User dedalus - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T05:38:29Z http://mathoverflow.net/feeds/user/7607 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/132912/what-analysis-should-i-know-for-studying-arakelov-theory What analysis should I know for studying Arakelov Theory? Dedalus 2013-06-06T07:09:27Z 2013-06-06T07:09:27Z <p>Hi!</p> <p>I have a fairly good background in Algebraic Geometry (say at the level of Hartshorne's book and some Intersection Theory from Fulton) and since I think working over $\text{Spec } \mathbb{Z}$ is fun, I would like to learn some Arakelov Theory.</p> <p>My background in differential geometry and analysis is not that good, though - I know basic definitions in both fields and have taken some courses, but I have forgot a lot, and what more, I seem to need complex differential geometry, which I have never studied. From what I understand, residuce currents is important in Arakelov Theory. </p> <p>So my question is:</p> <p><strong>What books, what articles should I read to get a good analytical / complex differential geometric background (covering for example, residue currents) sufficient to study Arakelov Theory?</strong> </p> http://mathoverflow.net/questions/69737/contest-problems-with-connections-to-deeper-mathematics Contest problems with connections to deeper mathematics. Dedalus 2011-07-07T18:09:47Z 2013-05-10T09:29:26Z <p>Hi,</p> <p>I already posted this on math.stackexchange, but I'm also posting it here because I think that it might get more and better answers here! Hope this is okay.</p> <p>We all know that problems from, for example, the IMO and the Putnam competition can sometimes have lovely connections to "deeper parts of mathematics". I would want to see such problems here which you like, and, that you would all add the connection it has.</p> <p>Some examples:</p> <ol> <li><p>Stanislav Smirnov mentions the following from the 27th IMO: "To each vertex of a regular pentagon an integer is assigned in such a way that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y &lt;0$, then the following operation is allowed: the numbers $x,y,z$ are replaced by $x+y$, $-y$, $z+y$, respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps". </p> <p>He mentions that a version of this problem is used to prove the Kottwitz-Rapoport conjecture in algebra(!). Further, a version of this has appeared in at least a dozen research papers.</p></li> <li><p>(Taken from Gerry Myerson in this thread <a href="http://math.stackexchange.com/questions/33109/contest-problems-with-connections-to-deeper-mathematics" rel="nofollow">http://math.stackexchange.com/questions/33109/contest-problems-with-connections-to-deeper-mathematics</a>)</p> <p>On the 1971 Putnam, there was a question, show that if $n^c$ is an integer for $n=2,3,4$,… then $c$ is an integer.</p> <p>If you try to improve on this by proving that if $2^c$, $3^c$, and $5^c$ are integers then $c$ is an integer, you find that the proof depends on a very deep result called <a href="http://planetmath.org/encyclopedia/SixExponentialsTheorem.html" rel="nofollow">The Six Exponentials Theorem</a>.</p> <p>And if you try to improve further by showing that if $2^c$ and $3^c$ are integers then $c$ is an integer, well, that's generally believed to be true, but it hadn't been proved in 1971, and I think it's still unproved.</p></li> </ol> <p>The most interesting part would be to see solutions to these problems using both elementary methods, and also with the more abstract "deeper methods".</p> http://mathoverflow.net/questions/125834/homotopy-colimits-over-a-certain-subset-category Homotopy colimits over a certain subset category. Dedalus 2013-03-28T15:10:55Z 2013-03-28T15:10:55Z <p>Hi! Let $I$ denote all the finite subsets of some set (infinite or finite) S. For each n, let $I_n$ be the subset consisting of objects of cardinality n, so that there are no morphisms between the objects in $I_n$. Suppose that we have a diagram $X:I \rightarrow M$ where $M$ denotes some model category (if it is neccessary, assume some nice properties on it too). My question is the following:</p> <p>Could we present $hocolim X$ as the homotopy colimit of some subdiagram $J \subset I$? Ideally, I would like to be able to find a $J$ that is directed, so that the homotopy colimit will be the mapping telescope. If so, why? Is there any additional properties on X I need for this to be true? </p> http://mathoverflow.net/questions/123005/essential-geometric-morphisms-on-the-etale-site Essential geometric morphisms on the étale site. Dedalus 2013-02-26T17:27:32Z 2013-03-05T08:49:18Z <p>Hi!