Topology, Higher Categories, and Quantum Field Theory.
I'm interested in the interactions of these subjects, and my research emphasizes how to exploit these interactions to learn something new and interesting. Here are a few projects I am currently working on:
The Comparison Problem in Higher Category Theory. The history of higher category theory has lead to a wealth of definitions, each built on differing ideas and principles. Until recently there has been very little in the way of machinery to compare them. My ongoing work with Clark Barwick changes this. Our first paper is now available:
In the above work we give four axioms that characterize the (∞,1)-category of all (∞,n)-categories. We show that the space of (∞,1)-categories satisfying our axioms is (RP∞)n (it is connected so all these ∞-categories are equivalent!). Finally we show that a host of proposed theories of (∞,n)-categories satisfy our axioms.
The Structure of Tensor Categories via 3D TQFTs
the structure of 3D TQFTs. In particular we show that fusion categories are fully-dualizable objects in a the 3-category of tensor categories, bimodule categories, functors, and natural transformations. We identify the induced O(3)-action on the `space' of fusion categories, as predicted by the cobordism hypothesis, in light of Hopkins' and Lurie's work, this provides a fully local 3D TQFT for arbitrary fusion categories. Moreover understanding various homotopy fixed point spaces uncovers how many familiar structures from the theory of fusion categories are given a natural explanation from the point of view of 3D TQFTs.
Some projects in the wings....
The Meta-Local Cobordism Hypothesis.
Higher Homological Algebra
Another direction my research has turned is to the higher categorical versions of homological algebra. I began by classifying certain central extensions of smooth 2-groups, by which I mean weak group objects in the bicategory of Lie groupoids, left principal bibundles, and bibundle maps. As a consequence, I obtained a a construction of a finite dimensional model of the String group. This material is now available as a preprint. Following this I have extended these results to classify all extensions of smooth 2-groups using some new results in bicategorical homological algebra. I'm currently in the process of writing up these results. I hope this will lead to a better understanding of the geometry of string structures and higher categorical geometry in general.
On occasion I post articles on the Secret Bloging Seminar, and sometimes I'm spotted on MathOverflow.
Co-organizer of the Summer QFT Seminar.
Dr. Christopher Schommer-Pries
Selected Blog Posts:
Currently I am a scientist researcher
Before that I was an NSF postdoc and C. L. E. Moore Instructor at MIT, and even earlier a Postdoc at Harvard.
You Tube Videos!
I recently attended two events as part of Notre Dame's Thematic Program on Topology and Quantum Field Theory. All the talks were filmed and put on You Tube.
I taught a mini-course on Low Dimensional Higher Category Theory (Part 1) (Part 2) (Part 3) (Part 4). I also gave a lecture on The Unicity of the Homotopy Theory of Higher Categories.
On the unicity of the homotopy theory of higher categories (current version). See the description on the left under "The Comparison Problem in Higher Category Theory". Also available at arXiv:1112.0040 (Nov 30, 2011 version).
Geometry & Topology 15 (2011) 609-676.
Also available in preprint form at arXix:0911.2483. This paper classifies a certain interesting class of smooth 2-groups in terms of a cohomology theory for topological groups invented by G. Segal in the late 60's. This is probably the largest natural class of 2-group extensions which can be understood in classical terms, without passing to a 2-categorical form of group cohomology. As a consequence we construct the first finite dimensional model of the String group as an central extension of 2-groups in smooth stacks. The entire bicategory of extensions can be understood in terms of this cohomology, and from this we prove a that the String extension is in fact unique in the strongest possible sense.
Ph.D. Dissertation. The Classification of Two-Dimensional Extended Topological Field Theories. (208 pages, 107 figures, with an index).
In this work I proved the cobordism hypothesis in dimension two. Specifically, this workclassifies 2-dimensional extended topological field theories in terms of generators and relations. In this context, a topological field theory is a functor from a bordism category to some target category and an extended field theory is a higher categorical version of this. Why use higher categories? They allow you to mathematically encode the locality of the theory, something which is predicted from the physical point of view and which is very helpful mathematically.
The methods combined algebraic results on symmetric monoidal bicategories with a generalization of Cerf theory. This provides and alternate approach to the one given by Hopkins and Lurie.
This has lead to new joint work with Christopher Douglas, Bruce Barlett, and Jamie Vicary where we extend these results to higher dimensions. I've given several talks on this work, and it was featured on John Baez's This Week's Finds in Mathematical Physics (Week 275).
Houston Journal of Mathematics 30 (2004), no. 1, 55--87.Weiqing Gu.
In this work, which lead to my undergraduate thesis, I studied Cayley 4-submanifolds of 8-dimensional Euclidean space. These are calibrated submanifolds, and play a role in string theory and its generalizations. The main technique was to use 3-dimensional subgroup of Spin(7) to simplify and solve the PDEs defining these Cayley 4-manifolds.
Not for publication:
"Exotic Holonomy and Symplectic Connections" . This is an expository term paper I wrote for Alan Weinstein's symplectic geometry class in the Fall of 2005.
Notes on Smooth Group Cohomology. [Warning: This is known to contain a couple errors. I am working on fixing them. Caveat Lector] These are some notes on Segal's derived functor cohomology of topological groups. Recently, ideas from this have been used and expanded upon by work of Friedrich Wagermann and Christoph Wockel (arxiv:1110.3304).