I’m a mathematician currently based at IHES as a post-doc in the Simons Collaboration on Homological Mirror Symmetry.
I mostly study cohomological invariants in algebraic and differential geometry, especially of spaces with singularities. If you’re interested, you can find a selection of some of my work below. Alternatively, here’s a full list of my papers and lecture notes.
My e-mail address is [email protected]
A typical way singularities arise is in case of non-transverse intersections of smooth ambient spaces. Derived algebraic geometry affords a powerful way to study such situations via the notion of quasi-smoothness. Here are a few papers illustrating this idea.
Recommended starting point. (Joint with David Rydh.)
Constructs “derived cycle classes” and proves a non-transverse Bézout theorem. Also generalizes Grothendieck–Riemann–Roch to the quasi-smooth setting.
Extends Orlov’s blow-up formula to the quasi-smooth situation, and gives a variant of Haesemeyer’s criterion that works without resolution of singularities.
Voevodsky’s stable motivic homotopy category is designed to give a sheaf-theoretic approach to cohomology theories like the (higher) Chow groups, algebraic cobordism, and algebraic K-theory. The papers below prove some structural results about this category.
Develops intersection theory à la Fulton in the language of Grothendieck’s six operations. This gives an ample supply of transfers in the stable motivic homotopy category. (Joint with Déglise and Jin.)
A motivic version of the recognition principle for infinite loop spaces. (Joint with Elmanto, Hoyois, Sosnilo, and Yakerson.)
Characterizes algebraic cobordism by the existence of certain transfers, and gives a derived cycle complex computing the algebraic cobordism of smooth schemes (in nonnegative cohomological degrees). (Joint with Elmanto, Hoyois, Sosnilo, and Yakerson.)
Here are some more resources related to the above topics that are available elsewhere on the web.
A seminar on virtual fundamental classes.
A seminar on derived algebraic geometry and algebraic cobordism.
A seminar on algebraic K-theory and derived algebraic geometry.
A seminar on motivic infinite loop space theory.
A paper by Toni Annala which in particular gives a model for the Chow cohomology of singular schemes (in characteristic zero). This is another nice application of the theory of quasi-smoothness.
A survey of Voevodsky’s work, and more recent developments since then, written by Marc Levine.