</p> <p>There is a risk that this question might be utterly trivial, and if it is, my sincerest apologies, but I haven't been able to find anything in the literature. </p> <p>Let $f:X \rightarrow Y$ be an étale morphism of schemes. This gives rise to a geometric morphism of topoi $f^\ast:Sh(Y) \rightleftarrows Sh(X) : f_\ast$, and my question is now: In what cases do $f^\ast$ have a left adjoint? Are there any explicit descriptions for this for say affine schemes?</p> http://mathoverflow.net/questions/114081/explicit-computations-of-the-etale-homotopy-type Explicit computations of the étale homotopy type? Dedalus 2012-11-21T17:27:14Z 2013-01-08T13:34:41Z <p>Hi,</p> <p>I'm currently trying to learn about etale homotopy for schemes as introduced by Artin-Mazur. I know that by the Artin-Mazur comparision theorem, it is possible to compute the etale homotopy type of certain class of varieties as the profinite completion of the complex points. However, in most other cases for schemes, it seems quite cumbersome to calculate the étale homotopy type of a locally noetherian scheme say. Are there any explicit computations of the étale homotopy type that are particularly helpful for understanding the general theory? Or am I missing something here?</p> <p>Sorry if my question is a bit vague.</p> http://mathoverflow.net/questions/74464/the-fundamental-groupoid-and-a-pushout-in-the-category-of-groupoids The fundamental groupoid and a pushout in the category of groupoids. Dedalus 2011-09-03T22:04:07Z 2012-11-23T17:15:08Z <p>Hi, Recently I've been looking at the more general version of Van Kampen's theorem, or R. Browns version of it, for the fundamental groupoid. It mentions that if a space X is the union of the interiors of $X_1$ and $X_2$ , then:</p> <p><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/9/91/SeifertvanKampenPO.PNG/244px-SeifertvanKampenPO.PNG" alt="alt text"></p> <p>is a pushout in the category of groupoids. In the category of groups, we have the concrete description that the pushout is just the free product with amalgation. Does something similar hold here? Is there any explicit description, like free product with amalgation?</p> http://mathoverflow.net/questions/43889/proof-synopsis-collection/109815#109815 Answer by Dedalus for Proof synopsis collection Dedalus 2012-10-16T13:18:52Z 2012-10-16T13:18:52Z <p>Yoneda Lemma:</p> <p>The natural transformation is determined in fully by where the identity goes.</p> http://mathoverflow.net/questions/95579/sheafification-why-does-twice-suffice Sheafification - Why does twice suffice? Dedalus 2012-04-30T14:49:29Z 2012-05-01T23:14:36Z <p>Hi, I'm currently reading through "Sheaves in Geometry and logic" by Mclane-Moerdijk and this one issue has been bugging me for a long time, which I hope you could help me resolve. </p> <p>It is known that for a general presheaf on a Grothendieck Topology, we must in general apply the plus construction twice to obtain a sheaf. The first application turns an arbitrary presheaf into a separated presheaf, and one more application gives a sheaf. So, exactly, intuitively, what obstructs us from getting a sheaf from just one application of the plus construction when the sheaf is non-separated? Let us take the example: <a href="http://mathoverflow.net/questions/616/what-is-an-example-of-a-presheaf-p-where-p-is-not-a-sheaf-only-a-separated-pre" rel="nofollow">http://mathoverflow.net/questions/616/what-is-an-example-of-a-presheaf-p-where-p-is-not-a-sheaf-only-a-separated-pre</a> given by Sherry. When we apply it once we get a separated presheaf, OK. But what exact component of that presheaf hindered us from getting the sheaf we wanted? I agree with what Sherry wrote in that case, namely that : "So in our example, 1 and 3, over ABC and BCD, in our original presheaf were compatible on a refinement of BC but not on BC" Can we generalize this notion to become rigorous in the case for arbitrary non-separated presheaves?</p> http://mathoverflow.net/questions/35288/undergraduate-roadmap-to-algebraic-geometry Undergraduate roadmap to algebraic geometry? Dedalus 2010-08-11T22:53:05Z 2011-03-03T03:01:02Z <p>Hello, I'm sorry if this question isn't posted correctly. I hope that it is (since other questions regarding roadmaps have been allowed). Now to my question:</p> <p>From what I've heard from professors and such, algebraic geometry seems like an interesting branch of mathematics. I'd like to learn some basic results and maybe do some kind of thesis in a few years on the subject. So, what I'm curious about is you have any tips on what books to read? Say that one has read Artin's Algebra and Herstein's Topics in Algebra, and also has the basic courses in real analysis and topology, complex variables etc. down, where should one go to learn? What books? I'm also curious if algebraic geometry (at an "easy level") requires deep knowledge about other fields of mathematics too, so that one might have to read books that at first seems to have no relevance to algebraic geometry?</p> <p>Best regards.