User thomas sauvaget - MathOverflow

Answer by Thomas Sauvaget for Learning through guided discovery

2013-01-26T15:03:15Z

The book Abel's Theorem in Problems and Solutions by Alekseev & Arnold is a great one to learn about group theory and complex analysis (see excerpts here)

Also, have a look at the following related MO questions: 12709, 28158 and 56314.

Recursivity of the primes

2012-01-03T18:43:20Z

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).

It is thus tempting to look at the recursive construction (in some sense, the maximal possible generation of primes):

$2$ produces $3,5,7$ in $[2+1;3^2]=[3;9]$ (that's 3 primes, i.e. $3=\pi(9)-\pi(3)$)

then $2,3,5,7$ produces $11,13\dots ,113$ in $[7+1;11^2]$ (that's 26 primes)

then $2,\dots ,113$ produces $127,\dots ,16127$ in $[113+1;127^2]$ (that's 1847 primes)

then $2,\dots ,16127$ produces $16139,\dots ,260 467 313$ in $[16127+1;16139^2]$ (that's 14 218 065 primes).

And so on...

I would be interested to know more about this recursive procedure, for instance obtain an asympotic equivalent if possible.

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.

I have looked at the obvious places (the OEIS, and elementary texts like Conway & Guy's The Book of Numbers or Stein's lectures on Elementary Number Theory) but to no avail. In particular none of them, including the OEIS, seem to mention the relevant sequences:

a) growth of the number of primes produced: 1, 3, 26, 1847, 14 218 065, ...

b) accumulated growth: 1, 4, 30, 1877, 14 219 942, ...

c) first new prime produced: 2, 3, 11, 127, 16139, 260 467 367, 67 843 249 271 912 789, ...

d) last new prime produced: 7, 113, 16127, 260 467 313,...

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).

Use of n-transitivity in finite group theory

2009-10-24T13:11:01Z

Hello, apparently finite groups which are n-transitive with n>5 are only the permutation groups S_n or the alternating groups A_n+2, see e.g. page 226 this book by Isaacs http://books.google.fr/books?id=pCLhYaMUg8IC&pg=PA226

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_n ?

Answer by Thomas Sauvaget for Who coined the name tensor and why?

2011-09-18T17:21:16Z

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 Die fundamentalen physikalischen Eigenschaften der Krystalle in elementarer Darstellung published in 1898. (I do not have access to this paper, but probably this deals with deformation tensors in crystals).

Answer by Thomas Sauvaget for Canonical examples of algebraic structures

2009-10-29T11:15:53Z

Not an algebraist myself, but this is interesting. Few things I do know:

Read ------------------------------- Quickly think of

algebraically closed field --------- $\mathbb C$

abelian group ---------------------- $(\mathbb R;+)$

non-abelian group ----------------- $(\operatorname{SO}(n,\mathbb R);\times)$ ; ($n>2$)

simple group ----------------------- $(A_n; \circ)$ ; ($n\ge5$)

Least number of charts to describe a given manifold

2009-10-26T14:00:44Z

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.

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).

Edit: 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 http://books.google.fr/books?id=vMREfNN-L4gC&pg=PP1

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?

Final edit: 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.

Answer by Thomas Sauvaget for Is symplectic reduction interesting from a physical point of view?

2010-09-23T17:30:17Z

Of course it is interesting, and the idea of factoring out symmetries goes back to Newton.

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 the blue book of Bates & 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.

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.

As for your comment, the topology of the reduced space is certainly a "new physical insight". Have a look at this recent paper by Yanguas, Palacian, Meyer & 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.

(Edited to correct an error; more links added in reply to comment.)

Covering the primes by 3-term APs ?

2009-10-23T23:03:29Z

Hello, the Green-Tao theorem says infinitely many k-term Arithmetic Progressions exist for any integer k.

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 ?

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).

Answer by Thomas Sauvaget for Work of ICM 2010 plenary speakers (and other humans)

2010-06-25T11:36:58Z

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).

