- Math Learning Center
- News & Events
- Giving to Math
The classical Schur-Horn theorem characterizes the set of diagonals of the unitary orbit of a self-adjoint matrix in terms of a set of linear inequalities called majorization. In 2002 Kadison discovered a characterization of the diagonals of orthogonal projections on infinite dimensional Hilbert spaces. Since Kadison's breakthrough there has been a great deal of work by several authors to extend the Schur-Horn theorem to all self-adjoint operators on infinite dimensional Hilbert spaces. In this talk we discuss our recent contributions to this effort.
Translation surfaces are oriented surfaces equipped with an atlas of local charts to R^2 for which the transition functions are translations. This atlas gives us a well defined notion of whether or not a map from one translation surface to another is affine (linear plus a translation). The Veech group of a translation surface is the group of Jacobians of orientation preserving affine automorphisms of the surface. The size of this group can inform us on the dynamics (periodic/ergodic) of the geodesic flow [Veech 1998].
Lehmer conjectured the existence of a lower bound on height of algebraic numbers (nonzero, not a root of unity). Although answering this conjecture is still an open problem, there have been partial results in certain cases. One case was investigated by Amoroso and Dvornicich, who found a lower bound for algebraic numbers which lie in abelian extensions of the rationals.
Water, energy and food are needed for the health and prosperity of humanity (a subset of article 25 in the universal declaration of human rights). Each resource must be managed, treated, refined and delivered prior to consumption. Human activity already uses 37% of the Earth’s net primary production, and water scarcity is an issue for many arid environments. The majority of the power consumed is not sustainable.
Inspired by the experimental results by Haggerty, Zinn et al from 2004, we built an upscaled model for flow and coupled transport in heterogeneous media. The model extends those known from homogenization theory, and in particular those known as double porosity models, and applies to a wide range of scales and contrasts between media. The model includes three types of nonlocal (or memory) terms, whose kernels are derived from local cell problems.