User dan kneezel - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:01:44Z http://mathoverflow.net/feeds/user/6005 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60478/hirzebruchs-motivation-of-the-todd-class Hirzebruch's motivation of the Todd class Dan Kneezel 2011-04-03T19:26:16Z 2012-09-27T19:10:46Z <p>In <a href="http://books.google.com/books?id=zK7jmvsI_2wC&amp;lpg=PP1&amp;ots=LpN2oUA88m&amp;dq=hirzebruch%20%22prospects%20in%20mathematics%22&amp;pg=PA7#v=onepage&amp;q&amp;f=false" rel="nofollow">Prospects in Mathematics (AM-70)</a>, Hirzebruch gives a nice discussion of why the formal power series $f(x) = 1 + b_1 x + b_2 x^2 + \dots$ defining the Todd class must be what it is. In particular, the key relation $f(x)$ must satisfy is that </p> <p>($\star$) the coefficient of $x^n$ in $(f(x))^{n+1}$ is 1 for all $n$.</p> <p>As Hirzebruch observes, there is only one power series with constant term 1 satisfying that requirement, namely $$f(x) = \frac{x}{1-e^{-x}} = 1 + \frac{x}{2}+\sum_{k\geq 2}{B_{k}\frac{x^{k}}{k!}} = 1 + \frac{x}{2} + \frac{1}{6}\frac{x^2}{2} - \frac{1}{30}\frac{x^4}{24} + \dots,$$ where the $B_k$ are the <a href="http://en.wikipedia.org/wiki/Bernoulli_number" rel="nofollow">Bernoulli numbers</a>. </p> <p>The only approach I see to reach this conclusion is: </p> <ol> <li>Use ($\star$) to find the first several terms: $b_1 = 1/2, b_2 = 1/12, b_3 = 0, b_4 = -1/720$.</li> <li>Notice that they look suspiciously like the coefficients in the exponential generating function for the Bernoulli numbers, so guess that $f(x) = \frac{x}{1-e^{-x}}$.</li> <li>Do a residue calculation to check that this guess does satisfy ($\star$).</li> </ol> <blockquote> <p>My question is whether anyone knows of a less guess-and-check way to deduce from ($\star$) that $f(x) = \frac{x}{1-e^{-x}}$.</p> </blockquote> http://mathoverflow.net/questions/33766/what-makes-the-stable-module-category-stable/73720#73720 Answer by Dan Kneezel for What makes the stable module category stable? Dan Kneezel 2011-08-26T02:07:56Z 2011-08-26T02:07:56Z <p>The answer to the first question is yes, the two categories are nontrivially related: Both the traditional stable homotopy category and the stable module category are examples of stable homotopy categories in the sense of Hovey-Palmieri-Strickland, <a href="http://hopf.math.purdue.edu//Hovey-Palmieri-Strickland/axiomatic.pdf" rel="nofollow"><em>Axiomatic Stable Homotopy Theory</em></a>. See the first few pages of Schwede's <a href="http://www.math.uni-bonn.de/~schwede/stablemodelcats.pdf" rel="nofollow"><em>Stable model categories are categories of modules</em></a> for a convenient overview. Of special interest are Definition 2.1.1 and the subsequent paragraph on page 107, and Example 2.4.(v) on page 111. You may also find it useful to consult Hovey's <a href="http://www.ericmalm.net/work/projects/symmetric-spectra/hovey--model-cats.pdf" rel="nofollow"><em>Model Categories</em></a>, especially Chapters 2 and 7, as well as Chapter I.2 in Quillen's <em>Homotopical Algebra</em>.</p> <p>As for the second question, observe the following extract from Example 2.4.(v):</p> <blockquote> <blockquote> <p>Fortunately, the two different meanings of ‘stable’ fit together nicely; the stable module category is the homotopy category associated to an underlying stable model category structure [21, Section 2].</p> </blockquote> </blockquote> <p>so it sounds like Schwede would agree with Mariano and Tom that, at least <em>originally</em>, the selection of the word 'stable' in stable module category likely had nothing to do with stable homotopy and all that.</p> <p>Corrections are welcome.