User jay - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T09:38:30Z http://mathoverflow.net/feeds/user/4874 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/130848/minimal-model-of-a-infty-structure minimal model of $A_\infty$ structure Jay 2013-05-16T16:08:31Z 2013-05-16T16:17:48Z <p>Hi all,</p> <p>I am reading about minimal model of $A_\infty$ structure. So far, I find two different ways of the construction, given by Kadeishvili and Kontsevich, Soibelman.</p> <p>1) The construction of these two minimal model looks quite different, although we know they should be quasi-isomorphic. Is there any direct homomorphism between this two model?</p> <p>2) Is there any other algebraic construction of minimal model of $A_\infty$ structure?</p> <p>Thanks!</p> http://mathoverflow.net/questions/127306/differential-form-with-empty-zero-locus differential form with empty zero locus Jay 2013-04-12T01:54:52Z 2013-04-14T13:33:22Z <p>Hi there,</p> <p>I am considering existence of holomorphic 2-form $\omega$ with empty zero locus on a complex manifold. Beside K3 and holomorphic symplectic manifold, is there any other example on general dimension case?</p> <p>Generally, if we consider a differential forms (may not closed) with empty zero locus on a closed manifold, what will be the obstruction?</p> <p>Thanks!</p> http://mathoverflow.net/questions/100904/2-cycle-of-k3-surface 2-cycle of K3 surface Jay 2012-06-29T03:02:22Z 2013-02-10T08:51:38Z <p>Hi there,</p> <p>I want to ask about the 2-cycle of K3 surface.</p> <p>As we know, its betti number $b_2$=22, so there will be 22 2-cycle generators.</p> <p>Is there any topological way to figure out such cycles direct?</p> <p>For example, in the best case, if the K3 surface is elliptic and has a global section, can we use combinations of fibre and section to represent all the 22 2-cycles?</p> <p>Thanks!</p> http://mathoverflow.net/questions/100904/2-cycle-of-k3-surface/121035#121035 Answer by Jay for 2-cycle of K3 surface Jay 2013-02-07T01:13:31Z 2013-02-10T08:51:38Z <p>i find a paper by Michael B. Schulz and Elliott F. Tammaro: <a href="http://arxiv.org/abs/1206.1070" rel="nofollow">http://arxiv.org/abs/1206.1070</a></p> <p>from page. 11, it gives an explicit description these cycles from the point of view of the resolution of $T^4/Z_2$, seems great!</p> http://mathoverflow.net/questions/119899/mirror-symmetry-for-hyperkahler-manifold Mirror symmetry for hyperkahler manifold Jay 2013-01-26T00:10:59Z 2013-02-06T18:50:30Z <p>Hi there,</p> <p>I have some questions about the mirror symmetry of hyperkahler manifold and K3 surface.</p> <p>The well-known result said: the mirror symmetry for K3 surface is just given by its hyperkahler rotation.</p> <p>1) In what sense, the rotation gives the mirror map?</p> <p>2) Does this means:</p> <p>if we start from $(M,\omega_I,I)$, here the k3 surface $M$ has a special lagrangian fibration structure with respect to $\omega_I$ and $I$, and also a special lagrangian section. Denote its SYZ mirror by $(M^{mirror},J_{mirror})$. Then exist a (fiberwised) diffeomorphism $\phi: M \rightarrow M^{mirror}$, s.t. $\phi^{*} (J_{mirror}) = K$?</p> <p>(Here $I,K$ are the standard base of the $S^2$-family of compatible complex structure of the hyperkahler metric on $M$.)</p> <p>Thanks!</p> http://mathoverflow.net/questions/116031/fixed-norm-problem-for-analytic-functions Fixed norm problem for analytic functions Jay 2012-12-10T22:53:51Z 2012-12-25T21:20:48Z <p>Hi there,</p> <p>I have the following problems on my hand:</p> <p>Given arbitrary positive sub-harmonic function $l(x,y)$ on plane, for all analytic functions: $f=a(x,y)+ib(x,y)$ on complex plane, consider the equation:</p> <p>$|f|^2=l$, i.e. $a^2+b^2=l$</p> <p>1) for what kind of $l(x,y)$, the equation has $C^2$ solution?