</p> http://mathoverflow.net/questions/50071/prepending-strings-to-primes Prepending strings to primes. Dedalus 2010-12-21T16:01:40Z 2011-01-02T09:30:56Z <p>Hello, we all know that 31,331,3331,33331,333331,3333331,33333331 all are primes, and that 333333331 is not. Here we prepend the digit 3 to 31, to get a list of 7 primes.This gives me the following thought:</p> <p>Let $$D = \{\text{all possible nonnull finite digit strings}\},$$ $$D' = \{\text{all things in D that do not start with 0}\}.$$<br> Define a function $m: D' \times D \to N \cup {\infty}$ by: $$m(A,B)= \min \{ k\geq 1 \colon A^kB \text{ is composite} \} - 1,$$ i.e., the number of consecutive primes at the beginning of the list $AB, AAB, AAAB, \dots$. For example, $m(3,1)=7$.</p> <blockquote> <p>Which values does $m$ take? Is it unbounded? Is it ever $\infty$?</p> </blockquote> <p>This question has been posted to math.stackexchange too, and I got one comment talking about that it might involve Tao-Ziegler extension to the Green-Tao theorem, and so I thought it might be more appropriate here. So please, excuse me if it's posted wrongly, or if one shouldn't post to both channels.</p> http://mathoverflow.net/questions/32925/when-is-a-statement-provable When is a statement provable? Dedalus 2010-07-22T09:33:11Z 2010-12-30T07:25:15Z <p>We all know that Gödel showed that there, in a formal system, are true statements that are non-provable (undecidable). In ZFC, there's Kaplansky's Conjecture, the Whitehead problem, etc.</p> <p>We can also agree that we're sure to find more non-provable statements in ZFC. What I'm curious about is the following:</p> <p>What is the defining characteristic of a non-provable statement in ZFC? Are they all "strong" in some sense? Is it a necessary condition that they are strong, then? What future theorems might turn out to be non-provable? Is it, through the characteristics of the theorems known to be non-provable possible to make a fairly accurate guess if a theorem will turn out to be non-provable in ZFC?</p> http://mathoverflow.net/questions/35503/how-to-study-a-math-text How to study a math text Dedalus 2010-08-13T15:39:12Z 2010-08-13T17:10:44Z <p>Hello, recently I've been trying various attempts regarding how to approach a math book to learn in the best way. Should one memorize the theorems and proofs so that one can recite them? I tend to sometimes forget proofs and such after some time, so my question is the following.</p> <p>When you're trying to learn from a book and it's a new subject, how do you usually do? Do you learn all theorems and proofs? Do you write it down? Or do you just read? This is a general question, I'm aware of that, I do however hope that this question will fit the community.</p> http://mathoverflow.net/questions/32666/when-is-something-too-big-to-be-a-set When is something too big to be a set? Dedalus 2010-07-20T18:02:26Z 2010-07-22T08:08:41Z <p>Hello, recently, I've been reading some algebra and sometimes I stumble up on the concept of something "being too big" to be a set. An example, is given in (http://www.dpmms.cam.ac.uk/~wtg10/tensors3.html) , where he writes, "Let B be the set of all bilinear maps defined on VxW. (That's the naughtiness - B is too big to be a set, but actually we will see in a moment that it is enough to look just at bilinear maps into R.)" (where V and W are vector spaces over R). This is, too big to be a set, but why?</p> <p>My general question is this, when is something too big to be a set? What is it instead? Why have we put these requirements on the definition? Do we run into any problems if we let, say, B as defined up there be a set? What kind of problems do we run into? </p> http://mathoverflow.net/questions/31842/tensor-product-and-category-theory/32130#32130 Answer by Dedalus for Tensor product and category theory Dedalus 2010-07-16T08:21:18Z 2010-07-16T08:21:18Z <p>Thank you all. Your documents has been most helpful. I also saw some papers on the tensor product of modules, especially: <a href="http://www.math.ucsb.edu/~mckernan/Teaching/05-06/Winter/220B/l_7.pdf" rel="nofollow">http://www.math.ucsb.edu/~mckernan/Teaching/05-06/Winter/220B/l_7.pdf</a> was helpful, and <a href="http://www.dpmms.cam.ac.uk/~wtg10/tensors3.html" rel="nofollow">http://www.dpmms.cam.ac.uk/~wtg10/tensors3.html</a> gave some good info too.</p> <p>Now I'm considering TeXing a file where I try to motivate <em>why</em> one defines the tensor product in the first place. I think that might help me learn the definition even more. I really like the definition in some strange way, even though I find it kind of hard. I want to learn.</p> <p>So once again, Thank you.