A quantum spin system is a lattice of finite-dimensional full matrix algebras equipped with a local Hamiltonian, i.e., a self-adjoint operator H which is a sum of self-adjoint terms which are localized to small areas in the lattice. The eigenspace of the smallest eigenvalue of H is called the ground state space.

Topological order is a notion in theoretical condensed matter physics describing new phases of matter beyond Landau's symmetry breaking paradigm. Bravyi, Hastings, and Michalakis introduced certain topological quantum order (TQO) axioms to ensure gap stability of a commuting projector local Hamiltonian and stabilize the ground state space with respect to local operators in a quantum spin system.

In joint work with Corey Jones, Pieter Naaijkens, and Daniel Wallick, we study nets of finite dimensional C*-algebras on a $\mathbb{Z}^{k}$ lattice equipped with a net of projections as an abstract version of a quantum spin system equipped with a local Hamiltonian. We introduce a set of local topological order (LTO) axioms which imply the TQO conditions of Bravyi-Hastings-Michalakis in the frustration free commuting projector setting, and we show our LTO axioms are satisfied by known 2D examples, including Kitaev's toric code and Levin-Wen string net models associated to unitary fusion categories (UFCs).

A C*-algebra is approximately finite dimensional (AF) it it can be written as the closure of an increasing union of finite dimensional C*-algebras. I will discuss the structure of C*-subalgebras of AF algebras.

Amenability for groups is a concept that was first introduced by von Neumann in 1929 to provide an explanation of the Banach-Tarski paradox. The concept has since been exported to many different areas of mathematics and continues to hold an important position in fields such as group theory, ergodic theory, and operator algebras. In the area of von Neumann algebras the concept plays a fundamental role, and the classification of amenable von Neumann algebras by Connes and Haagerup is considered a touchstone of the field. In this talk, I will give a survey of amenability in von Neumann algebras, with special emphasis on recent uses of the concept and highlighting some of my own contributions.

C*-algebras defined from directed graphs and their higher-rank counterparts, provide examples of operator algebras that can be understood by analysis of their underlying combinatorial objects. In this talk, we will define graph algebras, state some uniqueness theorems, and see that these algebras behave well under quotients by so-called regular ideals. The new results are joint work with Jonathan Brown, Adam Fuller, and David Pitts.

I will give an introduction to Fredholm index theory and explain how a simple index theorem on the Euclidean space can be used to solve an open problem in differential geometry. I will make this talk accessible to everyone in the audience.

Catered by Sharma's Kitchen. We will set up the food by the outdoor seating on the southern side of the Purdule Memorial Union.