Introduction to Floer Theory? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T05:23:52Z http://mathoverflow.net/feeds/question/95407 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95407/introduction-to-floer-theory Introduction to Floer Theory? Igor Rivin 2012-04-28T02:18:29Z 2012-05-01T17:50:07Z <p>Can anyone suggest a good overview/introduction of the Floer machine for a beginner? (Someone pointed out some intriguing connections to surface mapping class groups, which might be enough incentive to dip the toe in (though I suppose "total immersion" might be the only realistic option...)</p> http://mathoverflow.net/questions/95407/introduction-to-floer-theory/95410#95410 Answer by Chris Gerig for Introduction to Floer Theory? Chris Gerig 2012-04-28T03:07:53Z 2012-04-29T05:40:28Z <p>Michael Hutchings' lecture notes were precisely for this purpose; posted on his webpage: <a href="http://math.berkeley.edu/~hutching/" rel="nofollow">http://math.berkeley.edu/~hutching/</a></p> <p><em>Lecture Notes on Morse Homology (With an Eye Towards Floer Theory and Pseudoholomorphic Curves)</em></p> http://mathoverflow.net/questions/95407/introduction-to-floer-theory/95417#95417 Answer by Jonny Evans for Introduction to Floer Theory? Jonny Evans 2012-04-28T08:16:27Z 2012-04-28T08:16:27Z <p>I wholeheartedly agree with both of Chris Gerig's suggestions.</p> <p>The small McDuff-Salamon book on holomorphic curves ("J-holomorphic curves and quantum cohomology", available on <a href="http://www.math.sunysb.edu/~dusa/jholsm.pdf" rel="nofollow">McDuff's webpage</a>) has a small chapter on Floer homology. The ideas in the rest of the book are also useful for Floer theory.</p> <p>Audin and Damian have an introductory book called "Théorie de Morse et homologie de Floer". I haven't read it, but I hear good things about it.</p> <p>If you're going for total immersion, a good place to start is with Seidel's early papers (e.g. <a href="http://arxiv.org/abs/math/0105186" rel="nofollow">arXiv:math/0105186</a>, <a href="http://arxiv.org/abs/math/9803083" rel="nofollow">arXiv:math/9803083</a>, <a href="http://arxiv.org/abs/math/0309012" rel="nofollow">arXiv:math/0309012</a>) where you learn by watching him do things.</p> http://mathoverflow.net/questions/95407/introduction-to-floer-theory/95443#95443 Answer by Liviu Nicolaescu for Introduction to Floer Theory? Liviu Nicolaescu 2012-04-28T14:40:03Z 2012-04-28T14:40:03Z <p>A very good place is Kronheimer and Mrowka's monograph on monopole homology. Chapter 1 goes through the finite dimensional part.</p> http://mathoverflow.net/questions/95407/introduction-to-floer-theory/95577#95577 Answer by Jo Nelson for Introduction to Floer Theory? Jo Nelson 2012-04-30T14:07:42Z 2012-04-30T14:07:42Z <p>Dietmar Salamon's notes are my favorite:</p> <p><a href="http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf" rel="nofollow">http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf</a></p> http://mathoverflow.net/questions/95407/introduction-to-floer-theory/95585#95585 Answer by Jo Nelson for Introduction to Floer Theory? Jo Nelson 2012-04-30T15:44:11Z 2012-04-30T15:44:11Z <p>Also I should point out that the geometric intuition provided by Andreas Floer in some of his early papers is really quite beautiful and illuminating. For example read the introduction to his 1989 paper, Symplectic Fixed Points and Holomorphic Spheres, in Comm. Math. Phys (120) 575-611. </p> <p><a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.cmp/1104177909" rel="nofollow">http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.cmp/1104177909</a></p> <p>For the grittier details it's better to look in something like Dietmar's notes or the big book of J-holomorphic curves. </p> http://mathoverflow.net/questions/95407/introduction-to-floer-theory/95680#95680 Answer by Andre Carneiro for Introduction to Floer Theory? Andre Carneiro 2012-05-01T17:50:07Z 2012-05-01T17:50:07Z <p>I can vouch for Audin-Damian's Theorie de Morse et Homologie de Floer, read it cover to cover for my quals. They do Hamiltonian Floer theory with simplifying assumptions ($\omega$ and $c_1$ vanish on $\pi_2$ so there's no need to worry about bubbling, grading issues or caps, which one can learn from Dietmar's notes). They prove everything and provide intuition all along. The most technical estimates used for gluing are grouped into a Chapter that one can skip without loss of understanding. It also does Morse theory as a warm-up.</p>