Events

We will begin our discussion of the first jump inversion theorem for structures by covering some necessary background such as defining the jump of a structure and effective bi-interpretability.

MC 5301

In joint work-in-progress with Rahim Moosa, we explore the notion of internality in continuous logic and show that, whenever this phenomenon arises in a metric structure, a definable binding group arises together with it. This requires navigating some of the difficulties of what definability means in the continuous sense.

MC 5403

We discuss contact strata in the arc space of the flag variety. In the preseminar period, we briefly introduce arc spaces and discuss their relevance in birational geometry.

MC 5403

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

We will begin with a proof of the Ash-Knight-Manasse-Slaman and Chisholm theorem. We will then discuss some properties of effective bi-interpretability, and start the proof of the first jump inversion theorem for structures. If time permits, we will also show several consequences of the theorem.

MC 5403