David Aldous [Probability & Statistics, especially random flows on networks] http://www.stat.berkeley.edu/~aldous/Research/research.html
Artur Avila [Dynamical Systems & Spectral Theory] http://w3.impa.br/~avila/new.html
R. Balasubramanian [Number Theory & Cryptography]
Jean-Michel Coron [Control of PDEs] http://www.scholarpedia.org/article/Control_of_partial_differential_equations
Irit Dinur [Computational Complexity Theory, Graph Theory]
Hillel Furstenberg [Ergodic Theory] http://www.wolffund.org.il/full.asp?id=155
Thomas J.R. Hughes [Computational Fluid mechanics]
Peter Jones [Differential and Complex Geometry] http://www.pnas.org/content/105/6/1803.full
Carlos Kenig [Nonlinear PDEs, especially Schödinger and Wave types] http://www-news.uchicago.edu/releases/08/080108.kenig.shtml
Ngo Bao Chau [Algebraic Geometry, Langlands program] http://www.mfo.de/programme/prize/Ngo2008.pdf and http://www.institut.math.jussieu.fr/projets/fa/bpFiles/DatTuan.pdf
Stanley Osher [Scientific Computing, especially Level-Set Methods] http://www.levelset.com/sjo/Interview.htm
R. Parimala [Arithmetic Algebraic Geometry]
A. N. Parshin [Harmonic Analysis & Arithmetic Groups]
Shige Peng [Financial Mathematics]
Kim Plofker [History of Math, especially India]
Nicolai Reshetikhin [Mathematical Physics & Statistical Physics]
Richard Schoen [Global Differential Geometry]
Cliff Taubes [4-dimensional Geometry, Symplectic Topology, Contact Geometry] http://www.ams.org/notices/200805/tx080500596p.pdf
Claire Voisin [Algebraic Geometry, especially Hodge Conjecture and Mirror Symmetry] http://www.math.jussieu.fr/~voisin/Articlesweb/template.pdf
Hugh Woodin [Logic, especially the Continuum Hypothesis]

Answer by Thomas Sauvaget for Adaptive controllers for stiff ODE and DAE integrators

2010-08-05T15:48:27Z

Have a look at chapter 8 of Jackiewicz's book, especially section 8.10 for a general background. There's some matlab code by Podhaisky too, used to do this, but no control here.

And then, the theses of Butcher's recent student are here, which discuss implementation details, in particular Huang's chapter 3 should be very useful to you.

Older fortran code by Hairer do implement both order and stepsize control: see RADAU and DR_RADAU here, it's not for IRKS but gives a well-tested framework that could be suitably modified.

Answer by Thomas Sauvaget for Birkhoff conjecture about integrable billiards

2010-07-05T20:18:06Z

I'm no expert, but according to Tabachnikov the conjecture was still open as of 2005, while Delshams and Ramirez-Ros have a local result (i.e. the conjecture is true when considering symmetric entire perturbations). Probably Mathscinet would help more.

Answer by Thomas Sauvaget for Major mathematical advances past age fifty

2010-05-23T11:08:24Z

Poincaré's conjecture has been formulated in 1904, 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.

Answer by Thomas Sauvaget for What kind of uniqueness can I conclude for solutions to a simple functional equation?

2010-05-15T21:03:04Z

Here's a vague answer then: there's an old 1978 paper in Aequationes Mathematicae 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.

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 equation (33.32) of a chapter about the semiclassical propagator in a great physics textbook. Now there's been a lot of mathematical work on the topic which may, at least implicitely, help you with your specific question, namely this paper of Meinrenken in particular page 7 and beyond, and that paper of Combescure-Ralston-Robert.

Answer by Thomas Sauvaget for What are examples of mathematical concepts named after the wrong people? (Stigler's law)

2010-05-10T20:29:07Z

Euler's nine point circle was never discussed 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.

Published results: when to take them for granted?

