diversity of tasks are key conditions that permit improved statistical efficiency via multi-task representation learning.

These conditions provably improve the sample efficiency for functions with certain complexity measures as the representation. If time permits, I will also talk about leveraging the theoretical insights to improve practical performance.


Instituto Superior Técnico
Av. Rovisco Pais, Lisboa, PT