Homotopy theory and applications

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My research involves topology, algebra, logic, 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, physics and computer science. 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. Recently it has been discovered that there is a close connection between topology and logic. Type theory is a foundation for mathematics that has been studied by logicians and computer scientists since the 1970s, but in the last decade it has been found to be intimately related to topology, in a deep and important way. Part of my work involves studying this relationship between topology and logic and using it to understand both subjects at a more fundamental level. In the future, I expect most mathematics to be formally verified, but even today formal verification is in heavy use in industry, e.g. at Airbus, Intel, Toyota and Microsoft. I also work on topics in geometry, which generalize existing methods of studying smooth spaces in a way that lets us study spaces with singularities and infinite-dimensional spaces that arise naturally in applications. I work on the topology of such spaces and also study tangent vectors, which are important tools for geometrical applications.