Algebraic topology and algebraic K-theory
My primary research focuses on formalizing geometric cutting-and-pasting problems and studying them using methods from algebraic topology and algebraic K-theory. The main idea is to take the notion of "cutting and pasting" and replace it with an algebraic model that remembers the combinatorial aspects and forgets the geometric ones. This formalism can then, using the methods of algebraic K-theory, be turned into a topological space, whose properties we can study. These properties should then reflect back on the geometric problems by producing invariants of the original geometric problem. My main focus has been on applying this to studying scissors congruence (along the lines of Hilberts third problem) and the Grothendieck ring of varieties, although I am always looking for new applications.
- The annihilator of the Lefschetz motive. Duke Math. J. Vol 166 (11) (2017), 1989-2022.
- On K1 of an assembler, J. Pure Appl. Alg. Vol 221 (7), 1495-1898.
- The K-theory of assemblers. Adv. in Math. Vol 302 (2017), 1176-1218.
- Perspectives on scissors congruence, Bull. Amer. Math. Soc. Vol 53 (2016), 269-294