User b. bischof - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T23:49:18Z http://mathoverflow.net/feeds/user/348 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/13526/geometric-interpretation-of-trace Geometric Interpretation of Trace B. Bischof 2010-01-31T01:49:07Z 2013-04-05T22:43:23Z <p>This afternoon I was speaking with some graduate students in the department and we came to the following quandry;</p> <blockquote> <p>Is there a geometric interpretation of the trace of a matrix?</p> </blockquote> <p>This question should make fair sense because trace is coordinate independent.</p> <p>A few other comments. We were hoping for something like: </p> <p>"determinant is the volume of the parallelepiped spanned by column vectors."</p> <p>This is nice because it captures the geometry simply, and it holds for any old set of vectors over $\mathbb{R}^n$.</p> <p>The divergence application of trace is somewhat interesting, but again, not really what we are looking for.</p> <p>Also, after looking at the <a href="http://en.wikipedia.org/wiki/Trace%5F%28linear%5Falgebra%29" rel="nofollow">wiki</a> entry, I don't get it. This then requires a matrix function, and I still don't really see the relationship.</p> <p>One last thing that we came up with; the trace of a matrix is the same as the sum of the eigenvalues. Since eigenvalues can be seen as the eccentricity of ellipse, trace may correspond geometrically to this. But we could not make sense of this.</p> <p>Thanks in advance.</p> http://mathoverflow.net/questions/105727/isomorphisms-of-quantum-planes/106064#106064 Answer by B. Bischof for Isomorphisms of quantum planes B. Bischof 2012-08-31T19:21:12Z 2012-09-01T02:26:05Z <p>While the above is a good way to see it, I kinda like to use the natural representation of $U_q(sl_2)$ on the quantum planes.</p> <p>Recall that $U_q(sl_2)\simeq_{\mathfrak{Hopf}}U_p(sl_2)$ iff $q=\pm p^{\pm1}$. Now recall there are faithful Hopf representations $\rho_q:U_q(sl_2)\hookrightarrow End_{\mathbb{C}}(\mathbb{C}_q[x,y])$ and $\rho_p:U_p(sl_2)\hookrightarrow End_{\mathbb{C}}(\mathbb{C}_p[x,y])$ given by $E(1)=0$, $F(1)=0$, $K(1)=1$, $E(x)=0$, $E(y)=x$, $F(x)=y$, $F(y)=0$, $K(x)=qx$, $K(y)=q^{-1}y$. So the quantum groups are Hopf subalgebras endomorphism rings of the quantum planes. Now if there exists an isomorphism of these quantum planes it would carry one quantum group to the other.</p> <p>My only concern is that this uses the Hopf structure on the quantum plane.</p> <p>EDIT: As Mariano points out, I was a bit sloppy and need to think a bit more. I will try to revise this in the next couple days. Sorry for the stupidity.</p> http://mathoverflow.net/questions/102628/when-is-a-quantum-affine-space-mathbban-calabi-yau/102633#102633 Answer by B. Bischof for When is a quantum affine space $\mathbb{A}^{n}$ Calabi-Yau? B. Bischof 2012-07-19T09:58:45Z 2012-07-19T10:03:52Z <p>The answer to the question "is the functions on quantum affine space a Calabi-Yau algebra?" is yes.</p> <p>It is presented <a href="http://www-math.mit.edu/~trasched/deformations.pdf" rel="nofollow">here</a> (in particular 5.2) in Travis Schedler's notes from MSRI.</p> <p>However, if this coincides with your alternate definition of CY I do not know. The definition of CY and twisted CY as used in this are 3.27. If I remember correctly the discussion, this is supposed to coincide in some sense with the categorical definition. In particular, by making the Ext condition with respect to Hochschild Homology, but also adding a resolution length condition.</p> http://mathoverflow.net/questions/14680/the-real-use-of-quantum-algebra-non-commutative-geometry-representation-theor The 'real' use of Quantum Algebra, Non-commutative Geometry, Representation Theory, and Algebraic Geometry to Physics B. Bischof 2010-02-08T19:43:03Z 2012-04-01T12:07:33Z <p>In <a href="http://mathoverflow.net/questions/14518/applications-of-noncommutative-geometry" rel="nofollow">this question</a>, Orbicular made the following comment to Feb7 and my own answers;</p> <blockquote> <p>Please keep in mind that - even though it is stated very often - noncommutative geometry does not give "real" insight to physics. The reason is that they only have toy models, all of which are unphysical (in the sense that they predict things which differ from real world measurements). Furthermore even the toy models are usually extremely complicated, killing most expectations to get a "real" model (which is not toyish).</p> </blockquote> <p>First, I want to thank Orbicular for pointing this out, as it is something that I 'kinda' knew, but often forget. The purpose of <em>this</em> question, is to ask for a deeper explanation, either from Orbicular or someone else. In particular</p> <blockquote> <p>to what degree does Quantum Algebra, Non-commutative Geometry, Representation Theory, and Algebraic Geometry influence/assist 'real' models and actual physics related to the physical world?</p> </blockquote> <p>I don't wish for this question to turn into a debate about whether or not these maths will later be applied in some beautiful stringy-quantum-symmetry theory; I would much rather it be some explanation of the real use of these things. Specifically, I am interested in hearing about the use of Quantum Groups and their representations to Physicists along with some thoughts on the actual usefulness of the results in NC Algebraic Geometry of those articles I posted over <a href="http://mathoverflow.net/questions/10512/" rel="nofollow">here</a>. Another particularly interesting subject I would like to hear about is the usefulness of commutative algebraic geometry in physics.