## Overview

A central theme of my research is applications of higher category theory to topology and differential geometry, and most recently, to field theory. Specifically, I have constructed a higher category of derived smooth manifolds using differential graded supergeometry, I have developed a universal framework to study higher (derived) orbifolds applicable in differential topology, and algebraic geometry over commutative classical rings, simplicial rings, and E_{∞}-ring spectra, and finally, I am investigating the use of transgression in twisted generalized cohomology theories as a means to quantize field theories.

## Description of Research

Together with Dmitry Roytenberg, I have constructed a simple model for derived manifolds using differential graded manifolds. This model is geometric in nature and allows one to explicitly calculate derived pullbacks of smooth maps between manifolds by an easy formula. Moreover, we have nearly finished showing that our higher category of dg-manifolds is equivalent to the models of derived manifolds proposed by Spivak, and Borisov-Noel (these two models are known to be equivalent).

I have developed a universal framework to study smooth higher orbifolds, on the one hand, and (higher) algebraic Deligne-Mumford stacks, as well as their derived and spectral analogues, on the other. The framework yields a new characterization of classical Deligne-Mumford stacks, which extends to the derived and spectral setting as well. In the differentiable setting, this characterization shows that there is a natural correspondence between n-dimensional smooth higher orbifolds, and classical fields for n-dimensional field theories. I have also found a simple formula for the weak homotopy type of smooth higher orbifolds, which gives a new way of expressing the weak homotopy types of certain classifying spaces arising in foliation theory and classical differential topology (preprints to appear soon).

##

I recently became involved in a project with Joost Nuiten and Urs Schreiber on the quantization of local prequantum field theories. The latter are, in a precise sense, field theories arising from a process of *prequantization* of geometric data, and encode (fully transgressed) action functionals. They are modeled as n-dimensional fully extended topological quantum field theories. Following ideas of Freed, we are attempting to formalize Feynman's "path integral" approach to quantization through transgression in twisted generalized cohomology. It is our hope to arrive at field theories which describe actual physics, and at the same time lead to interesting manifold invariants.

# Positions

I will be a Visiting Scholar at Berkeley from Jan.-May of 2014 with Alan Weinstein.

I was a Visiting Scholar at MIT during the summers of 2009, 2010, 2012, and 2013

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"