User thomas sauvaget - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T06:34:07Z http://mathoverflow.net/feeds/user/469 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119621/learning-through-guided-discovery/119941#119941 Answer by Thomas Sauvaget for Learning through guided discovery Thomas Sauvaget 2013-01-26T15:03:15Z 2013-01-26T15:03:15Z <p>The book <em>Abel's Theorem in Problems and Solutions</em> by Alekseev &amp; Arnold is a great one to learn about group theory and complex analysis (see excerpts <a href="http://books.google.fr/books?id=GI_SmiYsh0UC&amp;printsec=frontcover&amp;hl=fr#v=onepage&amp;q&amp;f=false" rel="nofollow">here</a>)</p> <p>Also, have a look at the following related MO questions: <a href="http://mathoverflow.net/questions/12709" rel="nofollow">12709</a>, <a href="http://mathoverflow.net/questions/28158" rel="nofollow">28158</a> and <a href="http://mathoverflow.net/questions/56314" rel="nofollow">56314</a>.</p> http://mathoverflow.net/questions/84823/recursivity-of-the-primes Recursivity of the primes Thomas Sauvaget 2012-01-03T18:43:20Z 2012-01-03T18:43:20Z <p>As is well-known, with the finite set of the first $n$ primes $p_1,\dots ,p_n$ one can find all primes exactly (i.e. without false positives) with the Eratosthenes Sieve in the interval $[p_n+1;p_{n+1}^2]$. Of course, that requires computing $p_{n+1}$ first to know where $p_{n+1}^2$ is, but then one can sieve without further ado (I'm not looking at efficiency here, so Eratosthesnes is fine for the purpose of this discussion).</p> <p>It is thus tempting to look at the recursive construction (in some sense, the maximal possible generation of primes): </p> <p>$2$ produces $3,5,7$ in $[2+1;3^2]=[3;9]$ (that's 3 primes, i.e. $3=\pi(9)-\pi(3)$)</p> <p>then $2,3,5,7$ produces $11,13\dots ,113$ in $[7+1;11^2]$ (that's 26 primes)</p> <p>then $2,\dots ,113$ produces $127,\dots ,16127$ in $[113+1;127^2]$ (that's 1847 primes)</p> <p>then $2,\dots ,16127$ produces $16139,\dots ,260 467 313$ in $[16127+1;16139^2]$ (that's 14 218 065 primes).</p> <p>And so on...</p> <blockquote> <p>I would be interested to know more about this recursive procedure, for instance obtain an asympotic equivalent if possible.</p> </blockquote> <p>Notice that I'm not asking simply for an asymptotics of $\pi (p_{n+1}^2) - \pi (p_n+1)$, since I'm interested in only a certain subsequence of that, which grows much faster. One step forward would thus be e.g. a useful characterisation of that subsequence.</p> <p>I have looked at the obvious places (the <a href="https://oeis.org/" rel="nofollow">OEIS</a>, and elementary texts like Conway &amp; Guy's <a href="http://books.google.fr/books?id=0--3rcO7dMYC" rel="nofollow">The Book of Numbers</a> or Stein's <a href="http://wstein.org/edu/2007/spring/ent/textbook.html" rel="nofollow">lectures on Elementary Number Theory</a>) but to no avail. In particular none of them, including the OEIS, seem to mention the relevant sequences: </p> <p>a) growth of the number of primes produced: 1, 3, 26, 1847, 14 218 065, ...</p> <p>b) accumulated growth: 1, 4, 30, 1877, 14 219 942, ...</p> <p>c) first new prime produced: 2, 3, 11, 127, 16139, 260 467 367, 67 843 249 271 912 789, ...</p> <p>d) last new prime produced: 7, 113, 16127, 260 467 313,...</p> <p>I would be grateful for any help or useful reference (I do not have access to the Monthly, but wouldn't be surprised if this had been discussed there already).</p> http://mathoverflow.net/questions/2281/use-of-n-transitivity-in-finite-group-theory Use of n-transitivity in finite group theory Thomas Sauvaget 2009-10-24T13:11:01Z 2011-10-24T01:16:50Z <p>Hello, apparently finite groups which are n-transitive with n>5 are only the permutation groups S<sub>n</sub> or the alternating groups A<sub>n+2</sub>, see e.g. page 226 this book by Isaacs <a href="http://books.google.fr/books?id=pCLhYaMUg8IC&amp;pg=PA226" rel="nofollow">http://books.google.fr/books?id=pCLhYaMUg8IC&amp;pg=PA226</a></p> <p>Is this characterization useful at all? For instance, are there famous proofs (maybe in a geometric context), which use it to, say, show that a certain group is in fact some A<sub>n</sub> ?</p> http://mathoverflow.net/questions/75765/who-coined-the-name-tensor-and-why/75770#75770 Answer by Thomas Sauvaget for Who coined the name tensor and why? Thomas Sauvaget 2011-09-18T17:21:16Z 2011-09-18T17:21:16Z <p>I think the OP referes to the modern meaning of the word, in which case, according to that website, it first appeared in german physicist Woldemar Voigt's paper <em>Die fundamentalen physikalischen Eigenschaften der Krystalle in elementarer Darstellung</em> published in 1898. (I do not have access to this paper, but probably this deals with <a href="http://en.wikipedia.org/wiki/Deformation_tensor" rel="nofollow">deformation tensors</a> in crystals).</p> http://mathoverflow.net/questions/3242/canonical-examples-of-algebraic-structures/3245#3245 Answer by Thomas Sauvaget for Canonical examples of algebraic structures Thomas Sauvaget 2009-10-29T11:15:53Z 2011-06-20T15:55:12Z <p>Not an algebraist myself, but this is interesting. Few things I do know:</p> <p>Read ------------------------------- Quickly think of</p> <p>algebraically closed field --------- $\mathbb C$ </p> <p>abelian group ---------------------- $(\mathbb R;+)$</p> <p>non-abelian group ----------------- $(\operatorname{SO}(n,\mathbb R);\times)$ ; ($n>2$)</p> <p>simple group ----------------------- $(A_n; \circ)$ ; ($n\ge5$)</p> http://mathoverflow.net/questions/2615/least-number-of-charts-to-describe-a-given-manifold Least number of charts to describe a given manifold Thomas Sauvaget 2009-10-26T14:00:44Z 2011-06-17T12:19:56Z <p>Hello, I'm wondering if there is a standard reference discussing the least number of charts in an atlas of a given manifold required to describe it. </p> <p>E.g. a circle requires at least two charts, and so on (I couldn't manage to get anything relevant neither on wikipedia nor on google, so I guess I'm lacking the correct terminology).