research overview
- My research expertise is algebraic topology and, more specifically, stable homotopy theory. Homotopy theory studies algebraic invariants of geometric objects that are unchanged under continuous deformations, while stable invariants are independent of certain spacial dimension shifts. My work on stable invariants has two distinct components: Chromatic Homotopy Theory: Chromatic homotopy theory is a framework that bridges algebraic geometry and topology, allowing the use of sophisticated algebraic tools to enable calculations of stable invariants. My work in this area has implications for structural conjectures and a large computational component whose goal is to provide data to help study these fundamental invariants. My work exploits the interplay between chromatic and equivariant homotopy theory (the study of symmetries) in innovative ways. Phases of Matter: In a collaboration with mathematicians and physicists at CU Boulder, I also study the relationship between stable homotopy theory and condensed matter physics. Our group has a unique combination of expertise that allows us to study problems relevant to physics using the rigorous framework of stable homotopy theory and algebraic quantum mechanics. Our current work is inspired by a general belief that certain phases of matter can be classified by stable homotopical invariants.
