Welcome to Pure Mathematics

We will compute the index of the deformation operator of a BPS monopole on R^3. During this computation, we will encounter how classical index results fall short in the presence of symmetry breaking. A hybridization of the b-calculus and scattering calculus will be the remedy to bypass the symmetry breaking and perform the computation.

MC 5417

We discuss several noncommutative generalizations of the classical Wasserstein distance of two probability measures on R^d, or more general metric spaces, which measures the minimal L^2 distance of two random variables with the given distributions. The first motivation comes from free probability theory, where ideas from optimal transport theory have been used to show isomorphism of von Neumann algebras and appear in functional inequalities with free entropy, as well as providing a general notion of "perturbing generators" of a von Neumann algebra. The second motivation comes from quantum information theory, where optimal transport theory provides a way to assign distances for states on a von Neumann algebra. As we will see, Duvenhage's version of quadratic Wasserstein distance relates quite closely to the Biane and Voiculescu's free Wasserstein distance. We discuss these distances in a common framework and demonstrate several properties that make the noncommutative setting more challenging, including non-separability of the Wasserstein space, and a stark difference between the Wasserstein and weak-* topologies on certain state spaces.

MC 5403

We will continue working through Arulseelan and Hanson's recent paper on AW*-algebras.

MC 5301