## Culminating Workshop

Saturday, July 30th 2022

09:00-09:30 am | Coffee
09:30-09:50 am | Rolando de Santiago: Mini-talk

I will provide an introduction to some aspects of my research for the GOALS students.

10:00-10:50 am | Mehrdad Kalantar: On ideal and trace structures of C*-algebras of quasi-regular representations (Notes)

The talk is concerned with the general problems of determining ideal and trace structures of C*-algebras generated by quasi-regular representations of discrete groups. For certain classes of subgroups, we obtain complete description of traces and characterization of simplicity of the C*-algebras of the corresponding quasi-regular representations. These results are consequences of a more fundamental uniqueness property of a class of morphisms in the category of C*-dynamical systems, called noncommutative boundary maps'. The main technical tool in proving the uniqueness of these maps is the notion of boundary actions in the sense of Furstenberg. We will explain the general/categorical ideas behind the proofs, and give applications in concrete examples. This is joint work with Eduardo Scarparo.

11:00-11:30 am | Coffee Break
11:30-11:50 am | Kristin Courtney: Mini-talk

I will provide an introduction to some aspects of my research for the GOALS students.

12:00-01:30 pm | Lunch
01:30-01:50 pm | Brent Nelson: Mini-talk

I will provide an introduction to some aspects of my research for the GOALS students.

02:00-02:50 pm | Samantha Pilgrim: Coarse Geometry and Operator Algebras (Notes)

Coarse geometry is the study of metric spaces up to an equivalence which only sees large scale' properties (as opposed to metric space topology, which only sees small scale properties). We will give an introduction to the theory of coarse geometry and discuss some relations to the study of operator algebras.

03:00-03:30 pm | Coffee Break
03:30-04:20 pm | Priyanga Ganesan: Quantum graphs and colorings (Notes)

Quantum graphs are an operator space generalization of classical graphs that have appeared in different branches of mathematics including operator algebras, non-commutative topology and quantum information theory. In this talk, I will introduce a quantum input-classical output non-local game that captures the coloring problem for quantum graphs. Using this framework, we show that every quantum graph has a finite quantum coloring and is four-colorable in the algebraic model. The winning strategies of the quantum-to-classical non local coloring game leads to a combinatorial characterization of quantum graph coloring. We will use this to obtain lower bounds for the chromatic numbers of quantum graphs and show generalizations of well-known classical bounds, such as the Hoffman's bound, to the quantum graph setting.

04:30-04:50 pm | Elizabeth Gillaspy: Mini-talk

I will provide an introduction to some aspects of my research for the GOALS students.

06:30-08:30 pm | Conference Dinner

Catered by Masala in the Mathematics Building Courtyard.