Homotopy theory is the study of properties of mathematical objects which remain invariant under deformation. Historically, homotopy theory emerged from algebraic topology, which is the study of topological spaces (roughly, geometry in which there is a notion of closeness but not distance) through algebraic invariants; gradually, however, it has found applications much more broadly throughout many areas of mathematics, especially algebraic and differential geometry, and has many interesting connections to physics. Part of the reason for this is that focusing on deformation-invariant properties of mathematical objects makes both classifications and calculations much more tractable. The interaction goes both ways; recently algebraic and geometric methods have found powerful applications in homotopy theory, and much recent research has been concerned with the algebro-geometric and higher-categorical nature of homotopical objects, illuminating many structural aspects and inspiring further interactions between these various mathematical disciplines.
The purpose of this project is to employ algebro-geometric and higher-categorical techniques in the study of algebraic topology and homotopy theory. The first goal is to study algebraic K-theory and related theories such as topological Hochschild homology, as well as more general motives arising in algebraic geometry, with emphasis on developing computational methods through identification of localization sequences. The second aims to classify thick subcategories of certain stable higher categories, especially those which arise in geometry by formation of perfect complexes over various classes of derived schemes. The third involves elliptic cohomology and is concerned with the explicit description of the elliptic cohomology of various orbifolds and is an important example of the not yet fully understood stable homotopy theory of orbifolds, also known as global stable homotopy theory, another subject of much recent attention. The final goal is to understand the units of a derived scheme using the emerging tools of derived algebraic geometry, in particular the Picard and Brauer groups and their higher categorical analogues and possible connections to topological field theories.