Research
Research
My current research is primarily split into two parts:
I have an interest in unstable homotopy theory through the lens of algebraic geometry: In the same way that spectra can be modeled by certain endofunctors of pointed spaces, a mild generalization leads to the unstable theory of Gamma spaces. These carry much of the same algebraic structure of spectra, including a presentably symmetric monoidal tensor product induced by Day convolution. In the (1,1)-categorical setting, this has been studied by Alain Connes and Caterina Consani as a model for algebra "over the field with one element." A primary research project of mine is to advance their theory in two ways: the first is to carry out their program to develop a geometric framework for ordinary Gamma sets, and the second is to extend it to the full theory of Gamma spaces as a (higher) category. Progress on the former includes a novel Riemann-Roch framework for those Gamma rings which satisfy a monic version of the Segal condition, and the latter takes place within the Homotopical Algebraic Geometry (HAG) framework.
As part of this, I was led to reinterpret the functor of points of a scheme in terms of classical Isbell duality. Isbell duality is an adjunction between presheaves and copresheaves which restricts to those which preserve limits; a consequence of this is that models for Lawvere theories carry a canonical "functor of points" with many of the same properties as for commutative rings. However, there is an alternative way to view commutative rings. Namely, they can be viewed as commutative algebras in the symmetric monoidal category of models for the Abelian Lawvere theory. This latter interpretation in terms of symmetric monoidal categories does not lend itself neatly to Isbell duality a priori, but I have shown that an Isbell-type adjunction is induced by a monoidal functor into a presentable category. The ur-example is the Yoneda embedding of a monoidal category and there the induced adjunction is a form of Sweedler duality with values in coalgebras for Day convolution of presheaves.
The other part of my research is primarily in derived differential geometry and the applications it has for ordinary manifolds and their geometric structures. I am working towards building and understanding (generalized) Riemannian structures on derived manifolds and stacks, with a view towards understanding derived versions of isometry groups for potentially singular spaces. Related to this is the explicit computation of models for derived versions of Lie groups which naturally can occur in Lie theory as intersection problems.
Joint Work :)
On D-Manifolds and the Truncations of Derived Manifolds: Joint with David Carchedi
Affine Connections and Curvature in the Spectral Exterior Calculus: Joint with Tyrus Berry
Explicit Generators for the Cohomology Ring of Path Lie Algebras: Joint with Marco Aldi, Sergio Da Silva, Tomas Mejia Gomez, Sam Blevins, Eleftherios Chatzitheodoridis
Parametrized Lawvere Theories and Isbell Duality