Algebraic Topology: The Abel Symposium 2007 by John C. Baez, Danny Stevenson (auth.), Nils Baas, Eric M.

By John C. Baez, Danny Stevenson (auth.), Nils Baas, Eric M. Friedlander, Björn Jahren, Paul Arne Østvær (eds.)

The 2007 Abel Symposium happened on the college of Oslo in August 2007. The objective of the symposium was once to collect mathematicians whose examine efforts have resulted in contemporary advances in algebraic geometry, algebraic K-theory, algebraic topology, and mathematical physics. a standard topic of this symposium was once the advance of latest views and new structures with a express style. because the lectures on the symposium and the papers of this quantity exhibit, those views and buildings have enabled a broadening of vistas, a synergy among once-differentiated matters, and strategies to mathematical difficulties either outdated and new.

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Theorem 1 uses the HNN extension or the free product with amalgamation decomposition of the fundamental group associated to a simple element to compute in terms of a canonical form for the conjugacy class . Theorems 2–5 uses Combinatorial Group Theory and the presentation of the Goldman–Turaev Lie bialgebra given in [9]. Extending this algorithm and the Theorems 2–5 to closed surfaces is work in progress. The bracket and cobracket were defined as part of String Topology on the reduced circle equivariant homology of the free loop space of any manifold (generalizing ¼ ) (see the 2003 Abel Proceedings [10]).

L. Breen and W. Messing, Differential geometry of gerbes, Adv. Math. 198 (2005), 732–846. Available as arXiv:math/0106083. 9. K. S. Brown, Abstract homotopy theory and generalised sheaf cohomology, Trans. Amer. Math. Soc. 186 (1973), 419–458. 10. R. Brown and C. B. Spencer, G -groupoids, crossed modules, and the classifying space of a topological group, Proc. Kon. Akad. v. Wet. 79 (1976), 296–302. 11. -L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Birkhauser, Boston, 1993.

C. Baez, A. S. Crans, U. Schreiber and D. Stevenson, From loop groups to 2-groups, HHA 9 (2007), 101–135. Also available as arXiv:math/0504123. 3. J. C. Baez and A. Lauda, Higher-dimensional algebra V: 2-groups, Th. Appl. Cat. 12 (2004), 423–491. Also available as arXiv:math/0307200. 4. J. C. Baez and U. Schreiber, Higher gauge theory, in Categories in Algebra, Geometry and Mathematical Physics, eds. A. , Contemp. Math. 431, AMS, Providence, RI, 2007, pp. 7–30. Also available as arXiv:math/0511710.

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