User max m - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T05:18:30Z http://mathoverflow.net/feeds/user/375 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5364/what-is-a-deformation-of-a-category/6351#6351 Answer by Max M for What is a deformation of a category? Max M 2009-11-21T05:11:10Z 2009-11-21T05:37:55Z <p>Without looking over it again, I think Seidel's "Fukaya categories and deformations" <a href="http://arxiv.org/abs/math/0206155" rel="nofollow">http://arxiv.org/abs/math/0206155</a> would be in some sense both an example and a bit of general theory. And only 9 pages, too.</p> <p>I seem to recall a talk by Seidel where the deformation of $A_\infty$ category was controlled by appropriate Hochschild cohomology group -as Greg points out, small categories are just algebras over semisimple rings, and deformations of algebras are controlled by Hochschild groups (well, classically not of $A_\infty$ algebras, but it extends (though I have not personally checked)). Then there was the point that in the particular example some of the deformations were geometric (think deforming complex structure for DCoh, or symplectic form for DFuk) and some were not. Some of the "not" ones were appropriately viewed as coming from non-commutative deformations of the underlying manifold. I wish I remembered more details. Maybe I can edit this answer if I find notes...</p> <p>Edit: After looking it over: Section 5 in Seidel's paper looks to indeed be (prtial?) answer to your question. The referenced paper of Fukaya I have not read, but it looks very interesting (in general, and with regards to this question in particular).</p> http://mathoverflow.net/questions/3841/the-miracle-of-heegard-floer The "miracle" of Heegard Floer. Max M 2009-11-02T19:33:23Z 2009-11-12T20:53:50Z <p>Taking tori in symmetric products and "miraculously" proving that the Floer homology is independent of choices always seemed, well, miraculous. Some time ago Max Lipyanski explained to me the origins of this construction from gauge theory on surfaces, a la Atiyah-Floer conjecture, which I have then forgotten. What is the origin of Heegard Floer?</p> http://mathoverflow.net/questions/4982/look-into-delzant-polytope/5175#5175 Answer by Max M for look into Delzant Polytope Max M 2009-11-12T09:14:54Z 2009-11-12T09:14:54Z <p>As "all related comments are welcome" I should perhaps say that if you add a "reflexive" condition (the dual polytope is also lattice polytope) we get smooth Fano ones of which there are finitely many in each dimension, but the number grows pretty quickly. The attempts to classify these have been moderately successful (see www.imf.au.dk/publications/phd/2008/imf-phd-2008-moe.pdf for example), but they are still somewhat wild. </p> http://mathoverflow.net/questions/5148/what-are-the-possible-images-of-a-square-under-an-area-preserving-map/5173#5173 Answer by Max M for What are the possible images of a square under an area-preserving map? Max M 2009-11-12T08:53:35Z 2009-11-12T08:53:35Z <p>Your statement of Gromov's theorem is wrong. Of course you can map a square to any rectangle of the same area! Just by diagonal linear matrix with eigenvalues (k, 1/k) (i.e. shrink in one direction, stretch in another). </p> <p>Gromov nonsqueezing says $B^{2n}(1)$ can not be put inside $B(r)x\mathbb{C}^{n-1}$ for $B(r)$ - real 2-d ball in $\mathbb{C}$ with $r&lt;1$ by a symplectomorphism. This means by a map that preserve symplectic form, which in turn can be viewed as measuring the "sum of the areas of projections of a 2-d object on the planes x1-y1, x2-y2 ....xn-yn". The non-squeezing in some sense is a statement about not being able to trade these "complex plane" directions for each other. It says nothing about squeezing inside those planes. </p> <p>I believe by a general volume preserving map you can take any contractible domain to any other of the same volume...</p> http://mathoverflow.net/questions/2905/is-the-fukaya-category-defined/3723#3723 Answer by Max M for Is the Fukaya category "defined"? Max M 2009-11-02T01:22:11Z 2009-11-02T01:22:11Z <p>Mike has given a good answer (basically - bubbling), but perhaps I can elaborate/add.