</p> <p>2) for what kind of $l(x,y)$, the <em>number of solutions is finite</em>, after quotient of $S^1$ action: $e^{i\theta}\cdot f$ ?</p> <p>Thanks!</p> http://mathoverflow.net/questions/106553/why-is-the-base-of-slag-fibration-of-cy3-expected-to-be-s3/114726#114726 Answer by Jay for Why is the base of SLAG fibration of CY3 expected to be $S^3$? Jay 2012-11-28T04:17:46Z 2012-11-28T04:17:46Z <p>There are topological conditions for a manifold to be a base of lagrangian fibration.</p> <p>For example, in the real 4 dimensional case, if you want a smooth lagrangian fibration without singular fiber, then the base have to be the unique integral affine surface, that is T^2. If you allow certain singular fibres, then the base has to be some special ones, here is a good reference: <a href="http://arxiv.org/abs/math/0312165" rel="nofollow">http://arxiv.org/abs/math/0312165</a></p> <p>Similarly, in the real 6 dimensional case, there are affine structure condition on the base.</p> http://mathoverflow.net/questions/45881/whats-dual-torus-and-mirror-manifold/114723#114723 Answer by Jay for What's dual torus and mirror manifold? Jay 2012-11-28T04:04:34Z 2012-11-28T04:04:34Z <p>I think you are asking about the complex structure on tangent bundle of integral affine manifold, there we can cook up a simple one, as we did for symplectic structure on cotangent bundle.</p> <p>Here is a good reference: <a href="http://www.math.cuhk.edu.hk/~kwchan/HMS_An.pdf" rel="nofollow">http://www.math.cuhk.edu.hk/~kwchan/HMS_An.pdf</a></p> <p>you can find a quick review of the construction.</p> http://mathoverflow.net/questions/98579/deformation-of-lagrangian-manifolds/112502#112502 Answer by Jay for Deformation of Lagrangian manifolds Jay 2012-11-15T17:17:50Z 2012-11-15T17:17:50Z <p>generally calculation is like this:</p> <ol> <li><p>you write down the tubular neighborhood and the exp map there;</p></li> <li><p>you do re-parametrization, such that your symplectic form comes in the "darboux type" </p></li> </ol> <p>then the section of the normal bundle will be a nearby lagrangian.</p> <hr> <p>there are some simple examples you can do the calculation explicitly, for example: you consider the unit circle in R^2 with the standard symplectic form, then you choose the polar coordinate to write down the exp map in the tubular neighborhood, you will find you need a simple substitution to make the symplectic form in the "darboux type"</p> http://mathoverflow.net/questions/110724/radius-of-tubular-neighborhood radius of tubular neighborhood Jay 2012-10-26T06:13:15Z 2012-10-27T05:39:49Z <p>Hi there,</p> <p>Is there any result about the calculation of radius of tubular neighborhood of submanifold inside a Riemannian manifold?</p> <p>For example, given a simple smooth curve on R^2, what's the radius of its tubular neighborhood? (One upper bound is given by the minimal curvature, but general it is not the radius)</p> <p>Maybe that is what we can expect: if the curvature of the curve is always decreasing, then the radius of the tubular neighborhood is given by the injective radius of (left) end point, it seems true, right? </p> <p>For example, they curve is given by: $xy=1, x \in [1,+\infty)$</p> <p>Thanks!</p> http://mathoverflow.net/questions/91831/holomorphic-sections-on-elliptic-k3-surface holomorphic sections on elliptic K3 surface Jay 2012-03-21T15:51:21Z 2012-04-04T20:22:01Z <p>Hi all,</p> <p>I want to ask something about the holomorphic sections on elliptic K3:</p> <p>Is there any obstruction for an ellptic K3 (as an elliptic fibration) to have holomorphic sections? Is that always some number as 240? For example, how about E(2) or Fermat's quartic?