Advisor: Peter Teichner

Research:
I am working on the Stolz-Teichner program attempting to directly relate super quantum field theories and elliptic cohomology. What I like about this is that it seems to meld two radically different subjects; what do things like CFTs and String Theory have to do with (esoteric?) generalized cohomolgy theories coming out of homotopy theory? Well some of the ideas are discussed in the paper "What is an Elliptic Object?" by my advisor and Stephan Stolz . What I love most about this subject is that it allows me to dip my fingers into all sorts of other seemingly unrelated mathematics: Higher categories, derived algebraic geometry, quantum groups, groupoids, quantization, stable homotopy, von Neumann algebras, subfactors, Knot homologies, TQFTs, and vertex operator algebras to name just a few. You can read a (slightly) more detailed explanation below.
Office: 1060 Evans Hall
Office hours: Wednesday 9-10am in 1060 Evans. Discussion in 939 Evans Friday 12-1pm.
Email address: cpries@math.easy-to-guess.edu
Snail address: 970 Evans Hall, Berkeley, CA 94720

Teaching: Durring Spring 2008, I am the TA for M253 Homological Algebra, taught by Peter Teichner.

Recent/Upcoming Talks:
Conferences: I'll will be attending or have recently attended these conferences and workshops... Papers:
More detailed explanation of my interests:
I work on the Stolz-Teichner program attempting to find a geometric description of elliptic cohomology via conformal field theories. I like this because it gives me an excuse to be interested in so many different areas of mathematics. On the one hand there is elliptic cohomology, or more precisely topological modular forms (tmf). This is a generalized cohomology theory introduced by Hopkins and Miller whose existence and properties can be proven using homotopy theoretic methods or more recently using Jacob Lurie's methods from "brave new algebraic geometry" (algebraic geometry techniques where one replaces rings with spectra). Many other generalized cohomology theories have alternate descriptions in terms of geometric objects. For example ordinary cohomology can be described by singular cocycles or perhaps differential forms, K-theory can be described in terms of vector bundles, and bordism-type theories can be described by, well, bordisms. Elliptic cohomology on the other hand has no such description yet.

One of the fascinating aspects of this project is the seemingly mysterious connection to physics. For example the Witten genus ( which is non-rigorously defined in terms of field theories) naturally takes values in tmf. Moreover there has been a proposal by Graeme Segal to view geometric elliptic objects as conformal field theories in a space X. What does this mean? Well Graeme Segal has also introduced a notion of conformal field theory (CFT) which shares much of the formalism of Atiyah's topological quantum field theory (TQFT). Whereas a TQFT is a functor from the category whose objects are d-dimensional manifolds and morphisms are (d+1)-dimensional bordisms (The bordism category) to the category of vector spaces and linear maps, a CFT in Segal's sense is a functor from a slightly different bordism category to the category of Hilbert spaces and linear maps. In particular Segal's bordism category has objects 1-manifolds and morphisms are conformal bordisms (hence the name conformal). Now a CFT in a space X again changes the bordism category so that it's objects are 1-manifolds with a map to X and it's morphisms are conformal bordisms with maps to X.

One problem with Segal's proposal is it doesn't obviously satisfy excision. Essentially this is because if a circle lies in the union of two open sets, it can't always be decomposed into circles lying in each individual open set. Stephan Stolz and my advisor Peter Teichner have proposed a two-pronged way to remedy this. The first prong of their proposal is to add a higher categorical aspect to the CFT. Instead of just a category on the bordism side there should be a 2-category whose objects are 0-manifolds, 1-morphisms are 1-manifolds and 2-morphisms are conformal bordisms. Moreover the target should also be a 2-category, the category of von Neumann algebras, bimodules and intertwiners. The second prong is to do everything supersymmetrically. This ensures that (like tmf) there is a well defined map to the ring of modular forms.

I have been primarily focusing on higher categorical aspects of their proposal. In particular, I've been attempting to come up with examples of these "extended CFTs". This has lead me through all kinds of interrelated mathematics, and many related constructions involve not just 2-categories, but 3-categories and even higher. Also there are well known relationships classical CFTs and 2+1 TQFTs and Hopf algebras. Similar relationships should exist between extended CFTs and "extended TQFTs". Also the von Neumann algebra aspect leads to connections with the study of subfactors, which in turn are related to both traditional CFTs and TQFTs.
Here is a list of the courses I've taken.

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