</p> <p><em>Edit</em>: in the case of an open covering of a topological space by n+1 contractible sets (in that space) then n is called the Lusternik-Schnirelman Category of the space, see Andy Putman's answer. The following book seems to be the standard reference <a href="http://books.google.fr/books?id=vMREfNN-L4gC&amp;pg=PP1" rel="nofollow">http://books.google.fr/books?id=vMREfNN-L4gC&amp;pg=PP1</a></p> <p>Great, now I'm still interested by the initial question: does anybody know of another theory without this contractibility assumption (hoping that it allows more freedom)? e.g. would it lead to different numbers say for genus-g surfaces? </p> <p><em>Final edit</em>: yes different numbers for genus-g surfaces (see answers below), but not sure there is a theory without contractibility. Right, really lots of interesting literature on the LS category nevertheless, hence the accepted answer. For example there are estimates for non-simply connected compact simple Lie groups like PU(n) and SO(n) in Topology and its Applications, Volume 150, Issues 1-3, 14 May 2005, Pages 111-123. </p> http://mathoverflow.net/questions/39772/is-symplectic-reduction-interesting-from-a-physical-point-of-view/39777#39777 Answer by Thomas Sauvaget for Is symplectic reduction interesting from a physical point of view? Thomas Sauvaget 2010-09-23T17:30:17Z 2010-09-24T14:14:47Z <p>Of course it is interesting, and the idea of factoring out symmetries goes back to Newton.</p> <p>As for your point (1), yes it leads to more complicated geometry, and often some form of singular reduction is required (e.g. have a look at <a href="http://books.google.fr/books?id=TJzFoZHlvfcC&amp;lpg=PP1&amp;pg=PP1#v=onepage&amp;q&amp;f=false" rel="nofollow">the blue book</a> of Bates &amp; Cushman for the case of integrable Hamiltonian systems with finitely many degrees of freedom), but from a dynamical point of view it makes much more sense. </p> <p>For instance suppose you would like to study numerically a classical mechanical system (integrable or non-integrable all the same): working in reduced coordinates allows to easily distinguish between different Periodic Orbits (i.e. to count them only once), while in non-reduced dynamics the Periodic Orbits come in continuous families. </p> <p>As for your comment, the topology of the reduced space is certainly a "new physical insight". Have a look at this <a href="http://math.uc.edu/~meyer/siam08.pdf" rel="nofollow">recent paper</a> by Yanguas, Palacian, Meyer &amp; Dumas where they study periodic orbits of a non-integrable system as you ask, and where they discuss the issue of reduction and provide further references.</p> <p>(Edited to correct an error; more links added in reply to comment.)</p> http://mathoverflow.net/questions/2214/covering-the-primes-by-3-term-aps Covering the primes by 3-term APs ? Thomas Sauvaget 2009-10-23T23:03:29Z 2010-09-13T05:17:09Z <p>Hello, the Green-Tao theorem says infinitely many k-term Arithmetic Progressions exist for any integer k. </p> <p>My question is: can we actually partition the primes into 3-term APs only (or is there a simple reason why it cannot be expected) ? And if it were possible, then what would it mean for the set of primes ?</p> <p>For example fix a large integer (say M=10,000) then take the primes (except 2): 3, 5, 7, 11, 13... and remove the longest possible AP with common difference less than M as you go along. It provides the partition: 3 5 7 -- 11 17 23 29 -- 13 37 61 -- 19 31 43 -- 41 47 53 59 -- 67 73 79 -- 71 89 107 -- 83 131 179 227 -- 97 103 109 -- 101 137 173 -- ... Numerical data for the first 10,000 primes with that M shows that the average length of APs so defined is 3.2 (which is thus quite close to 3, so at least numerically there exists partitions where 3-terms APs cover a large fraction of the primes, hence the question).</p> http://mathoverflow.net/questions/29485/work-of-icm-2010-plenary-speakers-and-other-humans/29501#29501 Answer by Thomas Sauvaget for Work of ICM 2010 plenary speakers (and other humans) Thomas Sauvaget 2010-06-25T11:36:58Z 2010-08-06T13:34:01Z <p>This is not really an answer, but too long to fit in comments. Some plenary speakers appear to have interesting informal descriptions of their work on their webpage for the non-specialist (or on some prize acceptance paper); also I've added their broad area between brackets (feel free to edit and add more). </p> <ol> <li>David Aldous [Probability &amp; Statistics, especially random flows on networks] <a href="http://www.stat.berkeley.edu/~aldous/Research/research.html" rel="nofollow">http://www.stat.berkeley.edu/~aldous/Research/research.html</a></li> <li>Artur Avila [Dynamical Systems &amp; Spectral Theory] <a href="http://w3.impa.br/~avila/new.html" rel="nofollow">http://w3.impa.br/~avila/new.html</a></li> <li>R. Balasubramanian [Number Theory &amp; Cryptography]</li> <li>Jean-Michel Coron [Control of PDEs] <a href="http://www.scholarpedia.org/article/Control_of_partial_differential_equations" rel="nofollow">http://www.scholarpedia.org/article/Control_of_partial_differential_equations</a></li> <li>Irit Dinur [Computational Complexity Theory, Graph Theory]</li> <li>Hillel Furstenberg [Ergodic Theory] <a href="http://www.wolffund.org.il/full.asp?id=155" rel="nofollow">http://www.wolffund.org.il/full.asp?id=155</a></li> <li>Thomas J.R. Hughes [Computational Fluid mechanics]</li> <li>Peter Jones [Differential and Complex Geometry] <a href="http://www.pnas.org/content/105/6/1803.full" rel="nofollow">http://www.pnas.org/content/105/6/1803.full</a></li> <li>Carlos Kenig [Nonlinear PDEs, especially Schödinger and Wave types] <a href="http://www-news.uchicago.edu/releases/08/080108.kenig.shtml" rel="nofollow">http://www-news.uchicago.edu/releases/08/080108.kenig.shtml</a></li> <li>Ngo