Scientific

Quantum computing is believed to provide many advantages over traditional computing, particularly considering the speed at which computations can be performed. One of the challenges that needs to be resolved in order to construct a quantum computer is the transmission of information from one part of the computer to another. This transmission can be implemented by spin chains, which can be modeled as a graph, and analyzed using algebraic graph theory. The ideal situation is that of perfect state transfer, where there exists a time interval during which the information is perfectly moved from one location to another. As perfect state transfer is relatively rare, we also consider pretty good state transfer, where for any desired level of accuracy, there exists a time interval during which the information transfer achieves this accuracy. We will discuss determining whether graphs admit perfect or pretty good state transfer.

Recently there has been a lot of progress in classifying phases of gapped quantum many-body systems. From the mathematical viewpoint, a phase of a quantum system is a connected component of the “space” of gapped quantum systems, and it is natural to study the topology of this space. I will explain how to probe it using generalizations of the Berry curvature. I will focus on the case of lattice systems where all constructions can be made rigorous. Coarse geometry plays an important role in these constructions.

Suppose mu is a probability measure which is shift invariant on {0,1}^{Z^d} and we know that for almost every configuration x in {0,1}^{Z^d} there are connected components of 1s which are infinite. In this talk, we will follow a paper by Burton and Keane (generalising results by Aizenman, Kesten and Newman) to give an elegant proof of the fact that, under fairly general conditions (say full support), the number of connected components of infinite cardinality is at exactly one.

See attached PDF

Going back to the foundation work of von Neuman there is a question of whether there are smooth models of the models of classical ergodic theory. When both measure and map are required to be smooth there is only one known obstruction but essentially no general results. Within the class of zero entropy transformation we have a method called conjugation by approximation that can be used to realize many interesting properties. I will describe the method and some of the classical and modern consequences of this.