User muon - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T22:33:41Z http://mathoverflow.net/feeds/user/21640 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/125196/stability-of-t-mathbbp2-and-omega-mathbbp2 Stability of $T_{\mathbb{P}^2}$ and $\Omega_{\mathbb{P}^2}$? Muon 2013-03-21T19:48:06Z 2013-05-02T22:22:00Z <p>How can one prove that the tangent bundle $T_{\mathbb{P}^2}$ and its dual $\Omega_{\mathbb{P}^2}$ are stable vector bundles with respect to <code>$\mathcal{O}_{\mathbb{P}^2}(1)$</code>? Similarly, is it true that $T_{\mathbb{P}^n}$ and its dual $\Omega_{\mathbb{P}^n}$ are stable vector bundle for any $n \in \mathbb{N}$? By stable I mean polynomial stability. </p> http://mathoverflow.net/questions/128213/is-pi-1-widetildex-g-always-finite-if-pi-1x-is-finite Is $\pi_1(\widetilde{X/G})$ always finite if $\pi_1(X)$ is finite? Muon 2013-04-21T07:46:38Z 2013-04-21T12:55:36Z <p>Let $X$ be a smooth complex manifold with finite fundamental group. Suppose that a finite group $G$ acts on $X$ and let $\widetilde{X/G}$ be a resolution of singularities. Is $\pi_1(\widetilde{X/G})$ always finite?</p> <p>I think this is true but don't know a way to prove or a reference.</p> http://mathoverflow.net/questions/119139/are-rational-sections-of-a-vector-bundle-useful Are rational sections of a vector bundle useful? Muon 2013-01-17T06:28:39Z 2013-01-17T15:40:12Z <p>Let $X$ be a complex manifold or variety and $L$ a line bundle on it. Given a rational section $s$ of $L$, we get a divisor $D=Div(s)$ and may recover $L$ as $\mathcal{O}(D)$. What about vector bundles? I think it makes sense to speak about rational section $t$ of a vector bundle $E$, as a section is locally a collection of functions. My question is, is this $t$ useful? For example, can we say anything about Chern classes of $E$ from $t$? </p> http://mathoverflow.net/questions/117317/what-do-correlation-functions-compute-in-cft What do correlation functions compute in CFT? Muon 2012-12-27T10:46:16Z 2012-12-28T04:40:47Z <p>I would like to understand what correlation functions compute in Conformal Field Theory in mathematics. Let me begin with basic definitions. We define a free boson field $\phi(z)$ as a formal power series $$\phi(z)=q+a_0\log(z)-\sum_{n\ne0}\frac{a_n}{n}z^{-n},$$ where $q,a_n$ for $n\in \mathbb{Z}$ are operators satisfying relations $$[a_n,q]=\delta_{n,0}, \ \ \ [a_m,a_n]=m\delta_{m+n,0}.$$ Here these operators acts on the $space$ generated by $vacuum$ vector $|0\rangle$ with properties $$a_n |0\rangle=0\ (n\ge0), \ \ \langle0|q=\langle0|a_n=0\ (n&lt;0),$$ where $\langle0|$ is the dual of the vacuum, i.e. $\langle0|0\rangle=1$. We then define the current $J(z)$ of $\phi(z)$ as a formal derivative $$J(z)=\sum_{n\in \mathbb{Z}}a_nz^{-n-1}.$$</p> <blockquote> <p>Here is my question. In CFT one is interested in $correlation$ $functions$ such as $$\langle0|J(z_1)J(z_2)|0\rangle, \ \ \langle0|J(z_1)J(z_2)J(z_3)J(z_4)|0\rangle$$ What do they compute in this context? </p> </blockquote> <p>In physics, correlation function like $\langle0|\phi(x)\phi(y)|0\rangle$ computes the possibility of the field $\phi(x)$ to become $\phi(y)$ etc where $x,y$ are coordinate in for example Minkowski space. How should one interprete the correlation function in our case? </p> http://mathoverflow.net/questions/117287/motivation-of-virasoro-algebra Motivation of Virasoro algebra Muon 2012-12-26T23:37:33Z 2012-12-27T16:19:18Z <p>I have a question on definition/motivation of Virasoro algebra. Recall that Virasoro algebra is an infinite Lie algebra generated by elements $L_n$ $(n\in \mathbb{Z})$ and $c$ over $\mathbb{C}$ with relations $$[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n,0}.$$ A typical explanation of this definition is the following. </p> <blockquote> <p>Define vector fields $l_n=-z^n\frac{\partial}{\partial z}$ on $\mathbb{C}\setminus {0}$. They form a Lie algebra of infinitesimal conformal transformation $$[l_m,l_n]=(m-n)l_{m+n}.$$ So the Virasoro algebra is a central extension of this algebra by $c$. $c$ is called the central charge.