2010-05-06T18:06:06Z

Two kinds of papers. There are two kinds of papers: self-contained ones, and those relying on published results (which I believe are the vast majority).

Checking the result. 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.

Trusting peer-review. 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.

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.

How to decide? Some cases are clear-cut (e.g. most people would accept the classification of finite simple groups), while others are borderline.

My questions on that matter are:

Are there rules of thumb that you have come up with when deciding between checking a result, and taking it as an axiom ?

When accepting without checking, how do you phrase it?

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)?

EDIT (friday 7 may): 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.)

What would a graduate course on systolic geometry typically cover?

2010-04-02T08:11:19Z

It's all in the title basically. There's an interesting topic called systolic geometry that has grown a lot in the past 30 years, with a (first?) textbook on the subject by M.Katz (AMS 2007).

So I was wondering what would a semester-long graduate course typically cover, assuming knowledge of basic Riemannian geometry.

I haven't been able to find much information online: Katz has a second semester graduate course which doesn't quite cover the same ground as the book. Has there been other courses on the topic elsewhere?

Answer by Thomas Sauvaget for Biographic Data/Stories about André Néron

2010-04-06T08:31:44Z

There might be a little more information in an 1986 article published by the alumni magazine of ENS: google would only reveal that he was born in the small village of La Clayette and died of cancer aged 62.

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 have described how they fled).

One thing for certain: Néron was "promotion 1943" at ENS, which means he entered the school then, not that he graduated in 1943. In fact Fields medalist René Thom 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.

Answer by Thomas Sauvaget for candidate for rigorous _mathematical_ definition of "canonical"?

2010-04-02T10:22:35Z

There are scanned notes in french that were used for the initial text of Théorie des Ensembles on the Bourkaki Archives website.

In particular there are indeed notes by Chevalley named Livre I. Théorie des ensembles Chap. IV (état 7 ?) Structures (53 p.) 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).

Least number of non-zero coefficients to describe a degree n polynomial

2009-11-20T15:42:07Z

I'd be grateful for a good reference on this, it feels like a classic subject yet I couldn't find much about it.

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 it is apparently well-known that the Tschirnhaus transformation allows to bring any quintic into so-called Bring-Jerrard form $x^5+ax+b$, while for degree 6 one needs at least three coefficents $x^6+ax^2+bx+c$.

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 this lecture) 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 Hamilton numbers, and couldn't find a relevant one).

Answer by Thomas Sauvaget for What is so "spectral" about spectral sequences?

2010-03-07T19:17:33Z

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 page 6 of this (Leray was in fact chair of Mechanics at the Académie after WWII, and also chair of ODEs and Functional Equations at Collège de France). For instance in this document (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).

The first paper of Leray on the topic of spectral sequences where really the word spectral appears is the comprehensive one published in 1950 (here is its Zentralblatt review), 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, Serre's CRAS note, 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 seminar on those things).

Specializing early

2010-02-27T11:16:34Z

Topic: this is a mathematics education question (but applies to other sciences too).

Assumptions: 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.

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.

Early specialization: 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.

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.

Question: 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 Siemens Foundation Prizes, but I haven't been able to learn much about their specific curriculum if any.

Note: Skipping grades in school to enter university earlier is not the point, I'm really interested in a subtopic-oriented curriculum.

Answer by Thomas Sauvaget for How to figure out the type of the bifurcation in a dynamical system?

2010-02-27T20:53:37Z

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 this scholarpedia article and the other articles, books and programs mentionned there (start with saddle-node) .

Editors in peer-review systems

2010-02-20T13:41:25Z

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.

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.

I was wondering was difference it makes, as a junior author, to have an editor's name on a paper:

is that a strong endorsement of the paper?
a way to say that the journal is ultra-strict about the refereeing process?
simply a full-disclosure practise of the journal?
an incentive to publish there (when some editor is a "big name", or to make sure a specific person read your paper) ?

It's really not clear to me.

Answer by Thomas Sauvaget for Real-world applications of mathematics, by arxiv subject area?