Bao Chau [Algebraic Geometry, Langlands program] <a href="http://www.mfo.de/programme/prize/Ngo2008.pdf" rel="nofollow">http://www.mfo.de/programme/prize/Ngo2008.pdf</a> and <a href="http://www.institut.math.jussieu.fr/projets/fa/bpFiles/DatTuan.pdf" rel="nofollow">http://www.institut.math.jussieu.fr/projets/fa/bpFiles/DatTuan.pdf</a></li> <li>Stanley Osher [Scientific Computing, especially Level-Set Methods] <a href="http://www.levelset.com/sjo/Interview.htm" rel="nofollow">http://www.levelset.com/sjo/Interview.htm</a></li> <li>R. Parimala [Arithmetic Algebraic Geometry]</li> <li>A. N. Parshin [Harmonic Analysis &amp; Arithmetic Groups]</li> <li>Shige Peng [Financial Mathematics]</li> <li>Kim Plofker [History of Math, especially India]</li> <li>Nicolai Reshetikhin [Mathematical Physics &amp; Statistical Physics]</li> <li>Richard Schoen [Global Differential Geometry]</li> <li>Cliff Taubes [4-dimensional Geometry, Symplectic Topology, Contact Geometry] <a href="http://www.ams.org/notices/200805/tx080500596p.pdf" rel="nofollow">http://www.ams.org/notices/200805/tx080500596p.pdf</a></li> <li>Claire Voisin [Algebraic Geometry, especially Hodge Conjecture and Mirror Symmetry] <a href="http://www.math.jussieu.fr/~voisin/Articlesweb/template.pdf" rel="nofollow">http://www.math.jussieu.fr/~voisin/Articlesweb/template.pdf</a></li> <li>Hugh Woodin [Logic, especially the Continuum Hypothesis]</li> </ol> http://mathoverflow.net/questions/19528/adaptive-controllers-for-stiff-ode-and-dae-integrators/34644#34644 Answer by Thomas Sauvaget for Adaptive controllers for stiff ODE and DAE integrators Thomas Sauvaget 2010-08-05T15:48:27Z 2010-08-05T15:48:27Z <p>Have a look at <a href="http://books.google.fr/books?id=SjNfr7gfUZEC&amp;lpg=PP1&amp;dq=General%2520Linear%2520Methods%2520for%2520Ordinary%2520Differential%2520Equations.&amp;pg=PA417#v=onepage&amp;q&amp;f=false" rel="nofollow">chapter 8 of Jackiewicz's book</a>, especially section 8.10 for a general background. There's some <a href="http://www.math.auckland.ac.nz/~hpod/atlas/irks.m" rel="nofollow">matlab code</a> by Podhaisky too, used to <a href="http://www.math.auckland.ac.nz/~hpod/atlas/" rel="nofollow">do this</a>, but no control here.</p> <p>And then, the <a href="http://www.math.auckland.ac.nz/~butcher/theses/" rel="nofollow">theses of Butcher's recent student are here</a>, which discuss implementation details, in particular Huang's chapter 3 should be very useful to you.</p> <p>Older fortran code by Hairer do implement both order and stepsize control: see <a href="http://www.unige.ch/~hairer/software.html" rel="nofollow">RADAU and DR_RADAU here</a>, it's not for IRKS but gives a well-tested framework that could be suitably modified. </p> http://mathoverflow.net/questions/30655/birkhoff-conjecture-about-integrable-billiards/30665#30665 Answer by Thomas Sauvaget for Birkhoff conjecture about integrable billiards Thomas Sauvaget 2010-07-05T20:18:06Z 2010-07-05T20:18:06Z <p>I'm no expert, but <a href="http://books.google.fr/books?id=g8PdXL4ST4YC&amp;lpg=PR7&amp;ots=H2DaDdVh4z&amp;lr&amp;pg=PA95#v=onepage&amp;q=birkhoff&amp;f=false" rel="nofollow">according to Tabachnikov</a> the conjecture was still open as of 2005, while <a href="http://upcommons.upc.edu/e-prints/bitstream/2117/1189/1/9503delsh.pdf" rel="nofollow">Delshams and Ramirez-Ros have a local</a> result (i.e. the conjecture is true when considering symmetric entire perturbations). Probably Mathscinet would help more.</p> http://mathoverflow.net/questions/25630/major-mathematical-advances-past-age-fifty/25651#25651 Answer by Thomas Sauvaget for Major mathematical advances past age fifty Thomas Sauvaget 2010-05-23T11:08:24Z 2010-05-23T11:21:36Z <p><a href="http://en.wikipedia.org/wiki/Poincare_conjecture" rel="nofollow">Poincaré's conjecture</a> has been <a href="http://www.springerlink.com/content/l482x733387176u4/" rel="nofollow">formulated in 1904</a>, when he had just turned 50, while presenting a counter-example (the Poincaré homology sphere) to another earlier conjecture of his. Probably, given the impact it has had for a whole century, the precise formulation of the conjecture can be seen as a "major discovery" by itself.</p> http://mathoverflow.net/questions/24669/what-kind-of-uniqueness-can-i-conclude-for-solutions-to-a-simple-functional-equat/24817#24817 Answer by Thomas Sauvaget for What kind of uniqueness can I conclude for solutions to a simple functional equation? Thomas Sauvaget 2010-05-15T21:03:04Z 2010-05-15T21:03:04Z <p>Here's a vague answer then: there's <a href="http://www.springerlink.com/content/unj551702r542h82/" rel="nofollow">an old 1978 paper in Aequationes Mathematicae</a> mentionning the problem of uniqueness for the functional equation $\varphi (x)=h(x,\varphi [f_1(x)],\dots, \varphi [f_n(x)] )$ under some hypotheses. This seems to be of the same form as your equation (identifying $x=(q_0,t_0,q_1,t_1)$ and so on). Maybe what you're looking for is in there, or in later papers quoting that one.</p> <p>Having said that, since your inspiration is Hamilonian mechanics, I would imagine at least some variant of your problem to have been investigated already. Obviously you're looking at the stationnary phase approximation to the quantum propagator, such as in <a href="http://chaosbook.org/chapters/VanVleck.pdf" rel="nofollow">equation (33.32) of a chapter about the semiclassical propagator</a> in a <a href="http://chaosbook.org/" rel="nofollow">great physics textbook</a>. Now there's been a lot of mathematical work on the topic which may, at least implicitely, help you with your specific question, namely <a href="http://ccdb4fs.kek.jp/cgi-bin/img/allpdf?199201019" rel="nofollow">this paper of Meinrenken</a> in particular page 7 and beyond, and <a href="http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=97-599" rel="nofollow">that paper</a> of Combescure-Ralston-Robert.