</p> <h2>Some things I have found</h2> <p>Just two references that I have found that at least address these things to some degree are Peter Woit's lecture <a href="http://www.math.columbia.edu/~woit/repthy.html" rel="nofollow">notes</a> on Representation Theory, and in Shawn Majid's book on Quantum Groups he discusses some definite physical motivation for studying quantum groups.</p> <p>Thanks! </p> http://mathoverflow.net/questions/49384/tools-for-long-distance-collaboration/86720#86720 Answer by B. Bischof for Tools for long-distance collaboration B. Bischof 2012-01-26T14:13:15Z 2012-01-26T14:13:15Z <p>Since this has been stirred back up after so long, I will chime in;</p> <p><a href="http://asana.com/" rel="nofollow">http://asana.com/</a></p> <p>is a great tool for organizing to-do lists with lots of tools for making the lists more than just lists.</p> http://mathoverflow.net/questions/77885/2-completeness-analog-of-completeness-theorem 2-completeness analog of completeness theorem B. Bischof 2011-10-12T02:14:17Z 2011-10-12T04:07:03Z <p>It's not hard to see that a category is finitely complete if it has finite products and equalizers. In short, this is because one can write all limits as iterations of these two "operations". </p> <p>I wonder if there is a 2-version of this. In particular,</p> <blockquote> <p>Does a category have all finite 2-limits if it has all 2-equalizers and 2-products?</p> </blockquote> <p>My instinct is no, and that we will need another(or several more) limits to build all 2-limits. </p> <p>Of course the question can be generalized to n-limits, and I'm curious about that also. </p> http://mathoverflow.net/questions/77616/d-modules-and-algebraic-solutions-of-pdes/77659#77659 Answer by B. Bischof for D-modules and Algebraic Solutions of PDEs B. Bischof 2011-10-10T04:57:18Z 2011-10-10T04:57:18Z <p>Perhaps you are looking for something deeper, but right there at the beginning of Hotta, Takeuchi, and Tanisaki's book on D-mods in the introduction is the connection to Linear PDEs.</p> <p>I quote: </p> <blockquote> <p>Therefore, systems of linear partial differential equations can be identified with the D-modules having some finite presentations like (0.0.3), and the purpose of the theory of linear PDEs is to study the solution space HomD(M, O). Since the space HomD(M, O) does not depend on the concrete descriptions (0.0.2) and (0.0.3) of M (it depends only on the D-linear isomorphism class of M), we can study these analytical problems through left D-modules admitting finite presentations. In the language of categories, the theory of linear PDEs is nothing but the investigation of the contravariant functor HomD(•, O) from the category M(D) of D-modules admitting finite presentations to the category M(C) of C-modules.</p> </blockquote> http://mathoverflow.net/questions/69059/a-simple-proof-of-the-weyl-algebras-rigidity A simple proof of the Weyl algebra's rigidity. B. Bischof 2011-06-28T22:03:26Z 2011-06-29T01:11:42Z <p>I am wondering if there is a nice presentation of the Hochschild cohomology of $A_n$ the Weyl algebra. It is known that $H^m(A_n,A_n)=0$ for $m>0$, and thus it is rigid. A proof can be found <a href="http://www.ams.org/journals/tran/1961-100-03/S0002-9947-1961-0130900-1/S0002-9947-1961-0130900-1.pdf" rel="nofollow">in Sridharan</a>, but this proof seems to be doing a lot more and is fairly complicated. </p> <p>I was wondering if there was a simpler way to see this fact specifically. Essentially, I am being a bit lazy. </p> <p>Thanks!</p> http://mathoverflow.net/questions/25924/introducing-cryptology-to-undergraduates Introducing Cryptology to Undergraduates B. Bischof 2010-05-25T20:27:09Z 2011-05-07T02:42:54Z <p>This summer I am going to give some lectures to some REU students. I am still tossing around ideas for what I am going to talk about, but one thing I would at least like to give one or two lectures on, is Cryptology.</p> <p>I had a fairly standard undergraduate course on Number Theory where we learned basic cyphers and some things about encryption. However, I am hoping to talk about the relationship of elliptic curves to encryption. Is there an appropriate level book that covers this relationship? Many of the students are strong, but lack some background. Many will have some experience with number theory, but may lack Abstract Algebra and Advanced Calculus.</p> <p>In the absence of a nice book talking about elliptic curves relation to cryptology, I will probably talk about the excellent book by Ash and Gross.</p> <p>I was just hoping to add this topic into my mix of lectures so I thought the MO community could offer some suggestions.</p> <p>Thanks in advance! </p> <p>EDIT: I wanted to add that Diffe-Hulman will definitely be covered as one of the main research projects will focus on it. The elliptic curves comes by request of the students, who have heard cool things about them :D</p> http://mathoverflow.net/questions/63033/what-would-you-want-on-a-lie-theory-cheat-poster/63049#63049 Answer by B. Bischof for What would you want on a Lie theory cheat poster? B. Bischof 2011-04-26T14:51:41Z 2011-04-26T14:51:41Z <p>I know of a couple physicists with charts of 6j's hanging in their office. </p> http://mathoverflow.net/questions/43820/extremely-messy-proofs/58616#58616 Answer by B. Bischof for Extremely messy proofs B. Bischof 2011-03-16T05:50:36Z 2011-03-30T23:57:26Z <p><strong>Poincare Duality</strong></p> <p>The original formulation was not only messy, but <em>wrong</em>. The modern formulation is more powerful, more elegant, and don't forget correct.