The goal of this project is to explore some applications and potential applications of algebraic topology and algebraic K-theory. In algebraic K-theory, the project explores ideas applicable to the conjecture of Waldhausen on the K-theory chromatic tower, relating arithmetic and geometry. The project explores a number of potential contributions to the theory and computation of topological Hochschild homology and topological cyclic homology. The project explores some new stable homotopy tools applicable to the study of the algebraic K-theory of spaces (which is closely related to the differential topology of high dimensional manifolds). In unstable homotopy theory, the project explores the applications of homotopy algebras to computations related to fiber squares and mapping spaces.
Homotopy theory studies those properties of mathematical objects that do not change under small deformations. These mathematical objects are often of a geometric nature but the methods of homotopy theory have been increasingly applied to objects of an algebraic nature as well. Homotopy theoretic properties tend to be accessible to computation by taking advantage of the invariance under small changes. Since they also generally retain important information about the original mathematical objects, homotopy theory is an effective tool for a wide range of mathematical problems.