</p> </blockquote> <p>My questions are </p> <ol> <li>How can one see that the Lie algebra above is associate to infinitesimal conformal transformation?</li> <li>What is the central charge $c$ intuitively? Why are we interested in such a central extension?</li> </ol> <p>As to second question, I don't have enough physics background to check what the central charge $c$ means in physics literature. </p> <p>At this point, I don't have any intuition and have trouble in digesting the concept. I would really appreciate your help. </p> http://mathoverflow.net/questions/103651/global-dimensions-of-non-commutative-rings Global dimensions of non-commutative rings Muon 2012-08-01T02:33:00Z 2012-11-28T19:59:26Z <p>This is related to my previous question: <a href="http://mathoverflow.net/questions/102628/when-is-a-quantum-affine-space-mathbban-calabi-yau" rel="nofollow">When is a quantum affine space $\mathbb{A}^{n}$ Calabi-Yau?</a> I now would like to know the global dimension of the ring $R=\mathbb{C}\langle x_1,\dots,x_n\rangle/I$, where I is the two-sided ideal generated by $x_ix_j=a_{ij}x_jx_i$ for some $a_{ij}=a_{ji}^{-1}\in \mathbb{C}$ for $1≤i,j≤n$. Recall that the global dimension $gl\dim(S)$ of a ring $S$ is defined to be the supremum of the set of projective dimensions of all left $S$-modules. </p> <p>In case your ring $R$ is defined by a quiver with relations, there seem some techniques. Are there another approach to compute $gl\dim(R)$? Moreover, are there any standard way to compute $gl\dim(S)$ when $S$ is non-commutative? </p> <p>There are many characterization of $gl\dim(R)$ when $R$ is commutative and I am aware of similar questions, such as <a href="http://mathoverflow.net/questions/53138/commutative-ring-of-finite-global-dimension" rel="nofollow">Commutative Ring of Finite Global Dimension</a>.</p> http://mathoverflow.net/questions/112558/boundedness-of-modules-on-as-regular-algebras Boundedness of modules on AS regular algebras Muon 2012-11-16T07:10:47Z 2012-11-16T07:10:47Z <p>Let $k$ be an algebraically closed field and $A$ be an Artin-Shelter regular $k$-algebra. Fix a numerical polynomial $H(t)$. I would like to know whether or not semi-stable f.g. graded $A$-modules with Hilbert polynomial $H(t)$ is bounded. More precisely we regard g.f. graded modules $M$ and $N$ as isomorphic is they are isomorphic up to torsion modules. Semi-stability and boundedness are defined in the same manner as commutative case. </p> <p>The boundedness may not be hold for general AS regular algebras, but then I would like to know for which class of algebra this kind of boundedness result hold. I would appreciate any reference, comments and suggestion. </p> http://mathoverflow.net/questions/108183/strongly-noetherian-propoerty-when-is-the-tensor-a-otimes-kb-noetherian-for Strongly Noetherian propoerty. When is the tensor $A\otimes_{k}B$ Noetherian for Noetherian rings $A$ and $B$? Muon 2012-09-26T17:10:04Z 2012-11-12T19:00:32Z <p>Let $k$ be a field. It is well-known that $A\otimes_{k}B$ is not necessarily Noetherian even if $k$-algebras $A$ and $B$ are Noetherian. For example $\mathbb{R}\otimes_{\mathbb{Q}}\mathbb{R}$. </p> <ol> <li><p>When is the tensor $A\otimes_{k}B$ Noetherian for Noetherian "commutative" $k$-algebras $A$ and $B$? </p></li> <li><p>What if $A$ is noncommutative? Are there any good criteria for $A\otimes_{k}B$ to be Noetherian? If $B$ is a finitely generated $k$-algebra, Hilbert's basis theorem implies that $A\otimes_{k}B$ is again Noetherian. So we need to check this with quite nasty $B$. </p></li> </ol> <p>My primary motivation to ask these questions is the second question. Such ring $A$ is called a "strongly Noethrian ring" and has a lot of good properties, but I don't know many examples. Moreover I realized that things are not very clear even in commutative case and I need to understand commutative case first. I would appreciate it if experts on MO could let me know good criteria for this property and provide me with examples. </p> <p>Rings I have in my mind are weakly noncommutative in the sense that they are commutative