</p> <p>You can read about both approaches and some of the history on the <a href="http://en.wikipedia.org/wiki/Poincare_duality" rel="nofollow">wiki page</a>. </p> http://mathoverflow.net/questions/48299/more-open-problems/59934#59934 Answer by B. Bischof for More open problems B. Bischof 2011-03-29T03:44:34Z 2011-03-29T03:44:34Z <p>Ismar Volic from Wellesly has a list of problems in Calculus of Functors related to knot theory on his webpage <a href="http://palmer.wellesley.edu/~ivolic/pdf/Papers/KnotsLiteratureAndOpenProblems-May10.pdf" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/7320/heuristic-explanation-of-why-we-lose-projectives-in-sheaves Heuristic explanation of why we lose projectives in sheaves. B. Bischof 2009-11-30T23:55:55Z 2011-03-13T23:38:29Z <p>We know that presheaves of any category have enough projectives and that sheaves do not, why is this, and how does it effect our thinking?</p> <p><a href="http://mathoverflow.net/questions/5378/when-are-there-enough-projective-sheaves-on-a-space-x" rel="nofollow">This</a> question was asked(and I found it very helpful) but I was hoping to get a better understanding of why.</p> <p>I was thinking about the following construction(given during a course);</p> <p>given an affine cover, we normally study the quasi-coherent sheaves, but in fact we could study the presheaves in the following sense:</p> <p>Given an affine cover of X,</p> <p>$Ker_2\left(\pi\right)\rightrightarrows^{p_1}_{p_2} U\rightarrow X$</p> <p>then we can define $X_1:=Cok\left(p_1,p_2\right)$, a presheaf, to obtain refinements in presheaves where we have enough projectives and the quasi-coherent sheaves coincide. Specifically, if $X_1\xrightarrow{\varphi}X$ for a scheme $X$, s.t. $\mathcal{S}\left(\varphi\right)\in Isom$ for $\mathcal{S}(-)$ is the sheaffication functor, then for all affine covers $U_i\xrightarrow{u_i}X$ there exists a refinement $V_{ij}\xrightarrow{u_{ij}}U_i$ which factors through $\varphi$.</p> <p>This hinges on the fact that $V_{ij}$ is representable and thus projective, a result of the fact that we are working with presheaves. In sheaves, we would lose these refinements. Additionally, these presheaves do not depend on the specific topology(at the cost of gluing).</p> <p>In this setting, we lose projectives because we are applying the localization functor which is not exact(only right exact). However, I don't really understand this reason, and would like a more general answer. </p> <p>A related appearance of this loss is in homological algebra. Sheaves do not have enough projectives, so we cannot always get projective resolutions. They do have injective resolutions, and this is related to the use of cohomology of sheaves rather than homology of sheaves. In paticular, in Rotman's Homological Algebra pg 314, he gives a footnote;</p> <blockquote> <p>In <em>The Theory of Sheaves,</em> Swan writes "...if the base space X is not discrete, I know of no examples of projective sheaves except the zero sheaf." In Bredon, <em>Sheaf Theory</em>: on locally connected Hausdorff spaces without isolated points, the only projective sheaf is 0</p> </blockquote> <p>addressing this situation.</p> <blockquote> <p>In essence, my question is for a heuristic or geometric explanation of why we lose projectives when we pass from presheaves to sheaves.</p> </blockquote> <p>Thanks in advance!</p> http://mathoverflow.net/questions/55200/good-programs-for-drawing-graphs-directed-weighted-graphs/55223#55223 Answer by B. Bischof for Good programs for drawing graphs ( directed weighted graphs ) B. Bischof 2011-02-12T18:00:58Z 2011-02-12T18:00:58Z <p><a href="https://networkx.lanl.gov/trac" rel="nofollow">https://networkx.lanl.gov/trac</a> has a lot of options</p> http://mathoverflow.net/questions/44075/can-we-see-the-geometric-realization-of-u-qsl-2s-relations-as-schubert-condi Can we see the geometric realization of $U_q(sl_2)$'s relations as Schubert Conditions? B. Bischof 2010-10-29T07:05:44Z 2010-10-29T17:05:55Z <p>In Nakajima's <a href="http://www.kurims.kyoto-u.ac.jp/~nakajima/TeX/HongKong/hongkong.pdf" rel="nofollow">Geometric construction of algebras</a>(pages 3-7), he uses the subalgebra of the convolution algebra of $Gr(k^N)\times Gr(k^N)$ invariant under $GL_N$ action to construct $U_q(sl_2)$. To do this, he finds imposes relations to form some special operators, which he shows are isomorphic to the generators of $U_q(sl_2)$. In particular, he uses characteristic functions on certain subsets of $k^N$ with dimensionality conditions. These conditions smell like Schubert conditions.</p> <blockquote> <p>Can we see these as the Schubert conditions in some way?</p> </blockquote> <p>I have compared them, but I don't see the link.</p> <p>EDIT: Read the comments below for a related paper.</p> http://mathoverflow.net/questions/43579/examples-of-categorification/43624#43624 Answer by B. Bischof for Examples of categorification B. Bischof 2010-10-26T03:06:44Z 2010-10-26T03:06:44Z <p>Is it perverse to just quote the original inception by Crane?</p> <p>An obvious nice collection would be <a href="http://arxiv.org/abs/q-alg/9607028" rel="nofollow">the paper with Yetter</a> on examples of Categorification.</p> <p>However, I actually like another Paper of Yetter's better in this direction; <a href="http://math.ucr.edu/home/baez/yetter.pdf" rel="nofollow">categorical linear algebra</a>. </p> <p>Also, Rosenberg's Noncommutative spectrum is a categorification: <a href="http://www.mpim-bonn.mpg.de/preprints/send?bid=3948" rel="nofollow">pdf-link</a>. Not in the strict sense, but "morally". That would be undoubtedly my favorite. </p> http://mathoverflow.net/questions/36848/what-category-without-initial-object-do-you-care-about What category without initial object do you care about? B. Bischof 2010-08-27T07:04:17Z 2010-10-05T15:32:38Z <p>Recently I have been listening to some constructions that have been designed to accommodate categories without an initial object. The speaker has given some idea of a category or two that he cares about, and thus why he was thinking in this direction, but I am now wondering;</p> <blockquote> <p>As working mathematicians, what category are you concerned with that does not have an initial object?