# Spin networks, spin foams and loop quantum gravity

Loop quantum gravity is a theory that results from the canonical quantization of general relativity. A covariant (path integral) formulation of the theory is expected to be in the form of a spin foam model, such as the Barrett-Crane model. Spin networks are diagrams which provide a basis for the Hilbert spaces of these theories. They also describe representation-theoretic computations that are used in computing amplitudes. This page is a collection of links to information about loop quantum gravity, spin foams, and spin networks, with some focus on parts relevant for extracting predictions from the theory.

I haven't updated this page in a long time, but many of the links will still be of use.

## The group at Western

Dan Christensen (faculty)

Jeff Morton (postdoctoral fellow, summer 2007 to summer 2010)

Wade Cherrington (M.Sc. student, Jan 2002 to Dec 2004; Ph.D. student, Jan 2005 to April 2009; postdoc May 2009 to April 2011)

Igor Khavkine (Ph.D. student, Sept 2004 to October 2008)
Josh Willis (postdoctoral fellow, Sept 2004 to July 2006)
Eli Hawkins (postdoctoral fellow, Aug 2004 to Aug 2005)
Michael de Jong (M.Sc. student, Jan 2003 to Dec 2004)

## Overview of our work

The project involves both theoretical and computational work with discrete models of quantum gravity, an exciting and active area of current research. One nice aspect of this approach is that it has connections to lots of great mathematics (in particular, diagrammatic and computational methods in representation theory, as well as topology) and also to physics (the model has all the traditional aspects of a quantum field theory, such as renormalization issues, running of the coupling constants, a classical limit, etc) so it is a good way to learn about these fields in a hands-on manner. The theoretical challenges are significant, in a large part because we are studying background independent models, leading to the problem of time. From a computational point of view, the methods that are needed to handle even the smallest examples are interesting in their own right. Thus some projects involve parallel programming on a Beowulf cluster. Parallel computation is an area that is becoming very important with the trend towards supercomputers built as large clusters of smaller machines.

I gave a talk summarizing of our past work.

Quantum gravity is one of the focus areas at the Perimeter Institute in Waterloo, Ontario, Canada, which has recruited Lee Smolin as well as several other influential researchers in quantum gravity. Lee Smolin has written a nice article about the founding of the Perimeter Institute, and about the relationship between string theory and loop quantum gravity. And here is an earlier article by Smolin about the founding of Perimeter.

## Students and postdocs

While my work is currently more focused on mathematics, I still consider supervising graduate students on topics related to quantum gravity or more general mathematical physics, especially students with a mathematical background.

At the postdoctoral level, I encourage eligible people to apply for an NSERC Postdoctoral Fellowship, an NSERC - NATO Science Fellowship, an NSF PDF (e.g. a Mathematical Sciences Postdoctoral Research Fellowships or a Distinguished International Postdoctoral Research Fellowship) or similar to be held under my supervision. At the graduate level, I can supervise students in the Department of Mathematics, the Department of Applied Mathematics and the Program in Theoretical Physics at Western. Please contact me if you are interested.

## Resources

Note: I haven't kept this page up to date recently. You may find additional papers on the list of my publications.

#### Articles involving our group

J. Daniel Christensen and Igor Khavkine. q-deformed spin foam models of quantum gravity. arXiv:0704.0278 [gr-qc].

J. Wade Cherrington. Finiteness and Dual Variables for Lorentzian Spin Foam Models. arXiv:gr-qc/0508088.

J. Daniel Christensen. Finiteness of Lorentzian 10j symbols and partitions functions. gr-qc/0512004.

J. Wade Cherrington and J. Daniel Christensen. Positivity in Lorentzian Barrett-Crane Models of Quantum Gravity.

Louis Crane and J. Daniel Christensen. Causal sites as quantum geometry.

John Baez, J. Daniel Christensen and Greg Egan. Asymptotics of 10j symbols.

John Baez, J. Daniel Christensen, Thomas Halford and David Tsang, Spin foam models of Riemannian quantum gravity.

John Baez and J. Daniel Christensen, Positivity of spin foam amplitudes.