</p> http://mathoverflow.net/questions/24132/what-are-examples-of-mathematical-concepts-named-after-the-wrong-people-stigler/24152#24152 Answer by Thomas Sauvaget for What are examples of mathematical concepts named after the wrong people? (Stigler's law) Thomas Sauvaget 2010-05-10T20:29:07Z 2010-05-10T20:29:07Z <p>Euler's <a href="http://en.wikipedia.org/wiki/Nine-point_circle" rel="nofollow">nine point circle</a> was <a href="http://sunsite.utk.edu/math_archives/.http/hypermail/historia/jun00/0134.html" rel="nofollow">never discussed</a> by Euler. This is an error of the "argument by authority" type: Catalan propagated that incorrect attribution made by another scholar, the "learned Terquem", without checking it himself.</p> http://mathoverflow.net/questions/23758/published-results-when-to-take-them-for-granted Published results: when to take them for granted? Thomas Sauvaget 2010-05-06T18:06:06Z 2010-05-10T11:57:10Z <p><strong>Two kinds of papers.</strong> There are two kinds of papers: self-contained ones, and those relying on published results (which I believe are the vast majority).</p> <p><strong>Checking the result.</strong> Of course, one should check carefully other's results before using them. There are several incentives to do that: become a real specialist; expand one's knowledge of concepts and techniques; find and mention a gap in the proof should that happen; get the ability to interact with more people ("I read your paper..."). So ideally, in a sense, checking a result before using it should always be the case.</p> <p><strong>Trusting peer-review.</strong> Yet, the very idea of academic peer-reviewed publications is to allow readers to locate results deemed trustable. The implied degree of trustability varies among scientific disciplines, but one would expect mathematics to have to most stringent one: a proof is either correct or it is not.</p> <p>Given this, it is sometimes very tempting to use a result as a kind of "useful axiom", especially if that result has been proven with concepts very far from one's own area(s) of expertise, or if it is the culmination of several long papers: in those cases it would require a substantial amount of time, maybe even years, to personally check the results in their own right. Someone wanting to move forward quickly (or with a short-term position) may not want to go into this.</p> <p><strong>How to decide?</strong> Some cases are clear-cut (e.g. most people would accept the classification of finite simple groups), while others are borderline. </p> <p>My questions on that matter are:</p> <blockquote> <ol> <li>Are there rules of thumb that you have come up with when deciding between checking a result, and taking it as an axiom ? </li> <li>When accepting without checking, how do you phrase it?</li> <li>Has it ever occured to you that taking a result for granted actually backfired: what happened, and what would you do differently (job interview, retraction of publication)?</li> </ol> </blockquote> <hr> <p><em>EDIT (friday 7 may)</em>: many thanks to those who have replied, very interesting comments! (Also, please note that since there is no "best answer" to that kind of question I will not single out one over the others.)</p> http://mathoverflow.net/questions/20147/what-would-a-graduate-course-on-systolic-geometry-typically-cover What would a graduate course on systolic geometry typically cover? Thomas Sauvaget 2010-04-02T08:11:19Z 2010-04-10T00:57:27Z <p>It's all in the title basically. There's an interesting topic called <a href="http://en.wikipedia.org/wiki/Systolic_geometry" rel="nofollow">systolic geometry</a> that has grown a lot in the past 30 years, with a (first?) <a href="http://www.ams.org/bookstore-getitem/item=surv-137" rel="nofollow">textbook</a> on the subject by M.Katz (AMS 2007).</p> <p>So I was wondering what would a semester-long graduate course typically cover, assuming knowledge of basic Riemannian geometry. </p> <p>I haven't been able to find much information online: Katz has a <a href="http://u.math.biu.ac.il/~katzmik/egreg826.pdf" rel="nofollow">second semester graduate course</a> which doesn't quite cover the same ground as the book. Has there been other courses on the topic elsewhere?</p> http://mathoverflow.net/questions/20456/biographic-data-stories-about-andre-neron/20483#20483 Answer by Thomas Sauvaget for Biographic Data/Stories about André Néron Thomas Sauvaget 2010-04-06T08:31:44Z 2010-04-06T08:31:44Z <p>There might be a little more information in an 1986 article published by the alumni magazine of ENS: <a href="http://www.google.fr/#hl=fr&amp;tbo=p&amp;tbs=bks%253A1&amp;q=%2522andr%25C3%25A9+n%25C3%25A9ron%2522+1922+1943&amp;meta=&amp;aq=f&amp;aqi=&amp;aql=&amp;oq=&amp;gs_rfai=&amp;fp=b9f6a4052b5d3491" rel="nofollow">google would only reveal</a> that he was born in the small village of <a href="http://fr.wikipedia.org/wiki/La_Clayette" rel="nofollow">La Clayette</a> and died of cancer aged 62.</p> <p>Concerning the wartime hiatus: in february 1943 it was announced by french authorities that people born in either 1920, 1921, or 1922 were compelled to go and work in Germany (STO: Service de Travail Obligatoire). Since Néron was born in 1922, he either went to Germany or fled (some ENS students <a href="http://www.archicubes.ens.fr/docs/2007319122412_Cabannes11nov.htm" rel="nofollow">have described how they fled</a>).