</p> </blockquote> <p>I am sorry if this question is slightly strange, I have made it CW because it seems appropriate.</p> <p>Thanks!</p> http://mathoverflow.net/questions/5499/which-mathematicians-have-influenced-you-the-most/5530#5530 Answer by B. Bischof for Which mathematicians have influenced you the most? B. Bischof 2009-11-14T16:47:53Z 2010-09-26T12:49:49Z <p><strong>Erdős</strong></p> <p>Sophomore year when I decided that I didn't like physics classes I just happened to be reading "The Man Who Loved Only Numbers" by Hoffman. Between this and "How to Read and Do Proofs" by Solow, I saw mathematics as something much more beautiful. This combined with reading about Erdos style of mathematics made me really attracted to research and led to my first REU experience. It was all downhill from there.</p> http://mathoverflow.net/questions/38911/a-reference-for-calculus-of-functors-for-model-categories A reference for Calculus of Functors for Model Categories B. Bischof 2010-09-16T01:44:35Z 2010-09-17T00:04:43Z <p>I am wondering where I might look to see what has been done in terms of Calculus of Functors for more general weak equivalences and Model Categories.</p> <p>I am at least aware of some of the extended definitions of the main concepts in Calculus of Functors to weak equivalence such as homotopy limits, but I was wondering if a document existed that worked through the basic of COF in this setting. </p> <p>I am aware also of Lurie's work <a href="http://www.math.harvard.edu/~lurie/papers/GoodwillieI.pdf" rel="nofollow">here.</a>(Thanks Harry for pointing this out.)</p> <p>I appreciate your help.</p> http://mathoverflow.net/questions/33455/seeing-stacks-in-the-calculus-of-functors Seeing stacks in the Calculus of Functors B. Bischof 2010-07-27T00:18:02Z 2010-07-27T09:37:28Z <p>Recently I was told (by an algebraic geometer) that when algebraic geometers look at the Calculus of Functors, they think of stacks.</p> <p>When I look at the Calculus of Functors, I see a categorification of polynomial approximation. While I am at best a beginner at algebraic geometry, I would like to understand why he is saying this. </p> <p>My motivation is twofold. First, I want to know why he is saying this, and second, because I am beginning to learn about stacks, and I want to come at it with some intuition. I have pursued the obvious routes of reading about them in general (such as <a href="http://sbseminar.wordpress.com/2008/06/19/whats-a-stack/" rel="nofollow">Tolland's Blog Post</a>).</p> <p>Specifically, my question is </p> <blockquote> <p>How does one see Calculus of Functors as stacks?</p> </blockquote> <p>A secondary question, </p> <blockquote> <p>Is there some highly degenerate way to look at stacks to see polynomial approximation?</p> </blockquote> <p>Thanks!</p> http://mathoverflow.net/questions/29471/matrix-conjugates-over-finite-fields Matrix Conjugates over Finite Fields B. Bischof 2010-06-25T05:41:59Z 2010-06-25T06:19:24Z <p>Thinking about Diffe-Hillman for matrices brought me to the following question.</p> <p>Given $\mathbb{F}_{p^k}$ the finite field with $p^k$ elements when can we find non-trivial solutions to </p> <p>$$$AQ^rA^{-1}=BQ^sB^-1$$$</p> <p>for $A,B,Q\in Mat_n(\mathbb{F}_{p^k})$ and $r\neq s$?</p> http://mathoverflow.net/questions/8258/whats-a-nice-argument-that-shows-the-volume-of-the-unit-n-ball-in-rn-approaches What's a nice argument that shows the volume of the unit n-ball in R^n approaches 0? B. Bischof 2009-12-08T22:08:04Z 2010-06-23T12:48:05Z <p>Before you close for "homework problem", please note the tags.</p> <p>Last week, I gave my calculus 1 class the assignment to calculate the $n$-volume of the $n$-ball. They had finished up talking about finding volume by integrating the area of the cross-sections. I asked them to calculate a formula for $4$ and $5$, and take the limit of the general formula to get 0.</p> <p>Tomorrow I would like to give them a more geometric idea of why the volume goes to zero. Anyone have any ideas? :)</p> <p>Comm wiki in case people want to add/modify this a bit.</p> http://mathoverflow.net/questions/21881/how-should-one-present-curl-and-divergence-in-an-undergraduate-multivariable-calc/21990#21990 Answer by B. Bischof for How should one present curl and divergence in an undergraduate multivariable calculus class? B. Bischof 2010-04-20T17:56:49Z 2010-04-20T17:56:49Z <p>This was originally a comment that got too big. Since I wont address the real heart of your question, it is CW. I hope this is ok :/</p> <p>I am also teaching Multivar out of Stewart this semester. As it has been suggested, I stick fairly close to the book, even working through some of the same examples he does. I focus very hard on motivating the ideas from the ground up. </p> <p>For instance when talking about curvature, I made the students try to define curvature for themselves. Then proceeded to find little issues with their definitions, until we arrived at something that was pretty close to the standard defn(with lots of urging).</p> <p>Just yesterday Terry Tao posted a link to this video;</p> <p><a href="http://www.youtube.com/watch?v=BlvKWEvKSi8" rel="nofollow">http://www.youtube.com/watch?v=BlvKWEvKSi8</a></p> <p>which talks about this exact style of teaching. I think that it works very well for Calc 3, where many of the students are pretty decent at math to get to that point. In so far as specific examples and motivation, I really enjoy the notes by Oliver Knill;</p> <p><a href="http://abel.math.harvard.edu/~knill//teaching/math21a/index.html" rel="nofollow">http://abel.math.harvard.edu/~knill//teaching/math21a/index.html</a></p> <p>There are some nice diagrams, examples, and explanations of pretty much everything in Calc 3. In particular, he gives real physical applications of the ideas, and at the end he gives a "calc beyond calc" intro.