Speaker: Soumyashant Nayak, University of Pennsylvania

Title: Analyticity in Operator Algebras

Date: 08/20/2018

Time: 11:00 AM - 12:00 PM

Place: C517 Wells Hall

The title of this talk is borrowed from a seminal paper by Arveson discussing non-commutative analogues of the Hardy space H^∞(T) via the so-called subdiagonal algebras. Subdiagonal algebras are a family of non-self-adjoint operator algebras which give a common perspective to the study of some triangular operator algebras (for example, the algebra of block upper triangular matrices in M_n(C)), Dirichlet function algebras, etc. The first part of the talk will be about a non-commutative version of inner-outer factorization in finite maximal subdiagonal algebras. We will then discuss a proof of a version of Jensen's inequality in this setting which relates to some classical results by Szegö.

Speaker: Hyenkyun Woo, Korea University of Technology & Education

Title: Bregman-divergence for Legendre exponential families and data analysis

Date: 08/24/2018

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

Bregman-divergence is a well-known generalized distance framework in various applications, such as machine learning and image processing. In this talk, by using dual structure of the Bregman-divergence associated with the subclass of convex function of Legendre function, we analyze the structure of the Legendre exponential families whose cumulant function corresponds to the conjugate convex function of Legendre type. Actually, Legendre exponential families are the extended version of the regular exponential families to include non-regular exponential families, such as the inverse Gaussian distribution. The main advantage of the proposed Bregman-divergence-based approach is that it offers systematic successive approximation tools to handle closed domain issues arising in non-regular exponential families and the statistical distribution having discrete random variables, such as Bernoulli distribution and Poisson distribution. In addition, we also introduce the generalized center-based clustering algorithm based on the Tweedie distribution.

Speaker: Gerard Awanou, University of Illinois, Chicago

Title: Discrete Aleksandrov solutions of the Monge-Ampere equation

Date: 08/31/2018

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

A discrete analogue of the Dirichlet problem of the Aleksandrov theory of the Monge-Amere equation is derived in this paper. The discrete solution is not required to be convex, but only discrete convex in the sense of Oberman. We prove that the uniform limit on compact subsets of discrete convex functions which are uniformly bounded and which interpolate the Dirichlet boundary data is a continuous convex function which satisfies the boundary condition strongly. The domain of the solution needs not be uniformly convex. We obtain the first proof of convergence of a wide stencil finite difference scheme to the Aleksandrov solution of the elliptic Monge-Ampere equation when the right hand side is a sum of Dirac masses. The discrete scheme we analyze for the Dirichlet problem, when coupled with a discretization of the second boundary condition, as proposed by Benamou and Froese, can be used to get a good initial guess for geometric methods solving optimal transport between two measures.