</p> <p>One thing for certain: Néron was "promotion 1943" at ENS, which means he <em>entered</em> the school then, not that he graduated in 1943. In fact Fields medalist <a href="http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Thom.html" rel="nofollow">René Thom</a> also was from promotion 1943, and also completed his doctorate in 1951, so in that respect the dates are not unusual. The big difference is that Thom was born in 1923, and thus not subjected to STO. </p> http://mathoverflow.net/questions/20154/candidate-for-rigorous-mathematical-definition-of-canonical/20156#20156 Answer by Thomas Sauvaget for candidate for rigorous _mathematical_ definition of "canonical"? Thomas Sauvaget 2010-04-02T10:22:35Z 2010-04-02T10:22:35Z <p>There are scanned notes in french that were used for the initial text of <em>Théorie des Ensembles</em> <a href="http://mathdoc.emath.fr/archives-bourbaki/feuilleter.php?chap=2_REDAC_E1" rel="nofollow">on the Bourkaki Archives website</a>. </p> <p>In particular there are indeed notes by Chevalley named <em>Livre I. Théorie des ensembles Chap. IV (état 7 ?) Structures (53 p.)</em> which seem at first glance to define "canonique" in the broader context of "transport de structures, idendifications" (see exemple 1 at the bottom of page 19 of that file). </p> http://mathoverflow.net/questions/6276/least-number-of-non-zero-coefficients-to-describe-a-degree-n-polynomial Least number of non-zero coefficients to describe a degree n polynomial Thomas Sauvaget 2009-11-20T15:42:07Z 2010-03-26T22:25:20Z <p>I'd be grateful for a good reference on this, it feels like a classic subject yet I couldn't find much about it. </p> <p>Polynomials in one variable of the form $x^n+a_{n-1}x^{n-1}+\dots +a_1 x+a_0$ can be transformed into simpler expressions. For instance <a href="http://homepage.mac.com/ehgoins/ma598/lecture%5F27.pdf" rel="nofollow">it is apparently well-known</a> that the Tschirnhaus transformation allows to bring <em>any</em> quintic into so-called Bring-Jerrard form $x^5+ax+b$, while for degree 6 one needs at least three coefficents $x^6+ax^2+bx+c$.</p> <p>Is there a name for such "generalized Bring-Jerrard form", and what is known about it? In particular there is a cryptic footnote of Arnold (page 3 of <a href="http://www.pdmi.ras.ru/~arnsem/Arnold/arnlect1.ps.gz" rel="nofollow">this lecture</a>) where he says roughly that the degrees for which more coefficients are needed occur along "a rather strange infinite sequence": could someone please describe what those degrees are (I had a look at the OEIS but I believe that sequence is different from <a href="http://www.research.att.com/~njas/sequences/A000905" rel="nofollow">Hamilton numbers</a>, and couldn't find a relevant one).</p> http://mathoverflow.net/questions/17357/what-is-so-spectral-about-spectral-sequences/17406#17406 Answer by Thomas Sauvaget for What is so "spectral" about spectral sequences? Thomas Sauvaget 2010-03-07T19:17:33Z 2010-03-07T19:17:33Z <p>I'm not a specialist, but from browsing the french literature it appears that the interpretation mentionned is correct: Leray was motivated by studying general invariants of spaces and continuous functions, this from his interest in Mechanics and PDEs, see the quote <a href="http://www.math.sciences.univ-nantes.fr/~guillope/LG/UPN-Leray-2006-11-17.pdf" rel="nofollow">page 6 of this</a> (Leray was in fact chair of Mechanics at the Académie after WWII, and also <a href="http://www.college-de-france.fr/default/EN/all/ins_dis/jean_leray.htm" rel="nofollow">chair of ODEs and Functional Equations</a> at Collège de France). For instance in <a href="http://www.math.sciences.univ-nantes.fr/~guillope/LG/JLeray-2006.pdf" rel="nofollow">this document</a> (warning: 60MB) are his publication list and some scanned notes from pre-WWII meetings with Bourbaki where Leray is listened to precisely about spectral theory matters. Also, Leray learned from Elie Cartan a lot about Lie groups and representation theory, and knew its relationship to quantum mechanics (i.e. again the idea of invariants).</p> <p>The first paper of Leray on the topic of spectral sequences where really the word <em>spectral</em> appears is the comprehensive one published in 1950 (here is its <a href="http://www.zentralblatt-math.org/zmath/scans.html?volume_=038&amp;count_=363" rel="nofollow">Zentralblatt review</a>), so the paper was circulating earlier. Apparently a first note in CRAS by Leray dates from 1945, then in 1947 Koszul generalized the idea, but still without the word spectral I think. These were treating cohomology stuff. On the other hand, <a href="http://books.google.fr/books?id=eaUoAKOAbUsC&amp;lpg=PA8&amp;dq=serre%2520spectrale&amp;pg=PA8#v=onepage&amp;q=serre%2520spectrale&amp;f=false" rel="nofollow">Serre's CRAS note</a>, which predates his thesis, appeared in 1950, and it treats homology stuff. For cohomology matters, I've seen in early papers anything from "Leray spectral sequence", to "Leray-Koszul", to "Leray-Koszul-Cartan" (since Cartan had <a href="http://www.numdam.org/numdam-bin/feuilleter?id=SHC_1950-1951__3_" rel="nofollow">a seminar</a> on those things). </p> http://mathoverflow.net/questions/16587/specializing-early Specializing early Thomas Sauvaget 2010-02-27T11:16:34Z 2010-03-02T03:55:45Z <p><strong>Topic</strong>: this is a mathematics education question (but applies to other sciences too). </p> <p><strong>Assumptions</strong>: my first assumption is that most mathematical concepts used in research are not intrinsically more complicated to grasp than high-school and undergraduate maths, the main difference is the amount of prerequisites (and hence time and experience) involved. My second assumption is that some undergraduate topics currently taught compulsarily are a bit of a burden for someone focussed on a particular topic.