up to scalar multiplication such as quantum planes and their $good$ hypersurfaces. </p> http://mathoverflow.net/questions/111545/does-castelnuovo-mumford-regularity-hold-for-this-mathbbc-algebra Does Castelnuovo-Mumford regularity hold for this $\mathbb{C}$-algebra$? Muon 2012-11-05T10:48:43Z 2012-11-05T10:48:43Z <p>Let$R$be a noncommutative finitely generated$\mathbb{C}$-algebra such that its center$S$is smooth (in commutative sense) and$R$is finite over$S$. Is there Castelnuovo-Mumford regularity theorem for this kind of$\mathbb{C}$-algebra$R$? I have $$R=\mathbb{C}\langle x,y,z \rangle/(xy=\zeta_1yx,yz=\zeta_2zy,zx=\zeta_3xz,x^3+y^3+z^3),$$ where$\zeta_i$is a 3rd root of unity. I am aware of CM regularity theorem for AS regular algebra but not sure if it works for this example. Thank you in advance. </p> http://mathoverflow.net/questions/111543/balanced-dualizing-complex-vs-rigid-dualizing-complex Balanced dualizing complex vs rigid dualizing complex? Muon 2012-11-05T10:27:57Z 2012-11-05T10:27:57Z <p>In noncommutative projective geometry, there is a counterpart of dualizing complex in commutative world. It seems to me that they are called either a balanced dualizing complex or rigid dualizing complex. I am aware that they are different objects but cannot really understand how they are different. I would appreciate it if someone kindly explain to me the difference. </p> <p>P.S. I am reading <a href="http://hardy.uhasselt.be/personal/vdbergh/Publications/dualcompl.ps" rel="nofollow">this paper</a> by M. van den Bergh</p> http://mathoverflow.net/questions/103581/diophantine-theory-of-homogeneous-cubic-polynomials Diophantine theory of homogeneous cubic polynomials Muon 2012-07-31T04:33:51Z 2012-10-13T03:29:22Z <p>Arithmetic of quadratic forms over$\mathbb{Z}$(or lattices theory) has received much attention and there are many applications in broad area of mathematics (such as intersection forms on fourfolds). I now wonder whether or not a similar theory for cubic forms can be developed. I recently found a beautiful theorem about binary cubic forms: </p> <blockquote> <p><strong>Theorem (B. N. Delone and D. K. Faddeev,W.-T. Gan, B. H. Gross, and G. Savin)</strong> There is a canonical bijection between isomorphism classes of cubic rings and the set of$GL_{2}(\mathbb{Z})$-equivalence classes of integral binary cubic forms. Under this bijection, the discriminant of a cubic ring is equal to the discriminant of the corresponding binary cubic form.</p> </blockquote> <p>I don't know how useful this theorem is because I don't know how difficult to classify cubic rings. My question is, are there any classification theory when the number of variable is small? I would appreciate it if anyone could give me a reference for recent development of cubic forms. </p> http://mathoverflow.net/questions/109347/point-modules-of-quantum-projective-space-mathbbpn Point modules of quantum projective space$\mathbb{P}^n$Muon 2012-10-11T01:42:57Z 2012-10-11T14:57:03Z <p>Let$A$be a quantum$\mathbb{P}^n$defined by $$A=\mathbb{C}\langle x_1,x_2,\dots,x_{n+1}\rangle/(x_ix_j-r_{ij}x_jx_i)_{1\le i &lt; j\le n+1}.$$ I would like to know the set$X$of isomorphism classes of point modules for$A$. Here a point module is a cyclic graded right$A$-module$M$such that each graded piece of$M$is one-dimensional. </p> <p>The set$Y$of isomorphism classes of of point modules for the quantum$\mathbb{P}^2$$$\mathbb{C}\langle x_1,x_2,x_3\rangle/(x_ix_j-r_{ij}x_jx_i)_{1\le i &lt; j\le 3}$$ is known to be$\mathbb{P}^2$or a union of three lines in$\mathbb{P}^2$. </p> <p>It seems known that$X$is projective for$n=4$, but does anyone know explicit description of$X$? I tried to compute$X$in a similar manner as the quantum$\mathbb{P}^2$case, but it seems quite complicated. I would also appreciate it if someone give me a good description for higher dimensional$n$case (especially$n=3,4$). </p> <p>Thank you very much. </p> http://mathoverflow.net/questions/109362/complex-manifold-x-with-dim-h0x-omega-dim-mathbbc-x2 Complex manifold$X$with$\dim H^0(X,\Omega^{\dim_{\mathbb{C}} X})>2$Muon 2012-10-11T08:30:50Z 2012-10-11T08:53:41Z <p>Are there any complex surface or threefold$X$with $$\dim H^0(X,\Omega^{\dim_{\mathbb{C}} X})>2?