Homotopy theory, representation theory and mathematical physics

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My research involves topology, physics, computation, and various topics in between. Topology is the study of surfaces and higher-dimensional shapes, and the ways in which they can be stretched, deformed and mapped into each other. We study such spaces by probing them with loops, spheres and other simple geometric shapes. This generally results in algebraic invariants which provide qualitative and computational information about a space. Topology has applications in many other fields, such as algebra, algebraic geometry and physics. There are many questions and techniques that make sense in all of these settings, and the main theme of my work in topology is the study of such interdisciplinary problems. This interplay between disciplines has proven to be extremely fruitful, often providing insight that leads to solutions to open problems. I will continue to tackle such problems, with a focus on problems involving algebra and geometry. One of the most important unsolved problems in physics is the unification of general relativity, Einstein's theory of gravity, with quantum mechanics, which describes all of the other known forces. My work focuses on a promising approach to this problem, a theory called loop quantum gravity. I intend to study one of the main open problems in this field: how to incorporate matter into the theory. I will study this problem both analytically and through the use of computer simulations, with the goal of finding a viable theory of matter and seeing whether it makes testable predictions. I also work on lattice gauge theory, which is a computational approach to particle physics that uses a discrete approximation to space-time. Methods from representation theory and quantum gravity can be used to understand these computations in a gauge invariant way, and I am developing new algorithms that I hope will make them faster.