</p> <p>Now of course cognitive development is a constraint, but upon reaching the age of high-school, I would think that a fairly large proportion of the scientifically-enclined students could really understand things usually taught much later and indeed become active at research level within a few years, provided some shortcuts are introduced. </p> <p><strong>Early specialization</strong>: I'm wondering if a balanced curriculum already exists (or is planned) to provide such early specialization. What I'm looking for is this: a one-week panorama of maths (or physics, or biology) would be organized at the beginning, and then the students would decide which subtopic to study. For example someone interested by group theory (or quantum optics, or genetics) would thus start with basics at the age 15 or 16, and gradually learn more stuff and skills, but for a few years with a strong emphasis on things directly relevant for the chosen subtopic. </p> <p>So for example the student specializing in group theory would only learn differential calculus and manifolds in passing in the context of Lie groups, and would skip most undergraduate real and functional analysis until it becomes relevant for his/her research topic, if at all. Of course other general courses would still be taught (history, sciences, programming, foreign languages...), but at least 50% of the student's week would be devoted to the research topic, ensuring satisfying progress. </p> <p><strong>Question</strong>: do you know of any active or planned educative curriculum (at a high-school or university, or maybe a specific home-schooling program) as outlined above? As an example of successful early specialization see e.g. the winners of the <a href="http://www.siemens-foundation.org/en/" rel="nofollow">Siemens Foundation Prizes</a>, but I haven't been able to learn much about their specific curriculum if any. </p> <p><strong>Note</strong>: Skipping grades in school to enter university earlier is not the point, I'm really interested in a subtopic-oriented curriculum.</p> http://mathoverflow.net/questions/16588/how-to-figure-out-the-type-of-the-bifurcation-in-a-dynamical-system/16634#16634 Answer by Thomas Sauvaget for How to figure out the type of the bifurcation in a dynamical system? Thomas Sauvaget 2010-02-27T20:53:37Z 2010-02-27T20:53:37Z <p>It depends: if you know $f$ explicitely then working out critical points and normal forms will tell you so, otherwise you'd have to use a specialized program. Have a look at <a href="http://www.scholarpedia.org/article/Bifurcations" rel="nofollow">this scholarpedia article</a> and the other articles, books and programs mentionned there (start with <a href="http://www.scholarpedia.org/article/Saddle-node_bifurcation" rel="nofollow">saddle-node</a>) .</p> http://mathoverflow.net/questions/15893/editors-in-peer-review-systems Editors in peer-review systems Thomas Sauvaget 2010-02-20T13:41:25Z 2010-02-20T17:02:51Z <p>This is a strictly technical question on peer-review systems currently employed in the mathematical literature, not a subjective discussion of merits/drawbacks of such systems, so I think/hope it's suitable for MO.</p> <p>I have noticed that some journals (e.g. PNAS, CRAS, Nonlinearity...) always publish papers with the name of the editor who supervised the refereeing process ("Presented by X", "Recommended by X", "Communicated by X"). Most other journals, while having editors listed explicitly for each area (and hence in theory one could also know in most cases who supervised what), do not make this explicit.</p> <p>I was wondering was difference it makes, as a junior author, to have an editor's name on a paper: </p> <ul> <li>is that a strong endorsement of the paper?</li> <li>a way to say that the journal is ultra-strict about the refereeing process?</li> <li>simply a full-disclosure practise of the journal?</li> <li>an incentive to publish there (when some editor is a "big name", or to make sure a specific person read your paper) ? </li> </ul> <p>It's really not clear to me.</p> http://mathoverflow.net/questions/2556/real-world-applications-of-mathematics-by-arxiv-subject-area/2602#2602 Answer by Thomas Sauvaget for Real-world applications of mathematics, by arxiv subject area? Thomas Sauvaget 2009-10-26T12:19:55Z 2010-01-30T01:23:02Z <p><strong>Math.AP Analysis of PDE</strong></p> <ul> <li>Partial differential equations are used a lot for modelling systems in biology and medicine, and help describe e.g. animal coat pattern formation (zebras, leopards...), wound healing, tumor growth, spread of a virus in a population, predator-prey systems in ecology, predicting the variations of concentrations of chemicals (hormones, drugs...) within an organ over time...</li> <li>PDEs are used in climate modelling, from atmospheric dynamics to ocean currents. </li> <li>Radar imaging is based on solving an inverse problem. The recent buzz about metamaterials and invisibility is based on understanding variable-coefficient elliptic problems.</li> </ul> http://mathoverflow.net/questions/11113/how-important-are-publications-for-undergrads/11145#11145 Answer by Thomas Sauvaget for How important are publications for undergrads? Thomas Sauvaget 2010-01-08T14:11:02Z 2010-01-08T14:11:02Z <p>As for mathematics, it might be worth mentionning that there's an AMS prize rewarding undergraduates who published "outstanding" results (by themselves or as a joint work), the <a href="http://www.ams.org/prizes/morgan-prize.html" rel="nofollow">Frank and Brennie Morgan prize</a>. </p> <p>That said, there is no indication on the number of candidates considered each year for the prize. Perhaps the fact that they have honorable mentions means that really only one or two undergraduates each year manage to publish at research-level, the rest of those publications not being worth mentionning.</p> http://mathoverflow.net/questions/10403/zeta-function-for-curves-in-a-manifold/10841#10841 Answer by Thomas Sauvaget for Zeta function for curves in a manifold Thomas Sauvaget 2010-01-05T20:38:56Z 2010-01-05T20:38:56Z <p>There's been a lot of work since Smale's idea of a dynamical zeta function for general flows (in particular geodesic flows). A good starting point would be this 12 year old <a href="http://www.dma.ens.fr/~baladi/etds.ps" rel="nofollow">review by Baladi</a>. There is a large and more recent literature but I'm no expert, although this <a href="http://www.mat.uniroma2.it/~liverani/Lavori/live0602.pdf" rel="nofollow">other review by Liverani and Tsuji</a> is probably not far from current knowledge.