This is my second year as a postdoc in mathematics at the Max Planck Institute for Mathematics. I co-organize the Higher Differential Geometry seminar here, together with Christian Blohmann.
Before this appointment, I was a PhD student of Ieke Moerdijk at Utrecht University. My thesis title was "Categorical Properties of Topological and Differentiable Stacks."
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A prezi outlining some of my research areas. It was part of my presentation in MPI's Oberseminar
The focus of my research is on
Dmitry Roytenberg and I have made considerable progress in developing a model for derived smooth manifolds using differential graded manifolds (dg-manifolds). We have introduced the notion of a differential graded structure to the setting of C∞-rings and constructed a Quillen model structure on the category of these dg-algebras generalizing the well-known one on the category of classical dg-algebras. The opposite of this model category contains dg-manifolds as a full subcategory, and we hope to show an appropriate subcategory thereof provides a model for derived smooth manifolds.
I am also developing the theory of étale differentiable stacks. These model quotients of manifolds by certain local symmetries, and their points poses intrinsic (discrete) automorphism groups. They are closely related to the Deligne-Mumford stacks of algebraic geometry. I have shown that the classical sheaf theory for manifolds naturally extends to étale stacks and used this to prove an unexpected result: There is an equivalence between the bicategory of étale stacks and local diffeomorphisms and the bicategory of all stacks on the site of smooth manifolds and local diffeomorphisms.
Furthermore, I am actively researching structured ∞-topoi. This research is a generalization of ideas in Lurie's DAG V, and provides a unifying framework to study geometric objects that arise by gluing together local models, e.g. manifolds, schemes, derived schemes and spectral schemes. I am currently working out some technical details, however this theory appears to give a new characterization of classical Deligne-Mumford stacks and their derived and spectral analogues.
Additionally, Urs Schreiber and I have done some work developing a theory of ∞-quasitopoi.
An Étalé Space Construction for Stacks. Journal of Algebraic and Geometric Topology
Compactly Generated Stacks: A Cartesian closed theory of topological stacks. Advances in Mathematics, Volume 229, Issue 6, April 1 2012, Pages 3339-3397 arXiv link
Homological Algebra for Superalgebras of Differentiable Functions (with Dmitry Roytenberg)
On Theories of Superalgebras of Differentiable Functions (with Dmitry Roytenberg)
Sheaf Theory for Étale Geometric Stacks
Structured Infinity Topoi and Higher Étale Stacks (Draft to Appear Here Soon)
Informal notes about Étale stacks I wrote for Alan Weinstein: A Quick Note on Étale Stacks
I was a Visiting Scholar at MIT during the summers of 2009, 2010 and 2012.
Before starting my PhD at Utrecht, I spent a year in Utrecht as a participant in the Master Class in Symplectic Geometry from 2006-2007.
I also participated quite a bit in the Master Class in Calabi-Yau Geometry from 2008-2009.
In Fall of 2002, I was a participant in the "Math in Moscow Program"
Higher Differential Geometry Seminar (co-organizer, together with Christian Blohmann)
