User aston smythe - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T20:12:37Z http://mathoverflow.net/feeds/user/1977 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27827/genus-of-complex-projective-space Genus of complex projective space Aston Smythe 2010-06-11T15:44:57Z 2010-06-22T20:33:13Z <p>The complex projective line is isomorphic to the 2-sphere, and so, has genus $0$. Does this result for all $CP^N$, that is, is the genus of $CP^N$ equal to $0$, for all $N$?</p> http://mathoverflow.net/questions/27839/genus-of-grassmannians-and-flag-manifolds Genus of Grassmannians and Flag Manifolds Aston Smythe 2010-06-11T17:09:19Z 2010-06-12T06:49:56Z <p>In light of the answers given to <a href="http://mathoverflow.net/questions/27827/genus-of-complex-projective-space" rel="nofollow">this question</a>, I would like to pose a more general one: Do all Grassmannian spaces have genus 0? If so, do there exist any flag manifolds with non-zero genus? </p> http://mathoverflow.net/questions/19219/smooth-manifolds-that-dont-admit-a-partition-of-unity Smooth manifolds that don't admit a partition of unity Aston Smythe 2010-03-24T18:07:52Z 2010-03-24T21:41:17Z <p>When I first starting studying differential geometry, I asked my lecturer a question about smooth manifolds that didn't admit a partition of unity. He promptly told not to worry about such objects as they were only studied by the extremely eccentric. I would like to know if this is true, ie, does anyone study manifolds that don't admit a partition of unity (not whether such people are eccentric).</p> http://mathoverflow.net/questions/18948/invariant-vector-fields-for-homogenous-spaces Invariant Vector Fields for Homogenous Spaces Aston Smythe 2010-03-21T19:23:34Z 2010-03-21T20:22:52Z <p>As we all know, the space of invariant vector fields on a Lie group can be identified with the tangent space at the identity (or any other point for that matter). My question is: How does this generalize to homogeneous spaces? My guess would be that one can equate the space with the tangent space at any point point. However, it is not clear to me why everything should carry over smoothly to this more general setting.</p> http://mathoverflow.net/questions/17666/what-to-call-the-elements-of-a-tensor-product What to call the elements of a tensor product. Aston Smythe 2010-03-09T23:05:04Z 2010-03-09T23:25:37Z <p>What does one call the first (or second) factor of an element of a tensor product? For example, if $V,W$ are vector spaces, and $v \in V$, $w \in W$, with $v \otimes w \in V \otimes W$, how would one refer to $v$? First tensor factor? </p> http://mathoverflow.net/questions/11821/su2-and-the-three-sphere $SU(2)$ and the three sphere Aston Smythe 2010-01-15T04:52:08Z 2010-02-01T10:45:57Z <p>Can anyone give me an explicit isomorphism between $SU(2)$ and the three sphere?</p> <p>What about for higher spheres? This question <a href="http://mathoverflow.net/questions/11628/complex-projective-space-as-a-u1-quotient" rel="nofollow">link text</a> seems to indicate that there exists a homeomorphism from $SU(n)/SU(n-1)$ to the $(2n-1)$-sphere.</p> http://mathoverflow.net/questions/13303/hodge-index-theorem-for-complex-manifolds Hodge Index theorem for Complex Manifolds Aston Smythe 2010-01-29T00:10:09Z 2010-01-29T02:42:17Z <p>The Riemann-Roch theorem is a result about Riemann Surfaces that was extended to the Hirzebruch–Riemann–Roch theorem, a result about compact complex manifolds. The Hodge Index theorem is a result about Riemann surfaces (I'm just worried about the complex case) that is proved using Riemann-Roch. Has the Hirzebruch–Riemann–Roch theorem been used to extend the Hodge Index theorem to a result about compact complex manifolds.</p> http://mathoverflow.net/questions/13002/principle-bundle-connection-correspondence-for-two-descriptions-of-the-mathbbc Principle Bundle Connection Correspondence for two descriptions of the $\mathbb{CP}^2$ Aston Smythe 2010-01-26T01:43:39Z 2010-01-26T04:26:44Z <p>Consider the following pair of principle bundle descriptions of $\mathbb{CP}^2$: $$\mathbb{CP}^2 \simeq SU(3)/U(2) \simeq S^5/U(1).