</p> <p>There's also a whole branch of physics around those ideas, indeed related to the spectrum of the Laplacian and applications to quantum physics and statistical physics. <a href="http://chaosbook.org/" rel="nofollow">This nice physics book</a> is a good start (in particular if you read the quote of Smale at page 3 of <a href="http://chaosbook.org/chapters/det.pdf" rel="nofollow">this chapter</a>, and then remark 19.2 at the very end of that chapter you'll get a quick sense of the stuff you've aked for).</p> http://mathoverflow.net/questions/10574/how-do-i-make-the-conceptual-transition-from-multivariable-calculus-to-differenti/10579#10579 Answer by Thomas Sauvaget for How do I make the conceptual transition from multivariable calculus to differential forms? Thomas Sauvaget 2010-01-03T10:55:27Z 2010-01-03T10:55:27Z <p>The great book <em>The Geometry of Physics (2nd edition)</em> by Frankel could be exactly what you need, see excerpts <a href="http://books.google.fr/books?id=DUnjs6nEn8wC&amp;printsec=frontcover&amp;source=gbs%5Fv2%5Fsummary%5Fr&amp;cad=0#v=onepage&amp;q=&amp;f=false" rel="nofollow">here</a>. </p> http://mathoverflow.net/questions/9754/magic-trick-based-on-deep-mathematics/9801#9801 Answer by Thomas Sauvaget for Magic trick based on deep mathematics Thomas Sauvaget 2009-12-26T16:48:18Z 2009-12-26T16:48:18Z <p>Apart from tricks based on numbers, there are topological objects whose properties can seem quite magical, like the Möbius strip or the unknot. </p> <p>E.g. take a standard page of paper, show that it has two sides (number them with a pen, show that any straight pen path meets a boundary). Next, cut out a long strip from it (not needed of course, but adds to the drama), and ask the audience "and how many sides does this have?". They reply "two". Then you put the the two small ends of the strip together to form a ring and you ask "and now, how many sides?", they still reply "two!". At this point do a little diversion, like putting a pair of scissors on the table saying out loud "I'll use this in a minute". Now do a half-twist with the strip before putting the small ends together and ask again "for the last time people, how many sides?". They answer "twoo!!", and you say "the magic has worked people, there's only one side!" (you show that now the pen paths along the long direction never meet a boundary and come back). Most laymen are quite bemused. Now do two half-twists and ask again, some won't dare an answer...</p> http://mathoverflow.net/questions/6325/equations-for-integrable-systems/6469#6469 Answer by Thomas Sauvaget for Equations for Integrable Systems Thomas Sauvaget 2009-11-22T18:10:47Z 2009-11-22T18:10:47Z <p>I'm no expert on this, but since nobody answers: </p> <p>one book which should definitely help is <a href="http://books.google.fr/books?id=c6JN6Gp4RBQC&amp;pg=PP1&amp;dq=INTRODUCTION+TO+CLASSICAL+INTEGRABLE+SYSTEMS#v=onepage&amp;q=&amp;f=false" rel="nofollow">this big introductory one by Babelon, Bernard and Talon</a>. There's also a very technical older <a href="http://arxiv.org/abs/math.AG/9902068" rel="nofollow">paper by Ben-Zvi and Frenkel</a> where apparently some sort of general construction is made, and there's a more readable <a href="http://www-math.univ-poitiers.fr/~vanhaeck/art/art31/www31.pdf" rel="nofollow">paper by Inoue, Vanhaecke and Yamazaki</a> on algebraic complete integrability and integrable hierarchies of PDEs intended for your second question (especially section 6 for the relationship). </p> <p>Just glancing at all this I'm not so sure a general explicit method to obtain the PDEs exists (I could be wrong) but some cases seem to be understood. Hope this helps...</p> http://mathoverflow.net/questions/5485/how-many-mathematicians-are-there/5490#5490 Answer by Thomas Sauvaget for How many mathematicians are there? Thomas Sauvaget 2009-11-14T09:28:36Z 2009-11-14T09:28:36Z <p>In <a href="http://smf.emath.fr/Publications/ExplosionDesMathematiques/pdf/smf-smai%5Fexplo-maths%5F92-97.pdf" rel="nofollow">an article</a> written a few years ago, Jean-Pierre Bourguignon estimates that there are around 80 000 mathematicians worldwide, with the AMS having about 15 000 members. </p> <p>For France he says 4000 work in academia ("a reliable estimate") and about 2000 in the private sector. Since there are about 60 million inhabitants there, that's 1 mathematician per 10 000 inhabitants.</p> http://mathoverflow.net/questions/119621/learning-through-guided-discovery/119941#119941 Comment by Thomas Sauvaget Thomas Sauvaget 2013-01-27T08:40:56Z 2013-01-27T08:40:56Z The book you cited and the answers you got are complementary to 12709, so there's nothing to worry about, I think. http://mathoverflow.net/questions/117622/math-french-words Comment by Thomas Sauvaget Thomas Sauvaget 2012-12-30T20:18:42Z 2012-12-30T20:18:42Z A useful book in that regard is the bilingual edition of the Grothendieck-Serre correspondence (from 1956-1964): lots of typical french sentences, and technical math words, translated carefully side by side. A few more recent technical words are thus missing, to be found online. The book is edited by the AMS, and a preview is here <a href="http://books.google.fr/books?id=FBfygannPSUC&amp;printsec=frontcover&amp;hl=fr#v=onepage&amp;q&amp;f=false" rel="nofollow">books.google.fr/&hellip;</a> http://mathoverflow.net/questions/101917/ruelle-inequality-on-a-noncompact-space Comment by Thomas Sauvaget Thomas Sauvaget 2012-07-11T13:49:09Z 2012-07-11T13:49:09Z Chapter 6 of the thesis of van Bargen seems to be relevant in the case M=R^d <a href="http://opus.kobv.de/tuberlin/volltexte/2010/2571/pdf/vanbargen_holger.pdf" rel="nofollow">opus.kobv.de/tuberlin/volltexte/2010/2571/pdf/&hellip;</a> http://mathoverflow.net/questions/84823/recursivity-of-the-primes