most recent 30 from http://mathoverflow.net 2013-05-25T07:56:58Z http://mathoverflow.net/feeds/user/26222 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/129833/classification-of-higher-dimensional-manifolds Allen 2013-05-06T13:19:08Z 2013-05-06T14:11:12Z

<p>It is known that a 2-connected closed smooth 6-manifold is homeomorphic to S^{6} or connected sum of (S^{3}xS^{3}). My question is whether we have a similar statement for (n-1)-connected closed smooth 2n-manifold (at least when n=4). If not, do we have a clear classification theorem?</p>

http://mathoverflow.net/questions/118125/about-the-macmahon-functions-on-dimensions-bigger-than-three Allen 2013-01-05T13:59:01Z 2013-01-05T13:59:01Z

<p>As is well known, the generating functions of partitions counting in dimension 2 and 3 admit closed formulas. These functions show up in Euler characterisitics of Hilbert scheme of points in surfaces and threefolds respectively. But in dimension bigger than three, we do not have a closed formula. My question is whether there is any plausible reason for that.</p>

http://mathoverflow.net/questions/117692/universal-family-of-hibert-shemes-of-points Allen 2012-12-31T08:05:05Z 2012-12-31T08:05:05Z

<p>My question is how to explicitly compute the Chern class of the universal sheaf of the moduli space of ideal sheaves of two and three points on a given smooth projective variety X (no need to be algebraic surface) ? When we consider the two points case, the cohomology ring of the moduli can be explictly written down, I hope to represent the Chern class of the universal sheaf in terms of some element of cohomology ring of moduli and base manifold X. </p>

http://mathoverflow.net/questions/101220/is-there-a-seiberg-witten-version-of-donaldson-thomas-theory/106439#106439 Allen 2012-09-05T14:23:44Z 2012-09-05T14:23:44Z

<p>In 8 dimensional case we do not have direct analogous theory like 4 dimension case by strong weak duality. One can naively consturct the SW theory for CY4(line bundles with sections), but the theory turns out to be quite trivial since we need the virtual dimention of the moduli space is topological and we do not have much choice. In 4 dimensions, it is Kroheimer and Mrowka first showed that Donaldson polynomials have recurrence relations for simple type 4 mfds. Then Seiberg and Witten wanted to understand this from Physical perspective and finally got to SW theory. All Gauge theories in 4 dim are expected to be recovered by SW theory. But this is far from clear even for CY3(people seems only consider DT invs for curves(=GW by MNOP) and pts(computed by several groups) so far).</p>

http://mathoverflow.net/questions/106416/whether-hilbert-schemes-of-3-points-on-arbitrary-smooth-projective-varieties-are Allen 2012-09-05T11:26:49Z 2012-09-05T12:15:48Z

<p>As is well known, the Hilbert scheme of two points on a given smooth projective variety X are blow up along diagonal of product of X and then quotient the Z2 action. It is smooth. My question is whether Hilbert schemes of 3 points on arbitrary smooth projective varieties are smooth. If so, why and how to describe the geometry of them?</p>

http://mathoverflow.net/questions/9391/higher-dimensional-gromov-witten-theories/106418#106418 Allen 2012-09-05T11:53:48Z 2012-09-05T11:53:48Z

<p>As ABayer said since the domain is no longer a curve, there are higher obsturction and one does not know how to define the virtual cycle. By the way, one can consider a related problem: try to count special lagrangians (in general calibrated submanifolds). It turns out that the deformation of calibrated submanifolds is unobsturcted. But one meets another problem to define a invariant, i.e the compactification issue, one usually does not know how to compactify the moduli space of calibrated submanifolds. </p>

http://mathoverflow.net/questions/129833/classification-of-higher-dimensional-manifolds/129842#129842 Allen 2013-05-08T01:12:53Z 2013-05-08T01:12:53Z

Thanks for your reminding, Danny. I always mean classification up to homeomorphism instead of diffeomorphism.

http://mathoverflow.net/questions/129833/classification-of-higher-dimensional-manifolds/129842#129842 Allen 2013-05-07T09:22:58Z 2013-05-07T09:22:58Z

Hi,Scott. You mean his book: Algebraic and Geometric Surgery ?

http://mathoverflow.net/questions/129833/classification-of-higher-dimensional-manifolds Allen 2013-05-07T02:41:17Z 2013-05-07T02:41:17Z

Ryan, I found a good ref called &quot;A guide to the classication of manifolds&quot; by M. Kreck. It indicates the 3-connected closed 8-manifold already could be quite complicated. You can google &quot;surveys on surgery theory&quot; and the first PDF contains this paper.

http://mathoverflow.net/questions/129833/classification-of-higher-dimensional-manifolds/129842#129842 Allen 2013-05-06T14:28:24Z 2013-05-06T14:28:24Z

Thanks Danny. I have tried Wall's paper. But it seems not that clear to me. Meanwhile the paper did not give a classification when n is even, I suspect.

http://mathoverflow.net/questions/129833/classification-of-higher-dimensional-manifolds Allen 2013-05-06T14:23:27Z 2013-05-06T14:23:27Z

Not yet. Thanks for the comments.

http://mathoverflow.net/questions/117723/geometric-intuitive-interpretation-of-ext Allen 2012-12-31T13:56:32Z 2012-12-31T13:56:32Z

The first ext group of two modules(sheaves) can be explictly understood as the set of elements which fit with the given two modules into a short exact sequence (ref. Griffith Harris). For higher ext we have similar construction by going to a diagram of short exact sequences.

http://mathoverflow.net/questions/117691/example-of-special-lagrangian-fibration-of-compact-cy3 Allen 2012-12-31T13:39:09Z 2012-12-31T13:39:09Z

Mark Gross is an expert in this field. By the way, as far as I know consturcting a special Lag submanifold on a compact CY3 is too difficult, not to say you want a special Lag fibration. This is the reason why many people now only consider Lag fibration for the purpose of mirror symmetry.

http://mathoverflow.net/questions/6200/what-is-to-quantize-something/6220#6220 Allen 2012-09-28T11:23:19Z 2012-09-28T11:23:19Z

I think you mean quantization is some kind of deformation theory.

http://mathoverflow.net/questions/107972/reference-request-virtual-fundamental-class-of-moduli-of-stable-maps Allen 2012-09-28T08:23:33Z 2012-09-28T08:23:33Z

I think you may try Chapter 7 of the book &lt;&lt;Mirror symmetry and Algebraic geometry&gt;&gt; by Cox and Katz

http://mathoverflow.net/questions/106416/whether-hilbert-schemes-of-3-points-on-arbitrary-smooth-projective-varieties-are/106420#106420 Allen 2012-09-05T12:51:12Z 2012-09-05T12:51:12Z

Thank you for your detailed answer. I am curious about the global description too.