Eckhard Meinrenken has been teaching a course at Toronto on Lie groups and Clifford algebras, and he has lecture notes available. This is beautiful mathematics and brings together several of my favorite mathematical topics: Clifford algebras, spinors, and representation theory of Lie groups. Some of this material is quite new, and very possibly has interesting applications in physics.
Dennis Gaitsgory, a new young tenured member of the Harvard math department, has been teaching a course on Geometric Representation Theory, and also has lecture notes available. These explain the “D-module” point of view on the subject.
My Columbia math department colleague D. H. Phong and fellow ex-Princeton grad student Eric D’ Hoker have a new paper on the arXiv called Complex Geometry and Supergeometry. It is based on Phong’s recent lecture at this year’s Current Developments in Mathematics conference at Harvard last month. D’ Hoker and Phong have been able to explicitly write down and show finiteness of superstring amplitudes at two loops, with the problem still remaining open at higher loops.
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