Comment by Thomas Sauvaget Thomas Sauvaget 2012-01-04T10:04:00Z 2012-01-04T10:04:00Z Thanks for the comment. Yes, following your suggestion, the logarithm of sequence (c) is well-fitted, up to small errors, by the sequence $0.3023191518585 e^{0.6933439751085 n}$, while the logarithm of sequence (d) is less accurately fitted by $0.592994983 e^{0.696248232024512 n}$. This is certainly helpful, but unfortunately Plouffe's inverter doesn't seem to recognize either of these four constants, while I was hoping for something known and explicit. http://mathoverflow.net/questions/2281/use-of-n-transitivity-in-finite-group-theory/78803#78803 Comment by Thomas Sauvaget Thomas Sauvaget 2011-10-22T08:46:03Z 2011-10-22T08:46:03Z Very concrete indeed, thank you! http://mathoverflow.net/questions/76623/growing-random-trees-on-a-lattice-rightarrow-voronoi-diagrams/76646#76646 Comment by Thomas Sauvaget Thomas Sauvaget 2011-09-29T14:45:11Z 2011-09-29T14:45:11Z But wouldn't one expect those pockets to be filled by that same color later on, thanks to the untouched points inside them which still allow to go on ? http://mathoverflow.net/questions/60288/status-of-an-open-problem-about-semilinear-sets Comment by Thomas Sauvaget Thomas Sauvaget 2011-04-01T20:35:38Z 2011-04-01T20:35:38Z I've added the &quot;arithmetic-progression&quot; tag (even though you probably don't have only the r=1 case in mind). Perhaps even the &quot;nt.number-theory&quot; one might be appropriate (and would add the benefit of a substantial readership). http://mathoverflow.net/questions/56547/applications-of-mathematics/56552#56552 Comment by Thomas Sauvaget Thomas Sauvaget 2011-02-25T10:16:57Z 2011-02-25T10:16:57Z @Qiaochu: Sorry the link should be <a href="http://www.mpip-mainz.mpg.de/theory/events/namet2010/available_presentations/Friesecke.pdf" rel="nofollow">mpip-mainz.mpg.de/theory/events/namet2010/&hellip;</a> Also, the SO(4) symmetry is a property of the classical 2-body problem, which is thus inherited by the quantum one: for a natural conceptual explanation you can have a look at this wikipedia section <a href="http://en.wikipedia.org/wiki/Laplace-Runge-Lenz_vector#Rotational_symmetry_in_four_dimensions" rel="nofollow">en.wikipedia.org/wiki/&hellip;</a> (and that whole wikipedia page for more details on the corresponding additional conserved quantity). http://mathoverflow.net/questions/56547/applications-of-mathematics/56552#56552 Comment by Thomas Sauvaget Thomas Sauvaget 2011-02-25T10:07:09Z 2011-02-25T10:07:09Z @Qiaochu: That's fine as a basic example of the use of representation theory. But note that much more subtle aspects of how the Periodic Table is organized, which for a long time were only experimentally observed, have just recently been described mathematically: by detailed studies of asymptotic properties of the Schr&#246;dinger equation combined with some rep theoretical aspects, see &lt;a href=&quot;<a href="http://www.mpip-mainz.mpg.de/theory/events/namet2010/available_presentations/Friesecke.pdf&quot;&gt;Friesecke-Goddard-Mendl&lt;/a&gt" rel="nofollow">mpip-mainz.mpg.de/theory/events/namet2010/&hellip;</a>;. http://mathoverflow.net/questions/56495/what-to-do-when-your-work-is-dissed Comment by Thomas Sauvaget Thomas Sauvaget 2011-02-24T07:54:15Z 2011-02-24T07:54:15Z This also raises the issue of research announcements in general (such as Comptes Rendus notes, etc): I'm sure this is not the only case of claims in a research announcement that were not followed by a detailed publication. Maybe you should search for papers containing the words &quot;never appeared&quot;, and see how the other authors who did provide the missing details later have decided to present their situation. http://mathoverflow.net/questions/2615/least-number-of-charts-to-describe-a-given-manifold/54346#54346 Comment by Thomas Sauvaget Thomas Sauvaget 2011-02-05T21:19:03Z 2011-02-05T21:19:03Z Yes, sounds like interesting references, and the result is intuitively satisfying, thanks! I'll try to find a library that has them... http://mathoverflow.net/questions/30459/counting-the-groups-of-order-n-weighted-by-1-autg/35026#35026 Comment by Thomas Sauvaget Thomas Sauvaget 2010-08-09T17:51:23Z 2010-08-09T17:51:23Z And the sequence listing the number of groups of given order N up to 2047 is the very first sequence in Sloane's database <a href="http://www.research.att.com/~njas/sequences/A000001" rel="nofollow">research.att.com/~njas/sequences/A000001</a> the longer list being this <a href="http://www.research.att.com/~njas/sequences/b000001.txt" rel="nofollow">research.att.com/~njas/sequences/b000001.txt</a> http://mathoverflow.net/questions/33808/how-connected-are-you Comment by Thomas Sauvaget Thomas Sauvaget 2010-07-29T19:07:40Z 2010-07-29T19:07:40Z Also, there are things not captured by collaborations. For instance, Fields medalist Laurent Lafforgue has no coauthor, and on top of that his papers are said to be quite technical: so the probability to see his own work being related to other topics outside math by a string of collaborations is zero, yet its place in &quot;the scheme of things&quot; from a pure math perspective is not that limited. Probably a better measure would be a notion of &quot;greatly inspired by paper X&quot; (i.e. not just a citation). http://mathoverflow.net/questions/30655/birkhoff-conjecture-about-integrable-billiards/30665#30665 Comment by Thomas Sauvaget Thomas Sauvaget 2010-07-05T20:19:37Z 2010-07-05T20:19:37Z (By the way, hello Damien!) http://mathoverflow.net/questions/29485/work-of-icm-2010-plenary-speakers-and-other-humans/29501#29501 Comment by Thomas Sauvaget Thomas Sauvaget 2010-06-25T18:10:46Z 2010-06-25T18:10:46Z (Amusing side remark: upon inspection of various CVs, it appears that Coron and Voisin are actually husband and wife, surely a first.)