In the last ten years, the introduction of Heegaard Floer homology lead to important advances in low-dimensional topology.
We will introduce this theory with particular attention to its implications in knot theory.
We will also discuss other recent, homology theory-valued knot invariants of a different nature:
these arise as categorifications of earlier knot polynomials.
1.Morse homology on finite-dimensional closed manifolds
3.Heegaard Floer homology, including knot Floer homology and sutured Floer homology
4.Jones and Homfly polynomials 5. Khovanov and Khovanov--Rozansky homology
1.Morse homology, by Matthias Schwarz, Birkhauser, 1993
2.Lecture notes on Morse homology, by Michael Hutchings, math.berkeley.edu/~hutching/teach/276/mfp.ps
3.An introduction to Heegaard Floer homology, by Peter Ozsvath and Zoltan Szabo, www.math.princeton.edu/~szabo/clay.pdf
4.Floer homology and surface decompositions, by Andras Juhasz, Geom. Topol. 12 (2008), no. 1, 299--350
5.The decategorification of sutured Floer homology, by Stefan Friedl, Andras Juhasz, and Jacob Rasmussen, arXiv:0903.5287
6.Some differentials on Khovanov-Rozansky homology, by Jacob Rasmussen, math.GT/0607544
Knowledge of multivariable calculus, basic complex analysis and algebraic topology will be assumed.
Based on a small number of homework problems and participation.
Rather than addressing deep analytic issues, the emphasis will be on applications of the techniques.
Many examples will be included.