</p> <p>Thanks a lot! :)</p> http://mathoverflow.net/questions/87463/question-about-the-recursion-equation-x-n1x-n1x-n214x-n question about the recursion equation: $x_{n+1}x_{n−1}=x_n^2(1−4x_n)$ Jay 2012-02-03T18:15:12Z 2012-02-06T22:09:35Z <p>Hi all,</p> <p>I am trying to slove the recursion equation: $x_{n+1}x_{n−1}=x_n^2(1−4x_n)$ in the form of $x_n=x_n(x_1,x_2)$ or $x_n=x_n(c_1,c_2)$, and finally get the limit of the ratio: $\dfrac{x_n}{x_{n+1}}$.</p> <p>I tried the way of setting: $x_n=f(n)$, and use the 1st order taylor expansion of $f(n+1)$ and $f(n−1)$, then get the first differential equation: $(f')2=4f^3$, the general solution is: $f(x)=\dfrac{4}{(2x−c_1)^2}$, but it does not fit the original recursion equation.</p> <p>Then I write the equation in the form: $f''=6f^2$, the computer provides some approach by Weierstrass elliptic function (<a href="http://www.wolframalpha.com/input/?i=d%5E2y%252Fdx%5E2%253D6y%5E2" rel="nofollow">http://www.wolframalpha.com/input/?i=d^2y%2Fdx^2%3D6y^2</a>), but it seems the Weierstrass elliptic function still has no such property as the recursion formula.</p> <p>Any method I cam apply to get the final limit of ratio, maybe without solving the general soltuions? Thanks!</p> http://mathoverflow.net/questions/4778/singular-k3-mathematical-meaning/87703#87703 Answer by Jay for Singular K3 -- mathematical meaning? Jay 2012-02-06T18:49:19Z 2012-02-06T18:49:19Z <p>by the way, "singular k3 surface" has a quite different meaning it the literature: it refers to a smooth K3 surface with Picard number 20. reference: Shioda, "singular" K3 surfaces</p> http://mathoverflow.net/questions/87633/construct-the-elliptic-fibration-of-elliptic-k3-surface construct the elliptic fibration of elliptic k3 surface Jay 2012-02-06T05:35:25Z 2012-02-06T12:42:35Z <p>Hi all,</p> <p>As we know, every elliptic k3 surface admits an elliptic fibration over $P^1$, but generally how do we construct this fibration? For example, how to get such a fibration for Fermat quartic?</p> <p>Moreover, as we know all (elliptic) k3 surfaces are differential equivalent to each other, does this mean: topologically the elliptic fibration we get for each elliptic fibraion is the same, which is just the torus fibration over $S^2$ with 24 node singularities? Or, the totally space is the same, but different complex data(structure) provides different way or "direction" of projection onto $S^2$, thus induces different type of fibrations?</p> <p>Thanks!</p> http://mathoverflow.net/questions/15282/to-what-extent-can-i-think-of-a-lagrangian-fibration-in-a-symplectic-manifold-as/87504#87504 Answer by Jay for To what extent can I think of a Lagrangian fibration in a symplectic manifold as T*N? Jay 2012-02-04T03:27:23Z 2012-02-04T03:27:23Z <p>good reference: Duistermaat, J. J. (1980), On global action-angle coordinates. Communications on Pure and Applied Mathematics, 33: 687–706</p> http://mathoverflow.net/questions/127306/differential-form-with-empty-zero-locus Comment by Jay Jay 2013-04-12T14:39:41Z 2013-04-12T14:39:41Z maybe I should say on closed manifold.. thanks MTS http://mathoverflow.net/questions/14740/connection-between-bi-hamiltonian-systems-and-complete-integrability/14747#14747 Comment by Jay Jay 2013-03-16T05:51:39Z 2013-03-16T05:51:39Z thanks for the reference http://mathoverflow.net/questions/119899/mirror-symmetry-for-hyperkahler-manifold Comment by Jay Jay 2013-02-06T18:48:06Z 2013-02-06T18:48:06Z thank you so much! i have not expect such direct approach.. the kahler form gives the deformation of the complex structure directly