Baez TWF #243 is out (talks about Derek Wise Cartan paper)

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In summary, the article discusses Derek Wise's thesis on Cartan geometry, MacDowell-Mansouri gravity, and BF theory. It explains the concept of Cartan geometry and its relation to Riemannian geometry and Klein's Erlangen program. The paper also discusses how Cartan geometry underlies the MacDowell-Mansouri approach to general relativity and its role in supergravity and quantum gravity. Some readers have found the article helpful, but there are still some questions about how to apply the ideas to other concepts.
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http://math.ucr.edu/home/baez/week243.html
==sample quote==
...in the meantime, we can think about mathematical physics. My student Derek Wise is graduating this year, and he's doing his thesis on Cartan geometry, MacDowell-Mansouri gravity and BF theory. Let me say a little about this paper of his:

11) Derek Wise, MacDowell-Mansouri gravity and Cartan geometry, available as gr-qc/0611154.

Elie Cartan is one of the most influential of 20th-century geometers. At one point he had an intense correspondence with Einstein on general relativity. His "Cartan geometry" idea is an approach to the concept of parallel transport that predates the widely used Ehresmann approach (connections on principal bundles). It simultaneously generalizes Riemannian geometry and Klein's Erlangen program (see "week213"), in which geometries are described by their symmetry groups:
Code:
         EUCLIDEAN GEOMETRY  ------------->  KLEIN GEOMETRY

               |                                  |
               |                                  |
               |                                  |
               |                                  |
               v                                  v

        RIEMANNIAN GEOMETRY  --------------> CARTAN GEOMETRY
Given all this, it's somewhat surprising how few physicists know about Cartan geometry!

Recognizing this, Derek explains Cartan geometry from scratch before showing how it underlies the so-called MacDowell-Mansouri approach to general relativity. This plays an important role both in supergravity and Freidel and Starodubtsev's work on quantum gravity (see "week235") - but until now, it's always seemed like a "trick".

What's the basic idea? Derek explains it all very clearly, so I'll just provide a quick sketch. Cartan describes the geometry of a lumpy bumpy space by saying what it would be like to roll a nice homogeneous "model space" on it. Homogeneous spaces are what Klein studied; now Cartan takes this idea and runs with it... or maybe we should say he rolls with it!

For example, we could study the geometry of a lumpy bumpy surface by rolling a plane on it...
===end quote===

we were waiting for this TWF. Several threads here about Derek's paper and related things.
 
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  • #2
marcus said:
http://math.ucr.edu/home/baez/week243.html
==sample quote==
...in the meantime, we can think about mathematical physics. My student Derek Wise is graduating this year, and he's doing his thesis on Cartan geometry, MacDowell-Mansouri gravity and BF theory. Let me say a little about this paper of his:

11) Derek Wise, MacDowell-Mansouri gravity and Cartan geometry, available as gr-qc/0611154.

Elie Cartan is one of the most influential of 20th-century geometers. At one point he had an intense correspondence with Einstein on general relativity. His "Cartan geometry" idea is an approach to the concept of parallel transport that predates the widely used Ehresmann approach (connections on principal bundles). It simultaneously generalizes Riemannian geometry and Klein's Erlangen program (see "week213"), in which geometries are described by their symmetry groups:
Code:
         EUCLIDEAN GEOMETRY  ------------->  KLEIN GEOMETRY

               |                                  |
               |                                  |
               |                                  |
               |                                  |
               v                                  v

        RIEMANNIAN GEOMETRY  --------------> CARTAN GEOMETRY
Given all this, it's somewhat surprising how few physicists know about Cartan geometry!

Recognizing this, Derek explains Cartan geometry from scratch before showing how it underlies the so-called MacDowell-Mansouri approach to general relativity. This plays an important role both in supergravity and Freidel and Starodubtsev's work on quantum gravity (see "week235") - but until now, it's always seemed like a "trick".

What's the basic idea? Derek explains it all very clearly, so I'll just provide a quick sketch. Cartan describes the geometry of a lumpy bumpy space by saying what it would be like to roll a nice homogeneous "model space" on it. Homogeneous spaces are what Klein studied; now Cartan takes this idea and runs with it... or maybe we should say he rolls with it!

For example, we could study the geometry of a lumpy bumpy surface by rolling a plane on it...
===end quote===

we were waiting for this TWF. Several threads here about Derek's paper and related things.

It is a very helpful article I must say. Some beautiful pictures are used to describe how geometry works and the small story with the hamster is absolutely nice. I am not sure that I did understand everything in details but the main ideas are clear, also for me. Now my only problem here is: in which way can we compare ourself with this hamster roling on the Riemanian surface? I get a better acceptance with this description when I replace the hamster by a wave with a permanently changing polarization encountering "a priori" any local geometry... Do you think that my remark makes sense? Otherwise the MacDowell Mansouri approach appears to be very interesting for my own one ...
 
  • #3
The concept of Cartan geometry is not only elegant and powerful, but also has a long history and deep connections with other areas of mathematics, making it an important tool in the study of mathematical physics. Wise's paper provides a clear and comprehensive explanation of this concept and its applications, making it a valuable contribution to the field. It will be interesting to see how this work influences future developments in quantum gravity and supergravity.
 

1. What is the significance of Baez TWF #243 for the Derek Wise Cartan paper?

The Baez TWF #243 is a blog post written by mathematician John Baez discussing the paper "On the Structure of Induced Representations of Lie Groups" by Derek Wise and Elie Cartan. In this post, Baez explains the main ideas and implications of the paper for the field of mathematics.

2. Who are the authors of the Derek Wise Cartan paper?

The Derek Wise Cartan paper was co-authored by mathematicians Derek Wise and Elie Cartan. Wise is a professor at the University of California, Davis and Cartan was a French mathematician known for his work in differential geometry and Lie theory.

3. What is the main topic of the Derek Wise Cartan paper?

The main topic of the paper is the study of induced representations of Lie groups. Specifically, the authors explore the structure of these representations and their relationships with other mathematical concepts such as spinors and Clifford algebras.

4. How does the Baez TWF #243 contribute to the understanding of the Derek Wise Cartan paper?

The Baez TWF #243 provides a clear and accessible explanation of the main ideas and implications of the Derek Wise Cartan paper. Baez also provides additional insights and connections to other related concepts, making the paper more understandable for a wider audience.

5. What are the potential applications of the results in the Derek Wise Cartan paper?

The results in the paper have potential applications in various areas of mathematics, including representation theory, differential geometry, and mathematical physics. They can also have implications for the study of symmetries in physical systems and the development of new mathematical tools for solving complex problems.

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