</p> http://mathoverflow.net/questions/71704/computing-the-q-series-of-the-j-invariant Computing the q-series of the j-invariant Dan Kneezel 2011-07-31T03:19:12Z 2011-08-01T09:51:14Z <p>It is a fundamental fact, often quoted these days in its connection with Monstrous Moonshine, that the q-expansion (i.e., the Laurent expansion in a neighborhood of $\tau = i\infty$) of the <a href="http://en.wikipedia.org/wiki/J-invariant#The_q-expansion_and_moonshine" rel="nofollow">j-invariant</a> is given by</p> <p>$$j(\tau) = \frac{1}{q} + 744 + 196884q + 21493760q^2+\dots, \quad q = e^{2 \pi i \tau}.$$</p> <p>I do not recall, however, ever seeing a modern treatment, nor even a hint, of how one might go about obtaining this expansion. Does anybody know a nice way to compute these coefficients? (I mean a way which does not invoke Moonshine, not that I'd expect that to make the computation more pleasant.) Is there a standard way to do it?</p> <p>I did find one approach published by H.S. Zuckerman in the late 1930s*, which makes use of a "fifth order multiplicator equation" for $j(\tau)$ -- distilled from Fricke and Klein's <em>Vorlesungen uber die Theorie der elliptischen Modulfunktionen</em> -- and an identity of Ramanujan for the generating function of partition numbers of the form $p(25n + 24)$. Is this typical?</p> <p>*Zuckerman, Herbert S., <em>The computation of the smaller coefficients of $J(\tau)$</em>. Bull. Amer. Math. Soc. 45, (1939). 917–919.}</p> http://mathoverflow.net/questions/43249/what-are-d-branes-really What are D-branes, really? Dan Kneezel 2010-10-22T23:32:02Z 2010-10-23T14:01:22Z <p>In the past couple years, I've read many words pertaining to "D-branes" without feeling I have fully comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as habitats for the ends of open strings and can be conceived of as submanifolds (of the target manifold in a sigma model), possibly augmented with a vector bundle, or a sheaf of [somethings], or maybe some other kind of label/data. (Corrections welcome.)</p> <p>In the hopes of tightening my grasp on the concept, here are some of the questions that have been nagging me during my reading.</p> <ol> <li>What <em>specifically</em> is the definition of a D-brane, say in the context of a topological field theory? (Or what are the most promising provisional definitions?) What references are most accessible to a mathematical audience?</li> <li>What picture should I have in my head when an author talks about "the moduli space of D-branes"?</li> <li>What is the idea behind the "dynamics of D-branes" that researchers sometimes talk about? (Perhaps when I understand better how to think about these gadgets, it will be easier to conceive of how they should change over time.)</li> <li>What goes into verifying (or at least asserting/conjecturing) that the elements of twisted K-theory classify "D-brane charges"?</li> </ol> <p>(Question reposted from <a href="http://math.stackexchange.com/questions/7435/what-are-d-branes-in-a-topological-field-theory" rel="nofollow">here</a>.)</p> http://mathoverflow.net/questions/60478/hirzebruchs-motivation-of-the-todd-class/108276#108276 Comment by Dan Kneezel Dan Kneezel 2012-09-28T02:30:19Z 2012-09-28T02:30:19Z jmac, I think you're getting dinged mainly because it is considered Bad Form to ask questions in the Answers or Comments fields. In general, if a thread inspires you to ask a new question, you should click &quot;Ask Question&quot; (grey button, top right of page). Before you do so, though, you should make sure to explain the context of your question and demonstrate you've already made some effort to resolve it on your own. See here for tips on how to formulate a good MO question. <a href="http://mathoverflow.net/howtoask" rel="nofollow">mathoverflow.net/howtoask</a> My 