</p> http://mathoverflow.net/questions/10512/theories-of-noncommutative-geometry/14443#14443 Answer by B. Bischof for Theories of Noncommutative Geometry B. Bischof 2010-02-06T23:18:46Z 2010-04-18T09:43:26Z <p>In accordance with the suggestion of Yemon Choi, I am going to suggest some further delineation of the approaches to "Non-commutative Algebraic Geometry". I know very little about "Non-commutative Differential Geometry", or what often falls under the heading "&agrave; la Connes". This will be completely underrepresented in this summary. For that I trust Yemon's summary to be satisfactory. (<b>edit by YC</b>: BB is kind to say this, but my attempted summary is woefully incomplete and may be inaccurate in details; I would encourage anyone reading to investigate further, keeping in mind that the NCG philosophy and toolkit in analysis did not originate and does not end with Connes.)</p> <p>Also note that much of what I know about these approaches comes from two sources:</p> <ol> <li><p>The paper by <a href="http://arxiv.org/abs/math/0501166" rel="nofollow">Mahanta</a></p></li> <li><p>My advisor A. Rosenberg.</p></li> </ol> <p>Additionally, much useful discussion took place at <a href="http://mathoverflow.net/questions/7917/non-commutative-algebraic-geometry" rel="nofollow">Kevin Lin's</a> question (as Ilya stated in his answer).</p> <p>I think a better break down for the NCAG side would be:</p> <h2>A. Rosenberg/Gabriel/Kontsevich approach</h2> <p>Following the philosophy of Grothendieck: "to do geometry, one needs only the category of quasi-coherent sheaves on the would-be space" (<strong>edit by KL</strong>: Where does this quote come from?)</p> <p>In the famous <a href="http://www.numdam.org/item?id=BSMF_1962__90__323_0" rel="nofollow">dissertation of Gabriel</a>, he introduced the injective spectrum of an abelian category, and then reconstructed the commutative noetherian scheme, which is a starting point of noncommutative algebraic geometry. Later, A. Rosenberg introduced the left spectrum of a noncommutative ring as an analogue of the prime spectrum in commutative algebraic geometry, and generalized it to any abelian category. He used one of the spectra to reconstruct any quasi-separated (not necessarily quasi-compact), commutative scheme. (Gabriel-Rosenberg reconstruction theorem.)</p> <p>In addition, Rosenberg has described the NC-localization (first observed also by Gabriel) which has been used by him and Kontsevich to build NC analogs of more classical spaces (like the NC Grassmannian) and more generally, noncommutative stacks. Rosenberg has also developed the homological algebra associated to these 'spaces'. Applications of this approach include representation theory (D-module theory in particular), quantum algebra, and physics.</p> <p><strong>References in this area</strong> are best found through the MPIM Preprint Series, and a large collection is linked <a href="http://ncatlab.org/nlab/show/Alexander+Rosenberg" rel="nofollow">here</a>. Additionally, a book is being written by Rosenberg and Kontsevich furthering the work of their previous <a href="http://arxiv.org/abs/math/9812158" rel="nofollow">paper</a>. Some applications of these methods are used <a href="http://www.springerlink.com/content/e10865131k571857/" rel="nofollow">here</a>, <a href="http://journals.cambridge.org/action/displayAbstract?fromPage=online&amp;aid=2519608" rel="nofollow">here</a>, <a href="http://arxiv.org/abs/math.QA/0301090" rel="nofollow">here</a>, and <a href="http://arxiv.org/abs/math.QA/0403276" rel="nofollow">here</a>. The first two are focusing on representation theory, the second two on non-commutative localization.</p> <h2>Kontsevich/Soibelman approach</h2> <p>They might refer to their approach as "formal deformation theory", and quoting directly from their <a href="http://www.math.ksu.edu/~soibel/Book-vol1.ps" rel="nofollow">book</a></p> <blockquote> <p>The subject of deformation theory can be defined as the "study of moduli spaces of structures...The subject of this book is formal deformation theory. This means $\mathcal{M}$ will be a formal space(e.g. a formal scheme), and a typical category $\mathcal{W}$ will be the category of affine schemes..."</p> </blockquote> <p>Their approach is related to $A_{\infty}$ algebras and homological mirror symmetry. <strong>References that might help</strong> are the papers of <a href="http://xxx.lanl.gov/find/hep-th/1/au%3a+Soibelman_Y/0/1/0/all/0/1" rel="nofollow">Soibelman</a>. Also, I think this is related to the question <a href="http://mathoverflow.net/questions/13005/what-is-formal" rel="nofollow">here</a>. (Note: I know hardly anything beyond that this approach exists. If you know more, feel free to <em>edit</em> this answer! Thanks for your understanding!)</p> <p>(<strong>Some comments by KL</strong>: I am not sure whether it is appropriate to include Kontsevich-Soibelman's deformation theory here. This kind of deformation theory is a very general thing, which intersects some of the "noncommutative algebraic geometry" described here, but I think that it is neither a subset nor a superset thereof. In any case, I've asked some questions related to this on MO in the past, see <a href="http://mathoverflow.net/questions/5364/what-is-a-deformation-of-a-category" rel="nofollow">this</a> and <a href="http://mathoverflow.net/questions/385/deformation-theory-and-differential-graded-lie-algebras" rel="nofollow">this</a>.