Tuesday, April 23, 2019
04/23/2019 - 4:00pm
Abstract: I will present recent results on the related problems of denoising, covariance estimation, and principal component analysis for the spiked covariance model with heteroscedastic noise. Specifically, I will present an estimator of the principal components based on whitening the noise, and optimal spectral shrinkers for use with these estimated principal components. I will also show new results on the optimality of whitening for principal subspace estimation. This is joint work with Elad Romanov of the Hebrew University.
04/23/2019 - 4:00pm
Abstract: I begin with some general results on anaylsis on homogeneous spaces and restricting representations of reductive groups to their subgroups. Then I will focus on a concrete geometric question arising from conformal geometry, giving the complete classification of conformally covariant “symmetry breaking operators” for differential forms on the model space for codimension one submanifolds. Some of the symmetry breaking operators are given as differential operators, whereas some others include integral operators and its meromorphic continuation.
If time permits, I would like to discuss some applications and related questions including a conjecture of Gross and Prasad.
 T. Kobayashi. A program for branching problems in the representation theory of real reductive groups. Progr. Math. 312, pp. 277-322, 2015.
 T. Kobayashi, T. Kubo, and M. Pevzner, Conformal symmetry breaking operators for differential forms on spheres, viii+192 pages. Lecture Notes in Mathematics, vol. 2170, 2016.
 T. Kobayashi and B. Speh. Symmetry Breaking for Representations of Rank One Orthogonal Groups, Memoirs of Amer. Math. Soc. 238. 2015. 118 pages.
 T. Kobayashi and B. Speh, – II, xv+342 pages, Lecture Notes in Math. 2234, Springer-Nature, 2018.
04/23/2019 - 4:15pm
In the study of hyperbolic 3-manifolds cusps play an important role. The geometry of a cusp is determined by a similarity structure on the boundary of the cusp. In the finite volume case, the boundary is a torus and the similarity structure is determined by a complex number with positive imaginary part. Properly-convex real-projective manifolds are a generalization of hyperbolic manifolds. In dimension 3 the moduli space of generalized cusps is a bundle over the space of similarity structures on the torus, with fiber a subspace of the space of (real) cubic differentials. Conjecturally a similar statement is true in all dimensions for cusps with compact boundary. There is a 9-dimensional cusp with fundamental group the integer Heisenberg group, and the classification of cusps with non-compact boundary is unknown. Joint: Sam Ballas, Arielle Leitner.
04/23/2019 - 7:15pm
The space of traceless 2 by 2 matrices, SL(2, C), is the most basic example of a Lie algebra. In this talk we will discuss the structure of its finite dimensional representations and how they can be used to study representations of other Lie algebras.