</p> <p>In general, due to disk bubbling one gets a curved or obstructed A-infinity category. The reason that's bad is that in this case m1 does not square to zero - no Floer homology. In the case when m0 is a multiple of fundamental class, then for CF(L,L) the A-infinity relations still say m1 squares to zero, and more generally for L_1 and L_2 if it's the same multiple \lambda. So you get completely disjointed categories for each \lambda. This happens for fibers of toric symplectic manifolds, for example (and on their mirror Landau-Ginsburg model we get the categories of singularities D^b sing, \lambda corresponding to the value of the superpotential). As Mike said, the business of bounding cochains is understanding when A-infinity relations can be modified to get m0 to be zero (unobstructed L) or multiple of fundamental class (weakly unobstructed L).</p> <p>As for finding the "right" definition of Fukaya category, my understanding it is tricky business. Form the point of view of mirror symmetry, the current definition is a cheat, where we use passing to derived category to sweep under the rug all the problems - one of which is not dealing with immersed Lagrangians, and another is perhaps ignoring the suggestion of Kapustin-Orlov to include coisotropics. As I understand people are studying ways to do Floer theory for both of these new types of objects. This is however, a somewhat different question from the one you asked.</p> http://mathoverflow.net/questions/2147/most-helpful-math-resources-on-the-web/2252#2252 Answer by Max M for Most helpful math resources on the web Max M 2009-10-24T05:59:51Z 2009-10-24T05:59:51Z <p><a href="http://maths.dept.shef.ac.uk/magic/index.php" rel="nofollow">http://maths.dept.shef.ac.uk/magic/index.php</a></p> <p>Apparently UK has been building a depository/interactive system for graduate math courses. Click on "courses" to access archives. Many have lecture notes and other materials.</p> <p>I found this recently. Have not actually personally used it, but potentially very useful. </p> http://mathoverflow.net/questions/952/where-are-mathematics-jobs-advertised-if-not-on-mathjobs-e-g-in-europe-and-else/977#977 Answer by Max M for Where are mathematics jobs advertised if not on mathjobs (e.g. in Europe and elsewhere)? Max M 2009-10-18T01:52:02Z 2009-10-18T01:52:02Z <p><a href="http://www.jobs.ac.uk/jobs/mathematics" rel="nofollow">http://www.jobs.ac.uk/jobs/mathematics</a></p> <p>Not complete, but occasionally useful. </p> http://mathoverflow.net/questions/761/undergraduate-level-math-books/817#817 Answer by Max M for Undergraduate Level Math Books Max M 2009-10-17T00:26:04Z 2009-10-17T00:26:04Z <p>Ordinary Differential Equations by Vladimir I. Arnold</p> http://mathoverflow.net/questions/761/undergraduate-level-math-books/816#816 Answer by Max M for Undergraduate Level Math Books Max M 2009-10-17T00:22:52Z 2009-10-17T00:22:52Z <p>Lectures on Linear Algebra by I. M. Gel'fand</p> http://mathoverflow.net/questions/761/undergraduate-level-math-books/814#814 Answer by Max M for Undergraduate Level Math Books Max M 2009-10-17T00:19:39Z 2009-10-17T00:19:39Z <p>Real Mathematical Analysis by Charles Pugh</p> http://mathoverflow.net/questions/243/compact-kaehler-manifolds-that-are-isomorphic-as-symplectic-manifolds-but-not-as/575#575 Answer by Max M for Compact Kaehler manifolds that are isomorphic as symplectic manifolds but not as complex manifolds (and vice-versa) Max M 2009-10-15T06:19:59Z 2009-10-15T06:19:59Z <p>So here are some examples: When X has no continuous families of automorphisms (H^0(X, TX)=0), complex deformations of X to first order are given by H^1(X, TX). For compact Calabi-Yaus this is H^{(n-1, 1)} and moreover by Bogomolov-Tian-Todorov the deformations are unobstructed. </p> <p>Symplectic deformations as Ben noted are controlled by H^2(X, R) by Moser's trick. If we want to deform while staying Kahler, then in H^{(1,1)}(X, R). In mirror symmetry (where this discussion is stolen from) one allows a B-field and correspondingly a complexified space of deformations H^{(1,1)}. Then for mirror manifolds these two spaces of deformations are switched.