</p> <p>However, there is the approach of noncommutative geometry via categories, as elucidated in, for instance, <a href="http://arxiv.org/abs/0806.0107" rel="nofollow">Katzarkov-Kontsevich-Pantev</a>. Here the idea is to think of a category as a category of sheaves on a (hypothetical) non-commutative space. The basic "non-commutative spaces" that we should have in mind are the "Spec" of a (not necessarily commutative) associative algebra, or dg associative algebra, or A-infinity algebra. Such a "space" is an "affine non-commutative scheme". The appropriate category is then the category of modules over such an algebra. Definitively commutative spaces, for instance quasi-projective schemes, are affine non-commutative schemes in this sense: It is a theorem of van den Bergh and Bondal that the derived category of quasicoherent sheaves on a quasi-projective scheme is equivalent to a category of modules over a dg algebra. (I should note that in my world everything is over the complex field; I have no idea what happens over more general fields.) Lots of other categories are or should be affine non-commutative in this sense: <a href="http://mathoverflow.net/questions/9733/matrix-factorizations-and-physics" rel="nofollow">Matrix factorization categories</a> (see in particular <a href="http://arxiv.org/abs/0904.4713" rel="nofollow">Dyckerhoff</a>), and probably various kinds of Fukaya categories are conjectured to be so as well.</p> <p>Anyway I have no idea how this kind of "noncommutative algebraic geometry" interacts with the other kinds explained here, and would really like to hear about it if anybody knows.)</p> <h2>Lieven Le Bruyn's approach</h2> <p>As I know nearly nothing about this approach and the author is a visitor to this site himself, I wouldn't dare attempt to summarize this work.</p> <p>As mentioned in a comment, his website contains a plethora of links related to non-commutative geometry. I recommend you check it out <a href="http://win.ua.ac.be/~lebruyn/index.html" rel="nofollow">yourself</a>.</p> <h2>Approach of Artin, Van den Berg school</h2> <p>Artin and Schelter gave a regularity condition on algebras to serve as the algebras of functions on non-commutative schemes. They arise from abstract triples which are understood for commutative algebraic geometry. (Again edits are welcome!)</p> <p>Here is a nice report on <a href="http://www.birs.ca/workshops/2005/05w5035/report05w5035.pdf" rel="nofollow">Interactions between noncommutative algebra and algebraic geometry</a>. There are several people who are very active in this field: Michel Van den Berg, James Zhang, Paul Smith, Toby Stafford, I. Gordon, A. Yekutieli. There is also a very nice page of Paul Smith: <a href="http://www.math.washington.edu/~smith/Research/research.html" rel="nofollow">noncommutative geometry and noncommutative algebra</a>, where you can find almost all the people who are currently working in the noncommutative world.</p> <p><strong>References:</strong> <a href="http://books.google.com/books?hl=en&amp;lr=&amp;id=_BnSoQSKnNUC&amp;oi=fnd&amp;pg=PA33&amp;dq=%22Artin%22+%22Some+algebras+associated+to+automorphisms+of+elliptic+curves%22+&amp;ots=hRXnP7udMW&amp;sig=t77CnWnsYPHhuonQQffrSXedyj0#v=onepage&amp;q=%22Artin%22%20%22Some%20algebras%20associated%20to%20automorphisms%20of%20elliptic%20curves%22&amp;f=false" rel="nofollow">This</a> paper introduced the need for the regularity condition and showed the usefulness. Again I defer to <a href="http://arxiv.org/abs/math/0501166" rel="nofollow">Mahanta</a> for details. Serre's FAC is the starting point of noncommutative projective geometry. But the real framework is built by Artin and James Zhang in their famous paper <a href="http://www.ingentaconnect.com/content/ap/ai/1994/00000109/00000002/art01087" rel="nofollow">Noncommutative Projective scheme</a>.</p> <h2>Non-commutative Deformation Theory by Laudal</h2> <p>Olav Laudal has approached NCAG using NC-deformation theory. He also applies his method to invariant theory and moduli theory. (Please edit!)</p> <p><strong>References</strong> are on his page <a href="http://folk.uio.no/arnfinnl/" rel="nofollow">here</a> and <a href="http://folk.uio.no/arnfinnl/Noncom.alg.geom.pdf" rel="nofollow">this</a> paper seems to be a introductory article.</p> <h2>Apologies</h2> <p>Without a doubt, I have made several errors, given bias, offended the authors, and embarrassed myself in this post. Please don't hold this against me, just edit/comment on this post until it is satisfactory. As it was said before, the <a href="http://ncatlab.org/nlab/show/noncommutative+geometry" rel="nofollow">nlab</a> article on noncommutative geometry is great, you should defer to it rather than this post.</p> <p>Thanks!</p> http://mathoverflow.net/questions/21685/what-is-the-correct-category-of-multisets What is the "correct" category of multisets B. Bischof 2010-04-17T20:03:43Z 2010-04-18T06:15:26Z <p>During seminar the other day, when speaking about subobject classifiers, I asked if the subobject classifier for the category of multisets would have integer truth values, corresponding to the number of times and element is in the set. We attempted to show this, but quickly realized that we were not even sure of the "correct" category for multisets. </p> <p>To clarify, when I say correct I want my category to </p> <ol> <li>Have objects identified by multisets</li> <li>Have maps between the multisets be on the level of elements in the multiset, and forget the order of those elements, e.g. there is only one map {111223}->{55}</li> <li>The subobject classifier will behave as I had hoped, with {1} having truth value 3 in {111}</li> </ol> <p>My question is;</p> <blockquote> <p>Can you construct a category satisfying these properties?