J. Daniel Christensen and Greg Egan, An efficient algorithm for the Riemannian 10j symbols.

#### Code

A key ingredient of the spin foam models is a quantity called a 10j symbol. Greg Egan and I have developed a very efficient algorithm for computing Riemannian 10j symbols. You can read our paper describing this algorithm and you can download the code, either as a tar archive or as individual files.

#### Sample 10j symbols

You can view a table of Riemannian 10j symbols and a table of Lorentzian 10j symbols.

#### John Baez's This Week's Finds

John Baez's series of articles called This week's finds in mathematical physics is an excellent source of information. Here are some of the most relevant issues:

week 110 discusses Penrose's spin networks. A spin network is supposed to be a model of space, while a spin foam is suppose to model the evolution of space from one time to another, and so be like spacetime.

week 113 is a good starting point for spin foams.

week 128: entries 4 through 8 are most closely connected.

week 122 contains information about computer approaches, and connections to mathematical logic. See in particular entries 5 through 8. The idea of exploring the state space by making random moves is discussed, and this idea could be important for efficient computations. This week also contains references to good survey papers by Rovelli (items 2 and 4) and Loll (item 5, a review of over 200 papers studying discrete quantum gravity, predating spin foams).

week 168 describes the progress we have made with our own computational approach!

week 120 contains more information, which may make more sense after reading week 113.

week 134 is more of an overview of the various approaches to quantum gravity. It also contains a pointer to Baez's work on the quantum tetrahedron.

week 114 is a report on a conference at which spin foams were discussed.

week 140 is a somewhat more advanced look at some of the issues, and includes a bit more information about quantum triangles and quantum tetrahedra.

week55 discusses the issue of constraints, and talks about whether loops should be analytic or just smooth.

week56 discusses Kodama's Chern-Simon's state.

week57 discusses Smolin's paper Linking topological quantum field theory and nonperturbative quantum gravity about Chern-Simon's theory and loop quantum gravity.

week112 discusses the calculation of the entropy of black holes using loop quantum gravity.

#### Other things from Baez's web site

This note describes what it means for a theory to be background free. This is one of the main goals of this approach to quantum gravity.

One of the basic difficulties might be that various sums turn out to be infinite. To deal with this, we should understand renormalization. More about renormalization is in week 139. See also this posting about phase transitions, and the thread it is in.

The slides from a lecture John Baez gave on spin foams are available.

A non-technical introduction to C*-algebras in physics, with some good references for further reading.

The following google search gives more related articles from Baez's web site.

#### A random list of more formal references

(This list is far from being exhaustive. There are lots of excellent papers and books that I haven't got around to adding.)

#### Introductions, surveys and textbooks

Alejandro Corichi, Alberto Hauser. Bibliography of Publications related to Classical Self-dual variables and Loop Quantum Gravity, 2005. ps, pdf. Very thorough and complete!

Abhay Ashtekar, Jerzy Lewandowski. Background Independent Quantum Gravity: A Status Report, 2004. ps, pdf.

John C. Baez. An Introduction to Spin Foam Models of BF Theory and Quantum Gravity, 1999. What John refers to as his "big fat spin foam review article".

Thomas Thiemann. Introduction to Modern Canonical Quantum General Relativity, 2001, 301 pages. A detailed introduction. Also, Lectures on Loop Quantum Gravity, 2002, 90 pages.

Alejandro Perez. Spin Foam Models for Quantum Gravity, 2003. ps, pdf.

Seth Major. A Spin Network Primer, 1999.

Carlo Rovelli. Loop Quantum Gravity, 1998, 75 pages.

Carlo Rovelli, Quantum Gravity (draft of a book), pdf. 2003, 249 pages.

Daniele Oriti. Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity, 2001, 68 pages. A thorough and recent survey article.

John C. Baez and Javier P. Muniain. Gauge Fields, Knots and Gravity. World Scientific, 1994. A great introduction.

See also Chris Hillman's guide to books on relativity.

#### Other reading lists

Seth Major's Reading guide to quantum gravity.

There is a LQG group at National University of Singapore.

#### Research papers which go further

Carlo Rovelli and Lee Smolin. Loop representation for quantum general relativity, Nucl. Phys. B331 (1990), 80-152.

Louis Crane, Louis H. Kauffman, David N. Yetter. State-Sum Invariants of 4-Manifolds I, 1994. ps, pdf.

