Selected contributions
My work develops abstract tools that make previously inaccessible computations possible. The two long-term research programmes below remain active areas of my work.
01
Trace methods and the K-theory of ℤ/n
Together with Peter Scholze, I developed a homotopy-invariant framework for cyclotomic spectra and topological cyclic homology. With Benjamin Antieau, I further developed this theory through topological Cartier modules, giving an algebraic description of p-typical cyclotomic spectra. Joint work with Achim Krause on Bökstedt periodicity produced explicit calculations of the topological Hochschild homology of quotients of discrete valuation rings. Building on these computations and the preceding conceptual developments, joint work with Antieau and Krause gave an algorithm for all higher K-groups of ℤ/n, resolving a benchmark problem dating back to the early 1970s.
02
Grothendieck–Witt and L-theory
Together with collaborators, I am pursuing a long-term programme in hermitian K-theory and L-theory. Building on earlier work relating the L-theory of C*-algebras to topological K-theory and determining the L-spectra of ℤ, this programme led to a modern framework for hermitian K-theory of stable ∞-categories. The framework connects algebraic forms and stable homotopy theory with surgery and arithmetic groups, works integrally—including at the prime 2—and proves conjectures of Karoubi. A major application is the computation of the Grothendieck–Witt groups of ℤ—a longstanding open problem previously accessible only after inverting 2.