</p> </blockquote> <p>Thanks in advance!</p> <p>EDIT: First, sorry about not checking nLab, I forget about that site far too often. Second, I should say that I have a little bit of motivation for my property two. So let me clarify what I meant in property two. Given a multiset, it can be thought of as a pair $S\times\mathbb{N}$ for a set $S$. Now, when considering morphisms between multisets I want the maps $f,g:\lbrace 1122\rbrace\rightarrow\lbrace34\rbrace$ such that $f$ sends</p> <p>$\begin{eqnarray*} 1&amp;\mapsto&amp; 3,\ 1&amp;\mapsto&amp; 4,\ 2&amp;\mapsto&amp; 3,\ 2&amp;\mapsto&amp; 4\ \end{eqnarray*}$</p> <p>and $g$ sends</p> <p>$\begin{eqnarray*} 1&amp;\mapsto&amp; 4,\ 1&amp;\mapsto&amp; 3,\ 2&amp;\mapsto&amp; 4,\ 2&amp;\mapsto&amp; 3\ \end{eqnarray*}$</p> <p>to be the same morphism. But if $h$ sends </p> <p>$\begin{eqnarray*} 1&amp;\mapsto&amp; 4,\ 1&amp;\mapsto&amp; 4,\ 2&amp;\mapsto&amp; 4,\ 2&amp;\mapsto&amp; 3\ \end{eqnarray*}$</p> <p>then $h$ is not the same as $g$ or $f$. Further I would like it such that $\lbrace 112\rbrace$ is not a subobject of $\lbrace 12\rbrace$ but it is a subobject of $\lbrace 11122\rbrace$.</p> <p>Hopefully this will clear it up.</p> http://mathoverflow.net/questions/13847/research-experience-for-undergraduates-summer-programs/13911#13911 Answer by B. Bischof for Research Experience for Undergraduates Summer Programs B. Bischof 2010-02-03T01:59:55Z 2010-04-09T03:44:31Z <p>In my experience, it depends on the specific funding of the program. Some programs funded by the NSF will require the applicants be Americans. But this is not always the case. Last year at my university, a student from Cambridge took part in the REU. </p> <p>When applying for funding for these programs, the directors must include some explanation of who will be the participants. Additionally, many of these programs specifically encourage minorities to apply, sometimes including non-American applicants.</p> <p>I encourage you to look at the specific programs to see if they allow foreign applicants, emailing them if it is not stated on the webpage.</p> <p><a href="http://www.nsf.gov/crssprgm/reu/list_result.cfm?unitid=5044" rel="nofollow">Here</a> is the webpage that I used when searching for an REU as an undergrad.</p> <p>Additionally, just googling "NSF REU 2010" is helpful.</p> <p>EDIT: Shameless plug: <a href="http://www.math.ksu.edu/reu/sumar/" rel="nofollow">http://www.math.ksu.edu/reu/sumar/</a></p> http://mathoverflow.net/questions/20336/detailed-proof-of-cup-product-equivalent-to-intersection Detailed proof of cup product equivalent to intersection B. Bischof 2010-04-04T22:26:31Z 2010-04-07T04:49:28Z <p>Consider a smooth, closed, compact finite-dim manifold. We have Poincare Duality to relate the cocycles and cycles.</p> <blockquote> <p>I would like to know where I can find a reference for a proof that the cup product of the Cohomology Ring is given by the intersection of the corresponding cycles.</p> </blockquote> <p>Griffiths and Harris talk about intersection number, and discuss this result in chapter 0, Hatcher's book doesn't mention this explicitly as far as I can tell, Katz' little book on enumerative geometry alludes to this, Fulton's book on Young Tableaux dodges this, etc.</p> <p>I am preparing to give a talk on Schubert Cells and Schubert calculus, and I realized that I have not checked the details of this proof.</p> <p>Thanks in advance!</p> http://mathoverflow.net/questions/4279/interesting-applications-of-the-pigeon-hole-principle/18023#18023 Answer by B. Bischof for Interesting applications of the Pigeon-hole Principle B. Bischof 2010-03-13T00:04:18Z 2010-03-13T00:47:32Z <p>I remember hearing as an undergrad the "proof" that there are two human beings on the earth with the same number of hairs on their heads. This is done by a few estimations and then applying the pigeonhole principle. </p> <p>A number of examples including a version of this one can be found here: <a href="http://www.cut-the-knot.org/do_you_know/pigeon.shtml" rel="nofollow">http://www.cut-the-knot.org/do_you_know/pigeon.shtml</a></p> http://mathoverflow.net/questions/12498/motivation-for-cosuspended-category-axioms Motivation for Cosuspended Category Axioms B. Bischof 2010-01-21T04:51:57Z 2010-03-08T16:37:03Z <p>Today I was wondering about the axioms given by Bernhard Keller for Cosuspended Categories.</p> <p>The axioms of a triangle feel very much like exactness, but not quite. The last axiom about the large commutative diagram is particularly quizzical. While I am ok with understanding these axioms I was hoping to ask two questions about them.</p> <blockquote> <p>1) What was the classical motivation for these axioms? Was there a particular example in mind to conform to?</p> </blockquote> <p>and </p> <blockquote> <p>2) Is there a modern motivating example for these axioms that differs from the classical?</p> </blockquote> <p>I understand these things much better when I have specific examples to keep in mind, and since I am learning these in a general context, right now that is lacking. I was hoping you all could fill me in.</p> <p>Thanks in advance!</p> http://mathoverflow.net/questions/17452/expository-treatment-of-schubert-cells-paper Expository treatment of Schubert Cells Paper B. Bischof 2010-03-08T04:44:33Z 2010-03-08T06:11:09Z <p>I was wondering about the paper by Bernstein, Gel'fand, and Gel'fand on <a href="http://iopscience.iop.org/0036-0279/28/3/R01;jsessionid=E4D2896AE55597FA4F4F63297F63FBC6.c3" rel="nofollow">Schubert Cells</a>. This paper is fairly old(and often cited) so I figured someone must have represented this material. In particular, I was wondering if this was treated in an expository paper. More generally, I was wondering if there was a paper that explained the usefulness of the Schubert Calculus for representation theory, and even better one that talked about how Schubert Calculus came into the picture for BBD, again hopefully in an expository way.