$$ I am asking this because I don't know any example while there are complex curves of genus greater than one. I guess that there are no such example. If so, could someone kindly explain why? Any counter example is also welcome. </p> <p><strong>Edit</strong> My question turns out to be a silly question. Please ignore this. </p> http://mathoverflow.net/questions/108456/how-to-understand-ext-mathcalo-y-mathcalo-z-for-subvarieties-y-z-su How to understand$Ext(\mathcal{O}_{Y},\mathcal{O}_{Z})$for subvarieties$Y,Z\subset X$? Muon 2012-09-30T09:26:51Z 2012-10-01T10:13:27Z <p>By standard homological algebra we know that$Ext(A,B)$of$R$-modules classifies certain equivalence classes of short exact sequences$0\rightarrow B\rightarrow C \rightarrow A \rightarrow 0$of$R$-modules, where$R$is a commutative ring. I now would like to understand this fact in geometry. </p> <ol> <li>Let$X$be a variety (or a scheme if you want), how should I understand$Ext(O_{Y},O_{Z})$for subvarieties$Y,Z\subset X$? Of course it classifies extensions of$O_{Z}$by$O_{Y}$, but are there any geometric or intuitive way to understand$Ext(O_{Y},O_{Z})$? </li> <li>More generally are there any geometric way to understand$Ext(\mathcal{E},\mathcal{F})$for coherent$O_X$-modules$\mathcal{E},\mathcal{F}$? </li> </ol> <p>I would appreciate any idea about "seeing" these extensions. </p> http://mathoverflow.net/questions/107133/moduli-space-of-modules-over-non-commutative-rings Moduli space of modules over non-commutative rings Muon 2012-09-13T23:39:11Z 2012-09-14T09:10:57Z <p>Let$X=Proj(A)$be a projective scheme, one can the moduli space of coherent sheaves on$X$with fixed Hilbert polynomial and stability. Since coherent sheaves on$X$are all obtained as the sheafification$\widetilde{M}$of a graded$A$-module$M$, it is reasonable to ask whether we can construct the moduli space of sheaves on non-commutative space, i.e. the moduli space of graded right$B$-modules for some$good$non-commutative graded ring$B$with some fixed data. I am pretty sure that some works in this direction have been done (what condition on$B$and what data of modules should be fixed, etc). </p> <p>Could anyone tell me a reference or paper which discuss the construction? Thank you very much. </p> <p><strong>Edit</strong> I have for example a quantum plane in my mind. </p> http://mathoverflow.net/questions/106798/deformation-of-modules-over-noncommutaitve-rings Deformation of modules over noncommutaitve rings Muon 2012-09-10T09:33:35Z 2012-09-11T22:58:21Z <p>Let$M$be a finitely generated module over a commutative ring$R$. The first order deformation of module$M$is parametrized by$Ext^{1}(M,M)$and the obstruction is parametrized by$Ext^{2}(M,M)$. Is there a similar story for noncommutative$R$? I don't expect this to be true for any noncommutative rings but wonder if this is still true for some$good$ones. I would appreciate any reference suggestion, comments, and ideas. </p> <p><strong>Edit</strong> I would like naively to compute the tangent space of the moduli space$X$of module with some data if it exists. At$M\in X$, the tangent space can be understood as the set of extensions of the$R$-module$M$to some$R\otimes_{k} k[\epsilon]/(\epsilon^2)$-module. The obstruction is defined in the same manner. </p> http://mathoverflow.net/questions/89228/online-number-theory-video Online Number Theory Video? Muon 2012-02-22T21:30:05Z 2012-02-22T21:30:05Z <p>Are there any graduate level number theory course available on line ? The only video series I am aware of are some MSRI videos, and Ted Chinburg's courses <a href="http://www.math.upenn.edu/~ted/noframes.html" rel="nofollow">http://www.math.upenn.edu/~ted/noframes.html</a>, which are hard to see sometimes. I am also looking for good video lectures on algebraic k-theory. Thank you for your sharing information :)</p> http://mathoverflow.net/questions/128213/is-pi-1-widetildex-g-always-finite-if-pi-1x-is-finite/128220#128220 