$$ If I have a principle $U(2)$-bundle connection for $\mathbb{CP}^2$, will that correspond to a principle $U(1)$-bundle connection? (Where by correspond I suppose I mean give the same covariant derivative.) </p> <p>I am also interested in how this generalises to the $n$-case $$\mathbb{CP}^{n} \simeq SU(n+1)/U(n) \simeq S^{2n+1}/U(1).$$</p> http://mathoverflow.net/questions/11628/complex-projective-space-as-a-u1-quotient Complex Projective Space as a $U(1)$ quotient Aston Smythe 2010-01-13T06:10:22Z 2010-01-13T19:56:57Z <p>As is well known, one can view $\mathbb{CP}^n$ as a quotient of the unit $(2n + 1)$-sphere in $\mathbb{C}^{n+1}$ under the action of $U(1)$, since every line in $\mathbb{C}^{n+1}$ intersects the unit sphere in a circle. </p> <p>Moreover, we have $S^{2n + 1} = SU(n+1)/SU(n)$, where $SU(n)$ embeds into the bottom right-hand corner (say).</p> <p>My question is: Is there an embedding $j$ of $U(1)$ into $SU(n+1)/SU(n)$ that gives the $U(1)$ action as a left (right) multiplication, i.e. such that $A.e^{i \theta} = Aj(e^{i \theta})$, for all $A \in SU(n+1)/SU(n)$?</p> <p>For $n=1$, it's easy: $SU(2) = S^3$, and we embed $e^{i\theta}$ as<br /> $\left( \begin{array}{cc} e^{i \theta} &amp; 0 \\ 0 &amp; e^{-i \theta} \end{array} \right)$.</p> http://mathoverflow.net/questions/10724/relationship-between-algebraic-and-holomorphic-differential-forms Relationship between algebraic and holomorphic differential forms Aston Smythe 2010-01-04T17:48:10Z 2010-01-04T19:02:03Z <p>I'm a little confused and in need of some clarification about the relationship between algebraic and holomorphic differential forms:</p> <p>(1) What is the exact definition of the module of differential forms of a complex projective variety?</p> <p>(2) What is the definition of its differential?</p> <p>(3) Am I right in assuming that the algebraic forms are a submodule of the holomorphic forms with the two differentials coinciding in some obvious sense?</p> http://mathoverflow.net/questions/8285/badiou-and-mathematics Badiou and Mathematics Aston Smythe 2009-12-09T01:25:21Z 2009-12-09T14:07:33Z <p>Does anyone have an opinion on Alain Badiou's use of set theory? Is there anything interesting mathematically there? Also could anyone shed any light on the comment in the Wikipedia article <a href="http://en.wikipedia.org/wiki/Alain%5FBadiou" rel="nofollow">link text</a> that says: </p> <blockquote> <p>This effort leads him, in Being and Event, to combine rigorous mathematical formulae with his readings of poets such as Mallarmé and Hölderlin and religious thinkers such as Pascal.</p> </blockquote> http://mathoverflow.net/questions/7439/algebraic-varieties-which-are-also-manifolds Algebraic Varieties which are also Manifolds Aston Smythe 2009-12-01T14:58:56Z 2009-12-01T17:28:02Z <p>Any non-singular projective variety over $\mathbb{C}$ is easily seen to be a smooth manifold. Presumably the same is not true for algebraic varieties - one would not expect varieties with singular points to have a smooth structure. But do there exist non-singular varieties that are not smooth manifolds?</p> http://mathoverflow.net/questions/6332/the-2-sphere-and-mathbbcp1 The 2-sphere and $\mathbb{CP}^1$ Aston Smythe 2009-11-21T00:15:02Z 2009-11-21T00:28:42Z <p>As is very well known, the algebraic variety $S^2$ is isomorphic to projective variety $\mathbb{CP}^1$ as a complex manifold. As is also well known, the coordinate ring of $S^2$ is given by $&lt; x,y,z > / &lt; x^2 + y^2 +z^2 - 1 >$ and the function field of $\mathbb{CP}^1$ is $\mathbb{C}(\mathbb{CP}^1)$ (its coordinate ring being $\mathbb{C}$). My question is how the coordinate ring of $S^2$ and the function field of $\mathbb{CP}^1$ are related? Presumably this relation is a special case of a general variety-complex manifold relationship.