J. Fernando, G. Barbero. Real Ashtekar Variables for Lorentzian Signature Space-times, 1994.

John W. Barrett, Louis Crane. Relativistic spin networks and quantum gravity, 1997. ps, pdf.

John W. Barrett. The classical evaluation of relativistic spin networks, 1998. ps, pdf.

John W. Barrett, Ruth M. Williams. The asymptotics of an amplitude for the 4-simplex, 1998. ps, pdf.

David N. Yetter. Generalized Barrett-Crane Vertices and Invariants of Embedded Graphs, 1998. ps, pdf.

John W. Barrett, Louis Crane. A Lorentzian Signature Model for Quantum General Relativity, 1999. ps, pdf.

Robert Oeckl, Hendryk Pfeiffer. The dual of pure non-Abelian lattice gauge theory as a spin foam model, 2000. ps, pdf.

Hendryk Pfeiffer. Four-dimensional Lattice Gauge Theory with ribbon categories and the Crane-Yetter state sum, 2001. ps, pdf.

Hendryk Pfeiffer. Dual variables and a connection picture for the Euclidean Barrett-Crane model, 2001. ps, pdf.

John C. Baez, John W. Barrett. Integrability for Relativistic Spin Networks, 2001. ps, pdf.

Louis Crane, Alejandro Perez, Carlo Rovelli. A finiteness proof for the Lorentzian state sum spinfoam model for quantum general relativity, 2001. ps, pdf. A more concise published version of this appeared in Phys. Rev. Lett 87, 2001.

Laurent Freidel, David Louapre. Asymptotics of 6j and 10j symbols, 2002. ps, pdf.

John W Barrett, Christopher M Steele. Asymptotics of Relativistic Spin Networks, 2002. ps, pdf.

Lee Smolin. Quantum gravity with a positive cosmological constant, 2002. ps, pdf.

Karim Noui, Philippe Roche. Cosmological Deformation of Lorentzian Spin Foam Models, 2002. ps, pdf.

Laurent Freidel, David Louapre. Diffeomorphisms and spin foam models, 2002. ps, pdf. Argues that partition function should be divergent, because some infinite volume diffeomorphism invariance remains.

Louis Crane, David Yetter. A More Sensitive Lorentzian State Sum, 2003. ps, pdf.

Louise Crane, David Yetter. Measurable Categories and 2-Groups, 2003. ps, pdf.

Louis Crane, Marnie Sheppeard. 2-categorical Poincare Representations and State Sum Applications, 2003. ps, pdf.

Etera R. Livine, Alejandro Perez, Carlo Rovelli. 2d manifold-independent spinfoam theory, 2001/2003. ps, pdf.

Yuka U. Taylor, Christopher T. Woodward. 6j symbols for Uq(sl2) and non-Euclidean tetrahedra, 2003.

Martin Bojowald, Alejandro Perez. Spin Foam Quantization and Anomalies, 2003. ps, pdf. Includes discussion of the gauge fixing and divergence of partition function (see previous paper).

Florian Conrady. Geometric spin foams, Yang-Mills theory and background-independent models, 2005. ps, pdf. Suggests that spin foam amplitudes should only depend on the geometry of the spin foam as a surface, and not on its decomposition into faces. Shows that one version of the Riemannian Barrett-Crane model satisfies this. Also includes a discussion of lattice gauge theory and a review of relevant topics.

Wade Cherrington. Finiteness and Dual Variables for Lorentzian Spin Foam Models, 2005. ps, pdf.

#### Diagrammatic methods in representation theory

J. Scott Carter, Daniel E. Flath, Masahico Saito. The classical and quantum 6j-symbols. Princeton University Press, 1995. A good book on diagrammatic methods for SU(2) representation theory.

Predrag Cvitanović. Group Theory: Birdtracks, Lie's, and Exceptional Groups. Online.

#### Deriving a spin foam model from a group field theory

R. De Pietri, L. Freidel, K. Krasnov, C. Rovelli. Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space, 1999. ps, pdf.

Michael Reisenberger, Carlo Rovelli. Spin foams as Feynman diagrams, 2000. ps, pdf.

Alejandro Perez, Carlo Rovelli. A spin foam model without bubble divergences, 2000. ps, pdf.

Alejandro Perez, Carlo Rovelli. Spin foam model for Lorentzian General Relativity, 2000. ps, pdf.

