Abstract: This research-interaction seminar focuses on mathematical aspects of quantum information. In previous semesters we examined various applications of algebra, analysis, and geometry to quantum foundations, quantum cryptography, quantum computing, and other topics in theoretical physics. In this organizational meeting we'll discuss logistics, and then give some short advertisements for papers that we'd like to discuss during the fall semester. Suggestions and contributions are welcome! No previous experience in quantum theory is required, however linear algebra and (discrete) probability is a must. Seminar information is available at http://users.umiacs.umd.edu/~bclackey/QI-RIT2018Fall.html .
Abstract: It is well known that the quadratic-cost optimal transportation problem is formally equivalent to the Monge-Ampere equation, a fully nonlinear elliptic PDE. Instead of a traditional boundary condition, the PDE is equipped with a global constraint on the solution gradient, which constrains the transport of mass. Recently, several numerical methods have been proposed for this problem, but no convergence proofs are available. Viscosity solutions have become a powerful tool for analyzing methods for fully nonlinear elliptic equations. However, existing convergence frameworks for viscosity solutions are not valid for this problem. We introduce an alternative PDE that couples the usual Monge-Ampere equation to a Hamilton-Jacobi equation that restricts the transportation of mass. Using this reformulation, we develop a framework for
proving convergence of a large class of approximation schemes for the optimal transport problem. We describe several examples of convergent schemes, as well as possible extensions to more general optimal transportation problems.
Abstract: C*-algebras are a kind of operator algebras tailored to describe noncommutative (i.e., quantum) topological spaces via functional analytical means. A major source of examples throughout the history of C*-algebra theory lies in the construction of crossed products from topological dynamical systems. This bridge between operator algebras and dynamics, valid also in the measure-theoretical setting, has proven immensely fruitful. On the other hand, the dimension theory of C*-algebras, which studies analogs of classical dimensions for topological spaces, is young but has been gaining momentum lately thanks to the pivotal role played by the notion of finite nuclear dimension in the classification program of simple separable nuclear C*-algebras. The confluence of these two themes leads to the question: What type of topological dynamical systems give rise to crossed product C*-algebras with finite nuclear dimension? I will present some recent work on this problem.
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