Research
Description
A central or recurring theme in my work is Category Theory, in particular its applications to Algebraic Geometry and Homotopy Theory.
In more details, I am interested in higher algebra, which is, in a broad sense, the study of familiar algebraic structures—groups, rings, and others—suitably translated to (higher) categories.
Some instances where these gadgets concretely manifest themselves concern Intersection Theory and Theory, and the geometry of homotopy types. Here are some of the main themes:

The correspondence between (Cartier) divisors and line bundles is well known. Its analog in codimension 2 (think about a point on a surface) is that to such a cycle corresponds a gerbe bound^{1} by the sheaf , whose stalks are the Quillen groups. The construction of this gerbe, as well as the construction of the cupproduct corresponding to the intersection product via liftings to a central extension, is joint work with Niranjan Ramachandran.

Categorical rings are Picard groupoids^{2} equipped with a second monoidal structure turning them into the categorical analog of a ring. (Picard groupoids can conveniently model the 1type of connective spectra, such as the ones arising in Theory from, say, exact categories. A monoidal structure at this level turns the corresponding 1type into a categorical ring.) In general, Picard stacks have very convenient algebraic models—or presentations, as they are called. Some facts we can prove about them are: (1) for presented categorical rings the structure is encoded by a biextension; (2) for presented categorical rings whose underlying Picard stack is strict, the biextension is trivial and the presentation is a crossed bimodule—these categorical rings represent classes in AndréQuillen cohomology; (3) Picard groupoids comprise a multicategory in which categorical rings are the weak monoids, and this correspondence is transported to the presentations by way of a multifunctor.

Loosely speaking, a Homotopy Type classifies the connectivity of a space up to a certain degree, so you say Homotopy Type to consider everything up to the homotopy group. The interesting bit is to compute the space of morphisms between two homotopy types. For we are just talking about groups and the homomorphisms between them, but for resorting to (higher) categories and homotopy theory is essential. For we obtained relatively simpler answers by way of diagrams called Butterflies, with neat applications to nonabelian cohomology, with Behrang Noohi.
Projects

(With U. Bruzzo and V. Rubtsov) Extensions and cohomology of Lie Algebroids on a scheme .

(With Niranjan Ramachandran) Extend the correspondence between codimension 2 cycles and gerbes, as well as the intersection product via extensions, to any codimension. A second strand is to understand the infinitesimal theory, where the base scheme is replaced by its thickening , being an Artinian simplicial ring, in the sense of derived geometry.
Part of this project requires generalizing the Heisenberg central extension and its relation to the cup product to simplicial presheaves of abelian groups and presheaves of spectra. This is developed in part with my student Michael Niemeier.

Extend the functor from Waldhausen categories to the (homotopy) category of 1types, which assigns to a Waldhausen category a stable crossed module , a presentation for the 1type of , to a multifunctor.^{3} With my student Yaineli Valdes.

(With Niranjan Ramachandran and Denis Eriksson) Universal 2determinants, 2types, and .
Papers

Lie algebroid cohomology and Lie algebroid extensions (With U. Bruzzo and V. Rubtsov)
Journal of Algebra 505 (2018), 456–481
DOI
arXiv:1711.05156
local copy 
Biextensions, bimonoidal functors, multilinear functor calculus, and categorical rings
Theory and Applications of Categories 32 (2017), 889–969
arXiv:1501.04664
local copy 
Notes on Weak Units of Picard 1 and 2Stacks (with A. Emin Tatar)
Mathematical Proceedings of the Cambridge Philosophical Society (09 November 2016)
arXiv:1108.1922
local copy 
Cup products, the Heisenberg group, and codimension two algebraic cycles (with Niranjan Ramachandran)
Documenta Math. 21 (2016) 1313–1344
arXiv:1510.01825
local copy 
Stacks of AnnCategories and their morphisms
Theory and Applications of Categories 30 (2015), 1256–1286
arXiv:1501.07592
local copy 
Butterflies II: Torsors for 2group stacks (with Behrang Noohi)
Advances in Mathematics 225 (2010), 922–976
DOI: 10.1016/j.aim.2010.03.011
arXiv:0909.33450
Local copies: DVI PS PDF 
Butterflies I: Morphisms of 2group stacks (with Behrang Noohi)
Advances in Mathematics 221 (2009), 687–773
DOI: 10.1016/j.aim.2008.12.014
arXiv:0808.3627
Local copies: DVI PS PDF 
2gerbes bound by complexes of gr–stacks, and cohomology
Journal of Pure and Applied Algebra 212 (2008), 994–1038
DOI: 10.1016/j.jpaa.2007.07.020
arXiv:math/0512453 [math.CT]
Local copies: DVI PS PDF 
Hermitianholomorphic Deligne cohomology, Deligne pairing for singular metrics, and hyperbolic metrics
Int. Mathematics Research Notices 17 (2005), 1015–1046.
arXiv:math/0408118 [math.AG]
(Local copies in DVI PS PDF) 
Hermitianholomorphic (2)gerbes and tame symbols
Journal of Pure and Applied Algebra 200 (2005), 97–135.
arXiv:math/0310027 [math.CT]
(Local copies in DVI PS PDF) 
On hermitianholomorphic classes related to uniformization, the dilogarithm and the Liouville action
Commun. Math. Phys. 251 (2004) 27–64
arXiv:math/0211055 [math.CV]
(Local copies in DVI PS PDF) 
Homological algebra of multivalued action functionals
Letters in Mathematical Physics 60 (2002) 4758
arXiv:mathph/0112031 
Generating Functional in CFT on Riemann Surfaces II: Homological Aspects (with L. A. Takhtajan)
Commun. Math. Phys. 227 (2002) 303–348
arXiv:math/0006147 [math.AT]