K-Theory, Actions & stable Homotopy – KAsH

This project in algebraic topology builds upon spectacular recent advances in algebraic K-theory, stable homotopy theory, and equivariant or functorial techniques, using the framework of infinity categories and higher algebra.
The scope of algebraic K-theory has been considerably broadened with the development of continuous and Hermitian K-theory, and powerful tools such as the motivic or even filtrations of topological Hochschild homology have been discovered. The long-standing and important Telescope Conjecture in stable homotopy theory has been disproved, thanks to counter-examples coming from algebraic K-theory. Equivariant homotopy theory and more general functorial techniques are crucial players in these developments.
A major goal of the project is to develop trace invariants for Hermitian K-theory of general Poincaré categories, as well as for real K-theory (a theory encoding both algebraic and Hermitian K-theory).
Using (infinity, 2)-categories, a comprehensive equivariant stable homotopy theory will be developed, featuring higher Mackey functors.
Motivic cohomology theories and the motivic filtration of topological Hochschild homology will be studied, with the goal of expressing them as invariants of geometric objects.
Computations of (logarithmic) topological cyclic, Hochschild homology and their real versions will be performed, to investigate (trans-)chromatic phenomena in these settings.
The interaction of the cyclotomic structure of topological Hochschild homology with additional structure on the input, such as Calabi–Yau algebras, will be studied.
The theory of functors on additive categories, notably functor homology, will be applied to computations in topological Hochschild homology, stable cohomology of groups with twisted coefficients, and stable K-theory.
This array of powerful new techniques exploiting higher structures will lead to substantial new results, with significant and fruitful interactions between the different themes of the project.