</p> http://mathoverflow.net/questions/31842/tensor-product-and-category-theory Tensor product and category theory Dedalus 2010-07-14T12:17:46Z 2010-07-16T08:21:18Z <p>Hello. I'm trying to understand the definition of tensor product of two vector spaces. So far, I've read the one using free vector spaces and a quotient space (this one <a href="http://en.wikipedia.org/wiki/Tensor_product#Tensor_product_of_vector_spaces" rel="nofollow">http://en.wikipedia.org/wiki/Tensor_product#Tensor_product_of_vector_spaces</a>) , and I think I understand it well. However, I want to understand the other definitions I can find, and it seems as a very common way to define it is through the universal property (some category theory included, I suspect). Does anyone here know of a good treatment of this? I have no knowledge of category theory though, but would love to read some about it. I'm a second-year undergrad, so not too much of a high level would be nice.</p> http://mathoverflow.net/questions/132912/what-analysis-should-i-know-for-studying-arakelov-theory Comment by Dedalus Dedalus 2013-06-07T15:59:42Z 2013-06-07T15:59:42Z Damian R&#246;ssler: But that book already supposes that I have the neccessary background in analysis from what I have seen. http://mathoverflow.net/questions/132041/linear-algebra-textbook-recommendation/132048#132048 Comment by Dedalus Dedalus 2013-05-28T13:36:13Z 2013-05-28T13:36:13Z Halmos book is fantastic in almost every regard, except when he does tensors. I would suggest that the OP try some other source for that material, but otherwise, Halmos' can hardly be beaten. http://mathoverflow.net/questions/125834/homotopy-colimits-over-a-certain-subset-category Comment by Dedalus Dedalus 2013-03-29T10:48:46Z 2013-03-29T10:48:46Z Karol: Thank you! That was very helpful. I am working with simplicial sets with an action of a profinite group, so I think it should work in that case. I will look into the article you gave more explicitly. Dan: Thank you too! With cofinal - you mean homotopy cofinal , right? http://mathoverflow.net/questions/123005/essential-geometric-morphisms-on-the-etale-site Comment by Dedalus Dedalus 2013-02-28T19:28:11Z 2013-02-28T19:28:11Z Thanks, yes , I mean the small &#233;tale site! http://mathoverflow.net/questions/123005/essential-geometric-morphisms-on-the-etale-site Comment by Dedalus Dedalus 2013-02-28T18:20:03Z 2013-02-28T18:20:03Z I would mostly be interested in the case for affine schemes. Spec Z would be of interest, and some nice restrictions on X is OK. More generally, I am curious what I should look for to think if it is plausible that an adjoint exists. http://mathoverflow.net/questions/84474/which-math-workbooks-are-best-for-you-and-how-to-get-better-at-math Comment by Dedalus Dedalus 2011-12-28T20:58:17Z 2011-12-28T20:58:17Z SChang: This question is not a research-level question. Your question is much better suited for <a href="http://math.stackexchange.com/" rel="nofollow">math.stackexchange.com</a> ! http://mathoverflow.net/questions/84474/which-math-workbooks-are-best-for-you-and-how-to-get-better-at-math/84476#84476 Comment by Dedalus Dedalus 2011-12-28T20:57:04Z 2011-12-28T20:57:04Z -1 : Being rude towards an 8th grader? Come on. http://mathoverflow.net/questions/69737/contest-problems-with-connections-to-deeper-mathematics/83990#83990 Comment by Dedalus Dedalus 2011-12-21T17:22:56Z 2011-12-21T17:22:56Z As always, fantastic examples. Thank you so much! http://mathoverflow.net/questions/76580/famous-mathematicians-with-background-in-arts-humanities-law-etc/76683#76683 Comment by Dedalus Dedalus 2011-09-29T07:37:19Z 2011-09-29T07:37:19Z I was fortunate enough to see him lecture this summer and I was stunned by his enthusiasm. Great example! http://mathoverflow.net/questions/69737/contest-problems-with-connections-to-deeper-mathematics/69745#69745 Comment by Dedalus Dedalus 2011-07-08T18:32:03Z 2011-07-08T18:32:03Z That's really amazing. Thanks a lot for posting it! http://mathoverflow.net/questions/69737/contest-problems-with-connections-to-deeper-mathematics Comment by Dedalus Dedalus 2011-07-07T19:09:22Z 2011-07-07T19:09:22Z Right, changed it now! Thank you. http://mathoverflow.net/questions/49944/prepending-digits-to-obtain-an-arbitrarly-long-sequence-of-primes Comment by Dedalus Dedalus 2010-12-20T18:07:38Z 2010-12-20T18:07:38Z Sorry, the question is worded wrongly. Should I delete the question completly or just edit it? http://mathoverflow.net/questions/49944/prepending-digits-to-obtain-an-arbitrarly-long-sequence-of-primes Comment by Dedalus Dedalus 2010-12-20T12:11:51Z 2010-12-20T12:11:51Z Thanks Kevin for that remark! Do you know whether there exists a sequence of 8 primes generated this way? http://mathoverflow.net/questions/35503/how-to-study-a-math-text Comment by Dedalus Dedalus 2010-08-13T19:05:49Z 2010-08-13T19:05:49Z Noah: math.SE? What's that? http://mathoverflow.net/questions/35288/undergraduate-roadmap-to-algebraic-geometry Comment by Dedalus Dedalus 2010-08-12T09:19:08Z 2010-08-12T09:19:08Z I will Scott, thanks!