Andrea Oct 26, 2012 Do you mind doing a lecture on some basic topology, specifically homology groups and their physical content some time in the future (not necessarily soon) for math methods? I have taken some topology classes which seemed interesting but we never covered any physical significance. -------------------------------------------------------------------------- Alberto Fernandez De Las Nieves Some diff. geometry (curves and surfaces) and some topology would be great. Gauss-Bonnet, Euler theorem -------------------------------------------------------------------------- Shina Tan In my course I mentioned Lorentz group (for relativity), symplectic group (for canonical transformations), etc. Perhaps some mathematical aspects about these groups are interesting to students. -------------------------------------------------------------------------- Brian Kennedy For QM some homotopy rather than homology would be useful (see #2) : 1. SU(2) and S0(3): quotient spaces, covering spaces. 2. Fundamental groups for Aharonov-Bohm: e.g. $\Pi_1(S^1) = Z = \Pi_1(R^2 - {0}); \Pi_1(R^2) = 0; \Pi_1(T^2) = Z x Z$. 3. Dirac monopole - Wu-Yang approach. 4. Berry's phase: holonomy. -------------------------------------------------------------------------- Jean Bellissard Chern Numbers and at least the 2nd Chern class. Applications are, indeed Berry's phase, but also Quantum Hall Effect, the Chern-Simon field theories. Index Theory (Atiyah-Singer) is based upon computing these Chern classes. It had a huge impact in 1976 when the first calculation of instantons by Polyakov and t'Hooft was shown to be incomplete by Atiyah and co-authors, using the Index formula. Homotopy groups are also used for classification of defects (Bouligand-Toulouse-Poenaru 1979) in various fields, from crystalline solids, liquid crystals, stastitical mechanics, morphogenesis in biology, and the like. There is also now the huge Michaikow industry of computational cohomology to identify the singularities of orbits, the Conley index of various dynamical systems. In Gatech Michael Schatz and Alan Tannenbaum are involved (see http://chomp.rutgers.edu/people/) It would also be unfair if I would not mention the BRST cohomology to classify the anomalies in non abelian gauge field theories, since I shared my office with Rouet between 1971 and 1974, having the pleasure of inhaling the smoke of his pipe for years, risking my life, at the time when Carlo Becchi, Alain Rouet and Raymond Stora were building this BRST stuff that became seminal apparently in applying the standard model to fit the experimental results a obtained later when the vector boson were produced at CERN. If I think of other examples I'll let you know. I use a lot of cohomology to classify aperiodic solids, more precisely their tiling space (see http://people.math.gatech.edu/~jeanbel/talksjbE.html "On the Noncommutative Geometry of Aperiodic Solids" INDAM Meeting Noncommutative Geometry, Index Theory and Applications, Cortona, Italy, June 11-15 2012.) Well, this exercise made me realize that there is much more algebraic topology nowadays in Physics than I ever imagined. Good luck! -------------------------------------------------------------------------- file: PHYS-6124-12/topologyMM.txt svn: $Author: predrag $ $Date: 2012-11-27 09:20:43 -0500 (Tue, 27 Nov 2012) $ --------------------------------------------------------