Alejandro Perez. Finiteness of a spinfoam model for euclidean quantum general relativity, 2000. ps, pdf.

Alejandro Perez, Carlo Rovelli. 3+1 spinfoam model of quantum gravity with spacelike and timelike components, 2000. ps, pdf.

Alejandro Perez. Spin Foam Models for Quantum Gravity, 2003. ps, pdf.

#### Causal models, causal sets

Rafael D. Sorkin. Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School), 2003. ps, pdf.

Fotini Markopoulou, Lee Smolin. Causal evolution of spin networks, 1997. ps, pdf. A family of models based on spin foams and causal sets.

Fotini Markopoulou. Dual formulation of spin network evolution, 1997. ps, pdf.

Fotini Markopoulou, Lee Smolin. Quantum geometry with intrinsic local causality, 1997. ps, pdf.

Fotini Markopoulou. The internal description of a causal set: What the universe looks like from the inside, 1998. ps, pdf.

Fotini Markopoulou. Quantum causal histories, 1999. ps, pdf.

Fotini Markopoulou. An insider's guide to quantum causal histories, 1999. ps, pdf.

Eli Hawkins, Fotini Markopoulou, Hanno Sahlmann. Evolution in Quantum Causal Histories, 2003. ps, pdf.

Etera R. Livine, Daniele Oriti. Implementing causality in the spin foam quantum geometry, 2002. ps, pdf. A particular causal model. 39 pages.

Etera R. Livine, Daniele Oriti. Causality in spin foam models for quantum gravity, 2003. ps, pdf. 6 pages.

Hendryk Pfeiffer. On the causal Barrett--Crane model: measure, coupling constant, Wick rotation, symmetries and observables, 2002. ps, pdf. A different causal model.

David Rideout. Dynamics of Causal Sets, 2002. ps, pdf.

#### Dynamical Triangulations

R. Loll, J. Ambjorn, J. Jurkiewicz. The Universe from Scratch, 2005. ps, pdf.

J. Ambjorn, J. Jurkiewicz, R. Loll. Reconstructing the Universe, 2005. ps, pdf. Details about their positive results.

J. Ambjorn, J. Jurkiewicz, R. Loll. Emergence of a 4D World from Causal Quantum Gravity, 2004. ps, pdf. Announcement of some positive results.

J. Ambjorn, A. Dasgupta, J. Jurkiewicz, R. Loll. A Lorentzian cure for Euclidean troubles, 2002. ps, pdf.

J. Ambjorn. Simplicial Euclidean and Lorentzian Quantum Gravity, 2002. ps, pdf.

J. Ambjorn, J. Jurkiewicz, R. Loll. 3d Lorentzian, Dynamically Triangulated Quantum Gravity, 2002. ps, pdf.

J. Ambjorn, J. Jurkiewicz, R. Loll. Dynamically Triangulating Lorentzian Quantum Gravity, 2001. ps, pdf.

J. Ambjorn, J. Jurkiewicz, R. Loll. Lorentzian and Euclidean Quantum Gravity - Analytical and Numerical Results, 2000. ps, pdf.

#### Loop quantum cosmology

Martin Bojowald. Loop Quantum Cosmology: Recent Progress, 2004. ps, pdf.

Abhay Ashtekar, Martin Bojowald, Jerzy Lewandowski. Mathematical structure of loop quantum cosmology, 2003. ps, pdf.

Martin Bojowald, Kevin Vandersloot. Loop Quantum Cosmology, Boundary Proposals, and Inflation, 2003. ps, pdf.

Daniel Cartin, Gaurav Khanna, Martin Bojowald. Generating function techniques for loop quantum cosmology, 2004. ps, pdf.

Parampreet Singh, Alexey Toporensky. Big Crunch Avoidance in k = 1 Semi-Classical Loop Quantum Cosmology, 2003. ps, pdf.

Martin Bojowald, Kevin Vandersloot. Loop Quantum Cosmology and Boundary Proposals, 2003. ps, pdf.

Franz Hinterleitner, Seth Major. Isotropic Loop Quantum Cosmology with Matter II: The Lorentzian Constraint, 2003. ps, pdf.