</p> <p>This is discussed in Denis Auroux's notes on mirror symmetry (<a href="http://math.mit.edu/~auroux/18.969/" rel="nofollow">http://math.mit.edu/~auroux/18.969/</a>, any misinterpretation is my fault).</p> <p>Mirror symmetry is cool and all, but if we just stay on the same Calabi-Yau the deformation spaces for symplectic and complex structures can have different dimensions - with either one bigger, giving examples for both 1 and 2.</p> http://mathoverflow.net/questions/73603/how-to-calculate-a-fredholm-index-numerically Comment by Max M Max M 2012-12-01T06:12:18Z 2012-12-01T06:12:18Z <a href="http://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem" rel="nofollow">en.wikipedia.org/wiki/&hellip;</a> ? http://mathoverflow.net/questions/1861/looking-for-an-introduction-to-orbifolds/1930#1930 Comment by Max M Max M 2010-11-05T15:28:05Z 2010-11-05T15:28:05Z Never mind. <a href="http://library.msri.org/books/gt3m/" rel="nofollow">library.msri.org/books/gt3m</a> http://mathoverflow.net/questions/1861/looking-for-an-introduction-to-orbifolds/1930#1930 Comment by Max M Max M 2010-11-05T15:26:57Z 2010-11-05T15:26:57Z MSRI updated website. Link no longer works. Any other place to find these notes? http://mathoverflow.net/questions/5148/what-are-the-possible-images-of-a-square-under-an-area-preserving-map/5173#5173 Comment by Max M Max M 2009-11-12T17:03:37Z 2009-11-12T17:03:37Z For volume prserving embeddings, see &quot;Embedding problems in symplectic geometry&quot; by Felix Schlenk, Appendix B: volume is indeed the only constraint. http://mathoverflow.net/questions/3841/the-miracle-of-heegard-floer/3843#3843 Comment by Max M Max M 2009-11-03T02:03:36Z 2009-11-03T02:03:36Z I think you are right. I was hoping to get some info on the gauge theory. From what I gather, the idea is that the symmetric product is the space of solutions of vortex equations - explained <a href="http://front.math.ucdavis.edu/0606.5063" rel="nofollow">front.math.ucdavis.edu/0606.5063</a> This is U(1) gauge theory,and presumably &quot;monopole&quot; version of Atiyah-Floer is what produces the Heegard Floer, which Ozsvath-Szabo then went on to study directly. I wonder if anyone can flesh out some details (e.g. how to the Lagrangian tori arise). http://mathoverflow.net/questions/3841/the-miracle-of-heegard-floer/3843#3843 Comment by Max M Max M 2009-11-02T20:58:53Z 2009-11-02T20:58:53Z Yes, of course. Denis Auroux gave a talk about it <a href="http://www-math.mit.edu/~auroux/papers/slides-fuksymg.pdf" rel="nofollow">www-math.mit.edu/~auroux/papers/&hellip;</a> and it does remove the miraculousness. But this was a posteriori, in light of Lipshitz-Ozsvath-Thurston. Surely this is not how Ozsv&#225;th-Szabo came up with this? http://mathoverflow.net/questions/2905/is-the-fukaya-category-defined/3723#3723 Comment by Max M Max M 2009-11-02T06:37:50Z 2009-11-02T06:37:50Z Yes, roughly. The the cone on the morphism \lambda c is supposed to be equivalent to Lagrangian surgery on the intersection point c with an areа parameter controlled by \lambda (but perhaps only when there is one intersection). And even more vaguely the immersed Lagrangian should be like the cone on its self intersections. And yes, there probably should be a larger version of Fukaya category. http://mathoverflow.net/questions/3184/philosophical-meaning-of-the-yoneda-lemma/3208#3208 Comment by Max M Max M 2009-11-02T01:34:53Z 2009-11-02T01:34:53Z Could you explain the last statement on page 8? (Any functor C^op \mapsto Set can be built out of representables Hom(-,A) in very roughly the same way that any number is built as a product of primes). Should I make it a separate question? http://mathoverflow.net/questions/2147/most-helpful-math-resources-on-the-web/2252#2252 Comment by Max M Max M 2009-10-31T22:20:16Z 2009-10-31T22:20:16Z Now when all of these have unified search capabilities, that would be a great day!