Georgia Institute of Technology College of Sciences

The study of random graphs is a rich and well-developed field. However, the standard tools of this field break down when forbidding edge intersections. Our current understanding of random non-intersecting graphs is based on different methods, such as charging schemes. 

This talk is an introduction to this rather different theory of random graphs. We will survey the current best results (some by the speaker), applications, and common tools. 

The classification of quantum phases of matter is a fundamental topic in the study of quantum many-body systems. One main question in this study is whether or not there is a nonvanishing spectral gap above the ground state energy and, if so, whether or not this gap is stable under sufficiently short-range perturbations. In their seminal work, Affleck, Kennedy Lieb and Tasaki (AKLT) conjectured the existence of a spectral gap for the model they defined on the hexagonal lattice. Significant evidence supporting this long-standing claim was only recently achieved and naturally leads to the question of whether or not this gap is stable. In this talk, we review this and other recent progress on proving gaps for AKLT models and then turn to the question of whether these gaps are stable. One avenue for establishing gap stability pioneered by Bravyi, Hastings and Michalakis is to prove that the finite volume ground states are sufficiently indistinguishable by local observables in the bulk - a property known as Local Topological Quantum Order. We discuss a forthcoming work which uses cluster expansion techniques to prove that the ground states of the AKLT model on the hexagonal lattice and Lieb lattice satisfy LTQO. This talk is based on joint work with Thomas Andrew Jackson and Bruno Nachtergaele.

The colored Jones polynomial is a quantum knot invariant which can be constructed as a Reshetikhin–Turaev invariant using representations of $U_q(sl_2)$. Khovanov homology categorifies the Jones polynomial and by extension categorifies the representation theory of $sl_2$. Of particular interest is sutured annular Khovanov homology, which admits a structure as an $sl_2$-module. We will discuss a result of Grigsby–Licata–Wehrli that this structure is a representation-theoretic invariant of an annular link. Time permitting, we will discuss some of the structure of this representation, and extend the result to $sl_n$.

Quantum computing is concerned with harnessing the peculiar properties of quantum mechanics, in order to perform information-processing tasks beyond the capabilities of classical computers. Graph states are a family of quantum states, the resources for quantum computers. Graph states exhibit complex forms of quantum entanglement, implying for example that a quantum computer based on graph states is as powerful as any other quantum computer. But, unlike general quantum states, graph states are very easy to describe thanks to their one-to-one correspondence with mathematical graphs. This correspondence implies that many tools from graph theory can be applied to problems in quantum computing.

This talk aims to provide a gentle introduction to graph states, directed toward graph theorists. I will discuss two main applications of graph states, quantum networks and measurement-based quantum computing, and relate these applications to well-known graph-theoretical concepts, in particular vertex-minors. Finally, I will discuss the problem of classifying graph states, and the recent progress achieved through the development of new graph-theoretical tools.