2 cents: I think your question is reasonable, I had wondered about it myself. http://mathoverflow.net/questions/78950/what-is-the-origin-of-unit-vector-notation-i-j-k/78952#78952 Comment by Dan Kneezel Dan Kneezel 2011-10-24T05:05:02Z 2011-10-24T05:05:02Z For further discussion of this perspective, see <a href="http://en.wikipedia.org/wiki/Quaternion#Quaternions_and_the_geometry_of_R3" rel="nofollow">en.wikipedia.org/wiki/&hellip;</a> http://mathoverflow.net/questions/54430/video-lectures-of-mathematics-courses-available-online-for-free/54452#54452 Comment by Dan Kneezel Dan Kneezel 2011-09-14T21:08:11Z 2011-09-14T21:08:11Z Is anyone else having trouble getting audio from the videos? I'm pretty sure that the audio worked just fine for me back in '07 when I originally watched these, but now I don't hear anything. I tried streaming the videos in Chrome, in Firefox, downloading and opening with the standalone QuickTime application, as well as opening with VLC. The video shows up just fine but no audio from any of them. Running Windows 7. http://mathoverflow.net/questions/71704/computing-the-q-series-of-the-j-invariant Comment by Dan Kneezel Dan Kneezel 2011-08-01T19:57:27Z 2011-08-01T19:57:27Z @W. Zudilin: Thanks! http://mathoverflow.net/questions/71704/computing-the-q-series-of-the-j-invariant/71790#71790 Comment by Dan Kneezel Dan Kneezel 2011-08-01T19:34:16Z 2011-08-01T19:34:16Z Interesting! Thanks! http://mathoverflow.net/questions/71704/computing-the-q-series-of-the-j-invariant/71708#71708 Comment by Dan Kneezel Dan Kneezel 2011-08-01T19:16:17Z 2011-08-01T19:16:17Z @N. Elkies: Thank you for your detailed comments. They have been quite enlightening! http://mathoverflow.net/questions/71704/computing-the-q-series-of-the-j-invariant/71708#71708 Comment by Dan Kneezel Dan Kneezel 2011-07-31T07:52:39Z 2011-07-31T07:52:39Z @N. Elkies: Oh! The multiplicator in the Zuckerman reference is a relation between $h_5$ and $j$. In particular, $$j(\tau)=h_5+6\cdot 5^3+\frac{63\cdot 5^5}{h_5}+\frac{52\cdot 5^8}{h_5^2}+\frac{63\cdot 5^{10}}{h_5^3}+\frac{63\cdot 5^{13}}{h_5^4}+\frac{5^{15}}{h_5^5}.$$ http://mathoverflow.net/questions/71704/computing-the-q-series-of-the-j-invariant/71708#71708 Comment by Dan Kneezel Dan Kneezel 2011-07-31T04:14:22Z 2011-07-31T04:14:22Z Can this approach be extended to Thompson series $T_g(q)$ away from $g = id$? http://mathoverflow.net/questions/71704/computing-the-q-series-of-the-j-invariant/71706#71706 Comment by Dan Kneezel Dan Kneezel 2011-07-31T04:05:23Z 2011-07-31T04:05:23Z Nice. Thank you! http://mathoverflow.net/questions/60478/hirzebruchs-motivation-of-the-todd-class Comment by Dan Kneezel Dan Kneezel 2011-04-03T21:05:42Z 2011-04-03T21:05:42Z @Eric Interesting. Thanks for the link. http://mathoverflow.net/questions/60478/hirzebruchs-motivation-of-the-todd-class/60481#60481 Comment by Dan Kneezel Dan Kneezel 2011-04-03T20:45:58Z 2011-04-03T20:45:58Z That is very nice indeed. Thanks! http://mathoverflow.net/questions/43249/what-are-d-branes-really Comment by Dan Kneezel Dan Kneezel 2010-10-28T22:09:51Z 2010-10-28T22:09:51Z @Greg, @Urs - Thank you both for your explanations. I was really quite divided over which one to accept as an answer -- if I could pick two, I would -- but I decided to go with Urs' for including the observation that in 2d RCFTs, D-branes are Frobenius algebra modules. http://mathoverflow.net/questions/43249/what-are-d-branes-really Comment by Dan Kneezel Dan Kneezel 2010-10-23T02:55:37Z 2010-10-23T02:55:37Z @Jos&#233;, @Laie - Thank you both for your reference suggestions. @Jos&#233; - Thanks also for your comments. Refinements to my question are under construction.