</p> <p>Thanks in advance!</p> http://mathoverflow.net/questions/109347/point-modules-of-quantum-projective-space-mathbbpn Comment by B. Bischof B. Bischof 2012-10-11T12:44:20Z 2012-10-11T12:44:20Z Have you looked at Rogalski's notes? He mentions that in general X will live in the product of projective spaces. He also talks about how consider them for a finitely presented algebra with homogeneous ideal. http://mathoverflow.net/questions/109225/on-q-demazure-operators Comment by B. Bischof B. Bischof 2012-10-09T15:55:05Z 2012-10-09T15:55:05Z After looking at your profile, I worry that you knew already everything I said. http://mathoverflow.net/questions/109225/on-q-demazure-operators Comment by B. Bischof B. Bischof 2012-10-09T15:51:29Z 2012-10-09T15:51:29Z If you want these operators to compute KL multiplicities in the quantum group setting you need a lot more framework, if you just want to &quot;q-ify&quot; the formulas from the classical case, maybe this is already true from quantum Schubert polynomials? http://mathoverflow.net/questions/109225/on-q-demazure-operators Comment by B. Bischof B. Bischof 2012-10-09T15:51:03Z 2012-10-09T15:51:03Z As far as I can tell, you're being fairly abstract about what you mean about q-analog here. Perhaps you mean quantum Schubert calculus and what Demazure operators correspond to this. Perhaps you mean quantum K-theory and those Demazure operators. Perhaps you even mean Demazure operators associated to quantum groups. As far as I understand, there has been lots of work in the first two directions, and I am interested in the third direction. http://mathoverflow.net/questions/70320/for-mathfrak-g-a-lie-algebra-of-type-e-7-mathfrak-h-a-cartan-subalgeb Comment by B. Bischof B. Bischof 2012-09-07T14:47:23Z 2012-09-07T14:47:23Z Thank you for this comment Allen, it was very helpful in answering a question I hadn't been able to ask. http://mathoverflow.net/questions/105727/isomorphisms-of-quantum-planes/106064#106064 Comment by B. Bischof B. Bischof 2012-09-01T02:26:25Z 2012-09-01T02:26:25Z Thanks for your comments. I will think about this some more. http://mathoverflow.net/questions/105727/isomorphisms-of-quantum-planes/106064#106064 Comment by B. Bischof B. Bischof 2012-08-31T22:26:06Z 2012-08-31T22:26:06Z Yes I misspoke about the &quot;copy of Uq&quot;. I need to think more about your first comment. I was thinking that this would be the same elements making up the image of the other rep, but I don't know this. I'll think a bit more. http://mathoverflow.net/questions/105714/reference-request-for-character-formula-between-tensor-products-of-weyl-modules Comment by B. Bischof B. Bischof 2012-08-28T16:20:01Z 2012-08-28T16:20:01Z Sorry, I missed that you just wanted a reference. http://mathoverflow.net/questions/105727/isomorphisms-of-quantum-planes Comment by B. Bischof B. Bischof 2012-08-28T16:14:20Z 2012-08-28T16:14:20Z Note on derivations of graded rings and classification of differential polynomial rings by Awami, Van den Bergh, and Oystaeyen, Observation 2.1 and subsequent discussion should give you what you want I think. http://mathoverflow.net/questions/105714/reference-request-for-character-formula-between-tensor-products-of-weyl-modules Comment by B. Bischof B. Bischof 2012-08-28T13:56:22Z 2012-08-28T13:56:22Z Also, just to be clear, have you read this summary? <a href="http://www.encyclopediaofmath.org/index.php?title=Weyl_module#Character_formulas" rel="nofollow">encyclopediaofmath.org/&hellip;</a>. http://mathoverflow.net/questions/105714/reference-request-for-character-formula-between-tensor-products-of-weyl-modules Comment by B. Bischof B. Bischof 2012-08-28T13:50:17Z 2012-08-28T13:50:17Z Related: <a href="http://mathoverflow.net/questions/21875/multiplication-of-extreme-vector" rel="nofollow" title="multiplication of extreme vector">mathoverflow.net/questions/21875/&hellip;</a> http://mathoverflow.net/questions/105475/quasi-coherent-sheaves-on-x-g Comment by B. Bischof B. Bischof 2012-08-25T23:04:35Z 2012-08-25T23:04:35Z @Niels, what did you have in mind? As far as I remember he doesn't really go into these questions since these questions are asking about Qcoh on stacks, where as Vistoli talks about Qcoh on schemes forming stacks. Although he does talk a bit about torsors, so maybe this is what you were referring to. http://mathoverflow.net/questions/105475/quasi-coherent-sheaves-on-x-g Comment by B. Bischof B. Bischof 2012-08-25T17:56:51Z 2012-08-25T17:56:51Z Hello anon, I expect that you will find the following notes by Justin Hilburn very interesting <a href="http://pages.uoregon.edu/njp/hilburn.pdf" rel="nofollow">pages.uoregon.edu/njp/hilburn.pdf</a> These notes accompanied a very nice talk which discussed Qcoh for derived stacks, but don't let the word derived scare you off. Just as a brief response to your first two specific questions: A) Why not both? B) Justin's notes(and others) will take the Qcoh's to just be modules on &quot;affine pieces&quot; and glue them together. http://mathoverflow.net/questions/105221/quantum-groups-not-via-presentations/105240#105240 Comment by B. Bischof B. Bischof 2012-08-22T22:57:50Z 2012-08-22T22:57:50Z To be clear, are you saying there is a Tannakian reconstruction for Uq(g) up to non-unique isomorphism? http://mathoverflow.net/questions/103561/on-finding-a-polynomials Comment by B. Bischof B. Bischof 2012-07-31T02:16:19Z 2012-07-31T02:16:19Z I would suggest adding some tag referencing computation, implementation, Mathematica, or even qmultisum.