2009-10-26T12:19:55Z

Math.AP Analysis of PDE

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...
PDEs are used in climate modelling, from atmospheric dynamics to ocean currents.
Radar imaging is based on solving an inverse problem. The recent buzz about metamaterials and invisibility is based on understanding variable-coefficient elliptic problems.

Answer by Thomas Sauvaget for How important are publications for undergrads?

2010-01-08T14:11:02Z

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 Frank and Brennie Morgan prize.

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.

Answer by Thomas Sauvaget for Zeta function for curves in a manifold

2010-01-05T20:38:56Z

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 review by Baladi. There is a large and more recent literature but I'm no expert, although this other review by Liverani and Tsuji is probably not far from current knowledge.

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. This nice physics book is a good start (in particular if you read the quote of Smale at page 3 of this chapter, 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).

Answer by Thomas Sauvaget for How do I make the conceptual transition from multivariable calculus to differential forms?

2010-01-03T10:55:27Z

The great book The Geometry of Physics (2nd edition) by Frankel could be exactly what you need, see excerpts here.

Answer by Thomas Sauvaget for Magic trick based on deep mathematics

2009-12-26T16:48:18Z

Apart from tricks based on numbers, there are topological objects whose properties can seem quite magical, like the Möbius strip or the unknot.

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...

Answer by Thomas Sauvaget for Equations for Integrable Systems

2009-11-22T18:10:47Z

I'm no expert on this, but since nobody answers:

one book which should definitely help is this big introductory one by Babelon, Bernard and Talon. There's also a very technical older paper by Ben-Zvi and Frenkel where apparently some sort of general construction is made, and there's a more readable paper by Inoue, Vanhaecke and Yamazaki on algebraic complete integrability and integrable hierarchies of PDEs intended for your second question (especially section 6 for the relationship).

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...

Answer by Thomas Sauvaget for How many mathematicians are there?

2009-11-14T09:28:36Z

In an article 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.

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.

Comment by Thomas Sauvaget

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.

Comment by Thomas Sauvaget

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 books.google.fr/…

Comment by Thomas Sauvaget

2012-07-11T13:49:09Z

Chapter 6 of the thesis of van Bargen seems to be relevant in the case M=R^d opus.kobv.de/tuberlin/volltexte/2010/2571/pdf/…

Comment by Thomas Sauvaget

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.

Comment by Thomas Sauvaget

2011-10-22T08:46:03Z

Very concrete indeed, thank you!

Comment by Thomas Sauvaget

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 ?

Comment by Thomas Sauvaget

2011-04-01T20:35:38Z

I've added the "arithmetic-progression" tag (even though you probably don't have only the r=1 case in mind). Perhaps even the "nt.number-theory" one might be appropriate (and would add the benefit of a substantial readership).

Comment by Thomas Sauvaget

2011-02-25T10:16:57Z

@Qiaochu: Sorry the link should be mpip-mainz.mpg.de/theory/events/namet2010/… 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 en.wikipedia.org/wiki/… (and that whole wikipedia page for more details on the corresponding additional conserved quantity).

Comment by Thomas Sauvaget

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 Schrödinger equation combined with some rep theoretical aspects, see <a href="mpip-mainz.mpg.de/theory/events/namet2010/…;.

Comment by Thomas Sauvaget

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 "never appeared", and see how the other authors who did provide the missing details later have decided to present their situation.

Comment by Thomas Sauvaget

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...

Comment by Thomas Sauvaget

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 research.att.com/~njas/sequences/A000001 the longer list being this research.att.com/~njas/sequences/b000001.txt

Comment by Thomas Sauvaget

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 "the scheme of things" from a pure math perspective is not that limited. Probably a better measure would be a notion of "greatly inspired by paper X" (i.e. not just a citation).

Comment by Thomas Sauvaget

2010-07-05T20:19:37Z

(By the way, hello Damien!)

Comment by Thomas Sauvaget

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.)