V. Husain, O. Winkler. On Singularity Resolution in Quantum Gravity, 2003. ps, pdf.

Martin Bojowald, James E. Lidsey, David J. Mulryne, Parampreet Singh, Reza Tavakol. Inflationary Cosmology and Quantization Ambiguities in Semi-Classical Loop Quantum Gravity, 2004. ps, pdf.

#### Quantization

Paul A. M. Dirac, Lectures on Quantum Mechanics, Dover, 1964.

Sanjeev Seahra, The Classical and Quantum Mechanics of Systems with Constraints, 2002.

Victor Guillemin, Shlomo Sternberg, Sympectic Techniques in Physics, Cambridge University Press, Cambridge, 1984. This contains a section on the Groenewold-Van Hove theorem.

Paul Chernoff, Mathematical obstructions to quantization, 1980.

S. Twareque Ali, Miroslav Englis. Quantization Methods: A Guide for Physicists and Analysts, 2004. ps, pdf.

D. J. Simms, N. M. J. Woodhouse. Lectures on geometric quantization, Springer-Verlag, 1976.

Stephen Summers, On the Stone-von Neumann uniqueness theorem and its Ramifications, 1998.

Jonathan Rosenberg, A Selective History of the Stone-von Neumann Theorem, 2003.

C. J. Isham, Topological and global aspects of quantum theory, Les Houches 1983 Session XL: Relativity, Groups, and Topology II, North Holland.

Gerard Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley Interscience, 1972.

#### Immirzi parameter

Giorgio Immirzi. Real and complex connections for canonical gravity, 1996.

Alejandro Corichi, Kirill Krasnov. Loop Quantization of Maxwell Theory and Electric Charge Quantization, 1997. ps, pdf.

Carlo Rovelli, Thomas Thiemann. The Immirzi parameter in quantum general relativity, 1997. ps, pdf. A nice overview of the Immirzi parameter.

R. Gambini, O. Obregon, J. Pullin. Yang-Mills analogues of the Immirzi ambiguity, 1998.

Richard E. Livine. Immirzi parameter in the Barrett-Crane model?, 2001.

Matthias Arnsdorf. Relating Covariant and Canonical Approaches to Triangulated Models of Quantum Gravity, 2001.

S. Alexandrov, D. Vassilevich. Area spectrum in Lorentz covariant loop gravity, 2001.

Luis J. Garay, Guillermo A. Mena Marugan. Immirzi Ambiguity, Boosts and Conformal Frames for Black Holes, 2003.

Alejandro Perez, Carlo Rovelli. Physical effects of the Immirzi parameter, 2005. ps, pdf. Shows that in the presence of fermions, the Immirzi parameter appears in the equations of motion.

#### Ashtekar's "new" variables and the initial value formulation of GR

John C. Baez and Javier P. Muniain. Gauge Fields, Knots and Gravity. World Scientific, 1994. Chapters III.4 and III.5 talk about the ADM formulation of GR and the new variables. The rest of the book gives the necessary background.

Robert M. Wald. General Relativity. University of Chicago Press, 1984. Chapter 10 describes the initial value formulation of GR while Appendix E covers the Lagrangian and Hamiltonian formulations. This book is an excellent introduction to GR.

Abhay Ashtekar, Mathematical problems of non-perturbative quantum general relativity (lectures delivered at the 1992 Les Houches summer school on Gravitation and Quantization), 87 pages. Good intro.

New Perspectives in Canonical Gravity, lecture notes by Abhay Ashtekar and invited contributors, Bibliopolis, Napoli, Italy, 1988. (How to get it and errata.)

Abhay Ashtekar with Ranjeet Tate, Lectures on Non-perturbative Canonical Gravity, World Scientific Press, 1991. Thorough treatment.

Domenico Giulini, Ashtekar Variables in Classical General Relativity. A review article treating just the classical aspects.

Christopher Beetle and Alejandro Corichi, Bibliography of publications related to Classical and Quantum Gravity in terms of Connection and Loop Variables.

Seth Major (and Troy Schilling), Reading guide to quantum gravity.

C Rovelli, Ashtekar formulation of general relativity and loop space non-perturbative Quantum Gravity: a report, Classical and Quantum Gravity, 8, 1613-1675 (1991).