</p> http://mathoverflow.net/questions/29547/interesting-consequences-of-the-riemann-roch-formula-for-line-bundles-over-flag-m Comment by Aston Smythe Aston Smythe 2010-06-25T21:13:45Z 2010-06-25T21:13:45Z Yes. But I prefer to think about it as a special case of the Hirzebruch-Riemann-Roch formula. http://mathoverflow.net/questions/27839/genus-of-grassmannians-and-flag-manifolds/27840#27840 Comment by Aston Smythe Aston Smythe 2010-06-11T17:24:13Z 2010-06-11T17:24:13Z ... and the arithmetic genus? http://mathoverflow.net/questions/23627/group-and-hopf-algebra-structures-for-projective-varieties Comment by Aston Smythe Aston Smythe 2010-05-05T22:39:17Z 2010-05-05T22:39:17Z I suppose I mean when its coordinate ring when we consider it as a real affine variety - so scrap projective from the question and just consider affine. http://mathoverflow.net/questions/19219/smooth-manifolds-that-dont-admit-a-partition-of-unity Comment by Aston Smythe Aston Smythe 2010-03-24T18:29:52Z 2010-03-24T18:29:52Z I mean manifolds that aren't second countable but are locally euclidean and Hausdorff .... do we need to specify Hausdorff here, surely it should from the manifold being locally euclidean. http://mathoverflow.net/questions/18948/invariant-vector-fields-for-homogenous-spaces/18951#18951 Comment by Aston Smythe Aston Smythe 2010-03-22T20:44:40Z 2010-03-22T20:44:40Z Well, as a grad student I really like &quot;Grad student/homework&quot; answers. http://mathoverflow.net/questions/18948/invariant-vector-fields-for-homogenous-spaces/18951#18951 Comment by Aston Smythe Aston Smythe 2010-03-22T12:27:39Z 2010-03-22T12:27:39Z Can one construct a basis of all vector fields using the invariant ones, like in the Lie group case? http://mathoverflow.net/questions/17666/what-to-call-the-elements-of-a-tensor-product Comment by Aston Smythe Aston Smythe 2010-03-09T23:18:24Z 2010-03-09T23:18:24Z @ Andrea: But I've got a collection of about 116 $v \otimes w$'s, and I really don't want to refer to them individually. I need a way to say &quot;Take all the first tensor factors and consider the ideal they generate&quot;. http://mathoverflow.net/questions/13632/transition-functions-and-complex-structure/13677#13677 Comment by Aston Smythe Aston Smythe 2010-02-02T00:31:13Z 2010-02-02T00:31:13Z ... and a similar situation holds for $\Omega^{(1,0)}(M)$ and $\Omega^{(0,1)}(M)$? http://mathoverflow.net/questions/13632/transition-functions-and-complex-structure/13677#13677 Comment by Aston Smythe Aston Smythe 2010-02-01T23:05:17Z 2010-02-01T23:05:17Z To see if I understand I am going to take the example of $\mathbb{CP}^1$. The change-of-coordinate map from $\phi_1(U_1) \subset \mathbb{C}$ to $\phi_2(U_2) \subset \mathbb{C}$ is $z \mapsto z^{-1}$. The complex Jacobian of this map is $z \mapsto -z^{-2}$. This gives the bundle $\mathcal{O}(2)$? The conjugate is $z \mapsto \overline{z}^{-1}$, which is not $\mathcal{O}(-2)$ as I thought it should be. http://mathoverflow.net/questions/13303/hodge-index-theorem-for-complex-manifolds/13312#13312 Comment by Aston Smythe Aston Smythe 2010-01-29T14:33:51Z 2010-01-29T14:33:51Z How does this relate to versions that involve the rank of the Neron-Severini group? http://mathoverflow.net/questions/13002/principle-bundle-connection-correspondence-for-two-descriptions-of-the-mathbbc/13009#13009 Comment by Aston Smythe Aston Smythe 2010-01-26T23:39:47Z 2010-01-26T23:39:47Z I am I right in assuming then that any $U(1)$-connection induces a $U(2)$ connection via the embedding $\mathfrak{u(1)} \to\mathfrak{u(2)}$? http://mathoverflow.net/questions/11821/su2-and-the-three-sphere Comment by Aston Smythe Aston Smythe 2010-01-15T17:41:54Z 2010-01-15T17:41:54Z Yes, of course, I just mean homeomorphisn (or diffeomorphism if you want to get differential). http://mathoverflow.net/questions/11628/complex-projective-space-as-a-u1-quotient/11651#11651 Comment by Aston Smythe Aston Smythe 2010-01-13T17:44:46Z 2010-01-13T17:44:46Z @Andrew David's answer is what I was looking for. I knew that $\mathbb{CP}^n$ was a $U(1)$ quotient of the $(2n + 1)$-sphere because of the relationship between planes and the shpere explained above, but I wanted a concrete matrix version of the action. http://mathoverflow.net/questions/11628/complex-projective-space-as-a-u1-quotient Comment by Aston Smythe Aston Smythe 2010-01-13T06:47:26Z 2010-01-13T06:47:26Z I mean an embedding such that the quotient of $SU(n+1)/SU(n)$ under the resulting multplicative action of $U(1)$ is homeomorphic to $\mathbb{CP}^n$. http://mathoverflow.net/questions/10724/relationship-between-algebraic-and-holomorphic-differential-forms/10725#10725 Comment by Aston Smythe Aston Smythe 2010-01-04T18:33:02Z 2010-01-04T18:33:02Z @jvp So, just to be sure, you're saying that for projective smooth complex varieties: algebraic and holomorphic differential forms coincide?