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October 31, 2006
Bakovic on 2-Groupoid 2-Torsors
Posted by Urs Schreiber
Today Igor Bakovic gave a talk on his work on the categorification of the notion of principal groupoid fiber bundles.
Igor has a couple of nice results, of which his talk mentioned only few aspects and of which my report here will mention even fewer. When his thesis is finished there should be more to say.
In his talk Igor sketched the definition of torsors for bigroupoids . He indicated how a nonabelian cocycle with values in is obtained from any such torsor and how he hopes that these cocycles are in fact equivalent to -torsors.
Notice that reconstructing a 2-bundle from its local data is usually more subtle than going the other way around. Compare for instance Wirth’s work recently recalled by Stasheff.
Markl on Natural Differential Operators
Posted by Urs Schreiber
Just heard a talk by Martin Markl on
Natural Differential Operators and Graph Complexes.
He explained
- a way to make precise the idea that certain differential operators (like the Lie derivative, or the covariant derivative) are more natural than others,
- that all natural differential operators of a certain “type” arise as the 0th cohomology of a complex of graphs,
- where the graphs appearing here are like string diagrams representing the action of linear operators on tensor powers of vector spaces.
October 30, 2006
Foundations
Posted by David Corfield
Something that has long troubled me is the question of why philosophers have shown what to my mind is an unwarranted interest in ‘foundational’ mathematical theories which make little or no contact with mainstream mathematics. Two posts have made me reconsider the problem. Alexandre Borovik discusses Zilber’s work on the Schanuel conjecture, while Kenny Easwaran wonders whether string theorists may use mathematics which depends on conjectured axioms of set theory.
In comments to both I mention the model theorist Angus MacIntyre’s interest in foundational theories which do make contact with the mainstream. He personally wants to relate some of Gropthendieck’s ideas to model theory. To return to the philosophers, is their interest in disconnected theories just a vestige of earlier failed foundationalist projects, or is there a continuing rationale?
Puzzle #4
Posted by John Baez
This is the 100th blog entry at the n-Category Café! David, Urs and I thank all the people who have come together to make this a fascinating place to talk about philosophy, physics and math.
So, let’s celebrate with a puzzle.
What is the following sentence about?
“Such a brivla, built from the rafsi for the component gismu and cmavo, is called a lujvo.”
October 29, 2006
WZW as Transition 1-Gerbe of Chern-Simons 2-Gerbe
Posted by Urs Schreiber
As a kind of comment to M. Hopkins’ lecture on Chern-Simons theory (I, II, III) I want to describe how the Wess-Zumino-Witten 1-gerbe arises as the transition 1-gerbe of the Chern-Simons 2-gerbe (analogous to how a 1-gerbe itself has transition 0-gerbes, i.e. transition bundles).
I’ll advertise a point of view # where we consider the Chern-Simons 2-gerbe as a 3-bundle with structure 3-group
and use the fact that the 2-group
is #, as a groupoid with monoidal structure, nothing but the tautological bundle gerbe representation of the canonical “WZW” gerbe at level on .
October 27, 2006
Hopkins Lecture on TFT: Chern-Simons
Posted by Urs Schreiber
In the third part (following part I and part II) of his 2006 lecture series in Göttingen on topological field theory, Michael Hopkins considered the special case of 3-dimensional topological field theories characterized by classes
These are known as Chern-Simons field theories at level .
The first part of the talk reviewed some basic concepts in suitable language.
Then the seminal theorem in
D. Freed, M. Hopkins, C. Teleman
Twisted K-theory and loop group representations
math.AT/0312155,
which relates the modular tensor category encoding -Chern-Simons theory with the twisted -equivariant K-theory on - is used as a key for extracting topological information from Chern-Simons TFT 3-functors and reformulating everything in terms of K-theory.
October 26, 2006
Classical vs Quantum Computation (Week 4)
Posted by John Baez
Here are the notes for the latest installment of my course on Classical versus Quantum Computation:
- Week 4 (Oct. 26) - Currying and uncurrying, evaluation and coevaluation. Basic aspects of the "quantum lambda calculus": so far, the fragment of the lambda calculus that works in any monoidal closed category. The "name" of a morphism. Compact categories.
October 25, 2006
Lauda on Topological Field Theory and Tangle Homology
Posted by John Baez
Aaron Lauda an undergrad here at U.C. Riverside, but now he’s finished his Ph.D at Cambridge and is a postdoc at Columbia. I just got a copy of his thesis:
- Aaron Lauda, Open-closed topological quantum field theory and tangle homology, PhD thesis, Cambridge University, 2006.
It’s a great tour of these subjects!
Hopkins Lecture on TFT: Infinity-Category Definition
Posted by Urs Schreiber
In the second part of his lecture on topological field theory (notes on the first part were reproduced here) Michael Hopkins sketched what he currently sees as the emerging picture for the -tiered (aka “extended”) formulation of the definition of topological (quantum) field theory.
As I mentioned last time, in this picture one wants to refine the standard formulation in terms of 1-functors
by passing to -functors into something like -vector spaces.
In order to do so, M. Hopkins reviewed rudiments of the definition of weak -categories in terms of complicial sets, due to Street and Verity.
Ross Street
Weak omega-categories
(pdf).
He then sketched how he imagines forming an -category of -manifolds, , such that together with a suitable -category with an ring structure, one would say that
An -tiered -dimensional topological field theory is a morphism of -categories
(2)
Hopkins Lecture on TFT: Introduction and Outlook
Posted by Urs Schreiber
I am currently in Göttingen, attending the lecture series by M. Hopkins that I mentioned recently #, called
Topological Aspects of Topological Field Theory.
Yesterday we heard the first of three parts, titled Introduction to topological field theories. Michael Hopkins mentioned a couple of basic notions and concepts of topological field theories and, in closing, briefly indicated some of the more profound issues concerning the relation between Chern-Simons theory, categorification, K-theory # and elliptic cohomology #.
Here is a transcript of notes I have taken during this first lecture.
Puzzle #3
Posted by John Baez
For which 1998 referendum in Washington D.C. did the United States Congress pass a special bill to prevent the votes from even being counted?
Extra credit: do you know other cases where special laws were enacted to prevent votes from being counted?
October 23, 2006
Knowledge of the Reasoned Fact
Posted by David Corfield
In a comment I raised the question of what to make of our expectation that behind different manifestations of an entity there is one base account, of which these manifestations are consequences.
If I point out to you three manifestations of the normal distribution - central limit theorem; maximum entropy distribution with fixed first two moments; approached by distribution which is the projection onto 1 dimension of a uniform distribution over the -sphere of radius as increases - it’s hard not to imagine that there’s a unified story behind the scenes.
Perhaps it would encourage a discussion if I put things in a contextual setting.
This Week’s Finds in Mathematical Physics (Week 240)
Posted by John Baez
October 20, 2006
Classical vs Quantum Computation (Week 3)
Posted by John Baez
This week we had a guest lecture in our course on Classical versus Quantum Computation:
-
Week 3 (Oct. 19) - Guest lecture by James Dolan: Holodeck strategies and cartesian closed categories.
- James Dolan, Holodeck strategies and the lambda-calculus (a continuation of the above lecture).
- Todd Trimble, Holodeck games and CCCs (rigorous definitions and proofs).
If you read this stuff you’ll learn what Star Trek has to do with categories - and you’ll learn how to play