through Tian-Todorov coordinates. inspiring! http://mathoverflow.net/questions/110613/for-which-calabi-yau-threefolds-is-syz-conjecture-known-to-hold Comment by Jay Jay 2013-01-26T21:22:17Z 2013-01-26T21:22:17Z In the toric calabi yau case, which is not compact, Leung and his gropu have series of papers on this: <a href="http://arxiv.org/pdf/1006.3830.pdf" rel="nofollow">arxiv.org/pdf/1006.3830.pdf</a> http://mathoverflow.net/questions/119147/what-information-is-required-for-syz-mirror-symmetry/119203#119203 Comment by Jay Jay 2013-01-26T21:16:37Z 2013-01-26T21:16:37Z I find Leung and his group follow the similar idea and get an explicit construction of mirror complex manifold for toric CY by instant corrections of open Gromov-Witten invariants: <a href="http://arxiv.org/pdf/1006.3830.pdf" rel="nofollow">arxiv.org/pdf/1006.3830.pdf</a>. Do you have any comments on it? http://mathoverflow.net/questions/116031/fixed-norm-problem-for-analytic-functions/116032#116032 Comment by Jay Jay 2013-01-25T23:26:55Z 2013-01-25T23:26:55Z thanks a lot! the relation between harmonic function and holomorphic function is essentially important here... http://mathoverflow.net/questions/116031/fixed-norm-problem-for-analytic-functions/116032#116032 Comment by Jay Jay 2012-12-13T15:47:30Z 2012-12-13T15:47:30Z Thank you so much! In this case, we have $log l$ is indeed harmonic. But, how to prove it is sufficient condition? I still have no clue of this.. Do you have any reference? Thanks! http://mathoverflow.net/questions/28519/references-for-modern-proof-of-newlander-nirenberg-theorem/33929#33929 Comment by Jay Jay 2012-11-16T17:33:00Z 2012-11-16T17:33:00Z Hi Spiro, thank you for the reference. By the way, have you get some examples you can do the explicit calculation following there proofs? http://mathoverflow.net/questions/57520/examples-in-mirror-symmetry-that-can-be-understood/58101#58101 Comment by Jay Jay 2012-11-15T17:30:04Z 2012-11-15T17:30:04Z General do we have a natural map from a manifold to its its mirror? For example, in the $TB$ and $T^{*}B$ case, seems we need a legendre transform to do this, but that required extre data.. Also about K3 case, do we have such map between manifolds? http://mathoverflow.net/questions/110724/radius-of-tubular-neighborhood/110749#110749 Comment by Jay Jay 2012-10-26T16:28:38Z 2012-10-26T16:28:38Z so.. that is an extra restriction on $\theta \in [0, 2\pi]$? :) http://mathoverflow.net/questions/110724/radius-of-tubular-neighborhood/110749#110749 Comment by Jay Jay 2012-10-26T15:13:12Z 2012-10-26T15:13:12Z Thanks a lot! Maybe that is what I want: if the curvature of the curve is always decreasing, then the radius of the tubular neighborhood is given by the injective radius of (left) end point, it seems true, right? http://mathoverflow.net/questions/100904/2-cycle-of-k3-surface/100957#100957 Comment by Jay Jay 2012-06-30T14:39:38Z 2012-06-30T14:39:38Z we lose some data there? http://mathoverflow.net/questions/87633/construct-the-elliptic-fibration-of-elliptic-k3-surface/87655#87655 Comment by Jay Jay 2012-02-08T05:11:08Z 2012-02-08T05:11:08Z that is great~ thanks! by the way, do you know anything about the existence of sections? maybe just a topological section... http://mathoverflow.net/questions/87633/construct-the-elliptic-fibration-of-elliptic-k3-surface/87655#87655 Comment by Jay Jay 2012-02-07T05:35:25Z 2012-02-07T05:35:25Z Thanks a lot! It is quite helpful! http://mathoverflow.net/questions/87463/question-about-the-recursion-equation-x-n1x-n1x-n214x-n/87523#87523 Comment by Jay Jay 2012-02-06T02:31:15Z 2012-02-06T02:31:15Z thanks! could you try the way: x_0=1, x_1=s and find how their ratio goes? :)