Comment by Muon Muon 2013-04-22T04:42:06Z 2013-04-22T04:42:06Z Thank you for the nice answer, Francesco. I was a bit surprised to know that my question was not as easy as I had initially expected. http://mathoverflow.net/questions/125196/stability-of-t-mathbbp2-and-omega-mathbbp2/125201#125201 Comment by Muon Muon 2013-03-21T23:20:43Z 2013-03-21T23:20:43Z Thank you for the answer, but could you explain a bit more? I don't know why it is enough to check$H^0(T_{\mathbb{P}^2}(-2))=0$. I am not familiar with this field. http://mathoverflow.net/questions/86583/geometric-treatment-of-the-ward-takahashi-identity Comment by Muon Muon 2013-02-27T01:25:05Z 2013-02-27T01:25:05Z Could you explain what you mean by Ward-Takahashi identity in your question, please? http://mathoverflow.net/questions/121242/lifting-a-birational-map-of-x-g-to-a-birational-map-of-x/121250#121250 Comment by Muon Muon 2013-02-11T00:40:12Z 2013-02-11T00:40:12Z Doesn't your example have fixed locus$f=0$? http://mathoverflow.net/questions/119139/are-rational-sections-of-a-vector-bundle-useful/119177#119177 Comment by Muon Muon 2013-01-19T10:13:06Z 2013-01-19T10:13:06Z Thank ou for the answer. http://mathoverflow.net/questions/117317/what-do-correlation-functions-compute-in-cft Comment by Muon Muon 2012-12-28T17:50:11Z 2012-12-28T17:50:11Z @Kelly What is &quot;moment&quot; of the partition function? http://mathoverflow.net/questions/117317/what-do-correlation-functions-compute-in-cft/117386#117386 Comment by Muon Muon 2012-12-28T07:29:31Z 2012-12-28T07:29:31Z Thank you for the detailed answer providing physic background. I don't fully understand your answer, but now get a feeling about what CFT try to understand. http://mathoverflow.net/questions/117317/what-do-correlation-functions-compute-in-cft/117373#117373 Comment by Muon Muon 2012-12-28T02:54:03Z 2012-12-28T02:54:03Z Thank you for the answer, Igor. I am now happy to know that holomorphicity may be interpreted as a field equation. May I ask what the fields$\phi(z)$and$J(z)$stands for in this context? Is$\langle0| J(z)J(w)|0\rangle$the amplitude to observe a particle at$z$after a particle is created at$w$or something? http://mathoverflow.net/questions/117317/what-do-correlation-functions-compute-in-cft Comment by Muon Muon 2012-12-28T02:48:17Z 2012-12-28T02:48:17Z @Jeff I would rather want to know the physical interpretation. It would be nice if you could provide an answer. By the way, I don't know if there is any &quot;purely mathematical interpretation&quot; of correlation function. http://mathoverflow.net/questions/117317/what-do-correlation-functions-compute-in-cft Comment by Muon Muon 2012-12-28T02:45:40Z 2012-12-28T02:45:40Z @user1540 My question is a bit vague. I would like to know both the complex coordinates and the physical meaning of$J(z)$, and they must be related. My problem is that I don't have any intuition behind what I am computing. http://mathoverflow.net/questions/117287/motivation-of-virasoro-algebra/117336#117336 Comment by Muon Muon 2012-12-27T22:53:46Z 2012-12-27T22:53:46Z Thanks you for letting us know the projective representation point of view. That makes sense. http://mathoverflow.net/questions/117287/motivation-of-virasoro-algebra Comment by Muon Muon 2012-12-27T06:30:24Z 2012-12-27T06:30:24Z I don't quite understand what you try to mean in the second comment. I would appreciate it if you could post a bit more detail as an answer. http://mathoverflow.net/questions/117287/motivation-of-virasoro-algebra/117290#117290 Comment by Muon Muon 2012-12-27T06:29:05Z 2012-12-27T06:29:05Z Thanks for your writing &quot;why$c=D\$&quot; above. http://mathoverflow.net/questions/117287/motivation-of-virasoro-algebra/117290#117290 Comment by Muon Muon 2012-12-27T00:48:59Z 2012-12-27T00:48:59Z Thank you for the answer, Chris. May I ask why multiplication by the unit vector is related to the space dimension? http://mathoverflow.net/questions/117287/motivation-of-virasoro-algebra Comment by Muon Muon 2012-12-27T00:42:42Z 2012-12-27T00:42:42Z @Abdelmalek Thanks for pointing out the typo.