Baez, Strings, loops, knots and gauge fields, in Knots and Quantum Gravity, ed. J. Baez, Oxford U. Press, Oxford, 1994, pp. 133-168. Gives a careful treatment of the new variables starting on page 27 of the web version.

Baez's week7 gives an introduction, and week11 says more about the ADM formulation of GR and the constraints.

week37 discusses reality conditions, generalized measures and loop states.

week43 contains another intro to the Ashtekar variables, as well as lots of references to important papers.

In this 1997 s.p.r. message, Baez summarizes the Palatini and Ashtekar approaches, and variations.

#### Renormalization

Robert Oeckl (CPT). Renormalization for spin foam models of quantum gravity, 2004. ps, pdf.

Robert Oeckl (CPT). Renormalization of Discrete Models without Background, 2002. ps, pdf.

#### Connes-Kreimer Hopf algebra and renormalization

Dirk Kreimer. Renormalization and Knot Theory, 1996. 102 pages.

Dirk Kreimer. On the Hopf algebra structure of perturbative quantum field theories, 1997. 23 pages.

Dirk Kreimer. How useful can knot and number theory be for loop calculations?, 1998. 12 pages.

Alain Connes, Dirk Kreimer. Hopf Algebras, Renormalization and Noncommutative Geometry, 1998. 48 pages. Good intro.

D.J. Broadhurst, D. Kreimer. Renormalization automated by Hopf algebra, 1998. 21 pages.

Alain Connes, Dirk Kreimer. Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem, 1999. 35 pages.

Alain Connes, Dirk Kreimer. Renormalization in quantum field theory and the Riemann-Hilbert problem II: the $\beta$-function, diffeomorphisms and the renormalization group, 2000. 35 pages.

Dirk Kreimer, Knots and Feynman Diagrams, Cambridge University Press, Cambridge, 2000.

Dirk Kreimer. New mathematical structures in renormalizable quantum field theories, 2002. 26 pages.

Alain Connes, Matilde Marcolli. From Physics to Number Theory via Noncommutative Geometry, Part II: theory, 2004. ps, pdf. 97 pages, lots of background material, self-contained.

Baez:   week 122, week 123, week 125, week 179.

Other:

#### Discrete ordered calculus

Louis H. Kauffman. Non-Commutative Worlds -- A Summary, 2005. ps, pdf.

Louis H. Kauffman. Non-Commutative Worlds, 2004. ps, pdf.

Louis H. Kauffman. Non-commutative Calculus and Discrete Physics, 2003. ps, pdf.

Louis H. Kauffman. Noncommutativity and Discrete Physics, 1997. ps, pdf.

#### Misc

There are often discussions in the newsgroup sci.physics.research on spin foam models and related issues, so I encourage you to read that group and ask questions there. Here is an archive of one long discussion in this group. You can also search the archives.

Here are excerpts from an e-mail exchange with John Baez on the Barrett-Crane model.

#### Applets

Greg Egan, who writes fiction that I like a lot, has written a description of spin networks, and a Java applet that does some computations. He recently wrote a book, Schild's Ladder, which describes hypothetical physics based on spin foams and spin networks. Greg's description of decoherence is very good, and includes an interesting applet.

#### Estimating observables, Metropolis, etc

"Markov Chain Monte Carlo in Practice", edited by Gilks, Richardson and Spiegelhalter, Chapman & Hall/CRC Press, 1996. Written by statisticians, gives lots of good background about Metropolis, including both theory and methods for improving performance in practice. Examples are not physics related.

A course on Monte Carlo methods for statistical physics. Lots relevant, especially the chapter on Monte Carlo methods which contains a description of the Metropolis algorithm.

#### Parallel Computation

The Parallel Processing HOWTO.

Information about Beowulf clusters.

The Brahma beowulf cluster web site. Lots of good stuff.

A new HOWTO on building and maintaining Beowulf clusters, by Zouhir Hafidi.

Information about Western's Beowulf cluster, SHARCNET in the Applied Mathematics Department.

Information about the Message Passing Interface Standard (MPI) is available from the MPI homepage.

MPI: The Complete Reference . A book about MPI.

A user's guide to MPI.

Contact Dan Christensen at jdc@uwo.ca if you are interested in this project, or have any additions, corrections or other suggestions about the information on this page.

Dan Christensen's Home Page