Girard’s Transcendental Syntax

March 1, 2015

Jean-Yves Girard has recently put online his two papers on Transcendental Syntax.   As of early march 2015, part I is now said to be in final state while part II is still a blueprint (“This part is so new, the notions are so fresh, that can hardly be more than a blueprint; I am not yet familiar with the notions introduced, which may account for the sloppiness of notations. Hence the inclusion of this section in annex. I beg for the leniency of the reader in view of the absolute novelty of the ideas.“)

The aim is apprently a “fully independent approach to logic: no syntax, no semantics, i.e., no prejudice.

As usual with Girard, the papers introduce new terminology that probably requires a fair amount of time to grasp, but at the same time are written in appealing semi-discursive style, with motivations fully apparent.

Cédric Villani’s book tour

February 21, 2015

Cédric Villani, who is running an undergraduate MOOC with Diaraf Seck on evolution equations, is apparently about to tour the english-speaking world, now that his essay has been translated to english and is about to hit the stores (I did enjoy the original french version).

Some people from the book publishing world have started talking about it on twitter (e.g. here or there), while a review in The Times has just appeared (behind a paywall).

I’ve seen that events are planned next march in London, in Bristol, and in Oxford. Also, a BBC radio 4 program is scheduled, and then in april an event in Seattle. And perhaps other things elsewhere that I didn’t spot.

Update: apparently there’s a second event in London. Also, the Guardian now has a review of the book, and so has the New Scientist.

Update 2: the reviews are now abundant so I won’t care to track them all. Just to mention two more,there’s one in The Spectator, and another in THES.  Some like the book while others virulently dislike it.

Launch of the North-Western European Journal of Mathematics

February 20, 2015

As announced in the new issue of Gazette des Mathématiciens, a new (diamond-like) Open Access journal has been launched recently, the North-Western European Journal of Mathematics.

It is technically based in Lille and has support from the French Mathematical Society (SMF), the Dutch Mathematical Society (KWG), the Luxembourg Mathematical Society (SML) and the Fields Institute.

The editor-in-chief Serge Nicaise and technical director Nicolas Wicker say in the Gazette that :

Notre but est de mettre en place un journal académique de grande qualité en accès libre faisant concurrence aux trop nombreuses revues payantes détenues par les grandes maisons d’édition.

[…]

Évidemment, une telle initiative est un pari, et ne pourra être couronnée de succès qu’avec un mouvement d’ensemble de notre communauté vers les revues libres. Notre contribution à ce qui est une bataille contre les éditeurs privés est de joindre nos efforts
avec nos voisins du nord afin de faire déborder la lutte hors de la France, qui pour l’instant fait figure de pionnière.

Rough translation :

Our goal is to set up an academic open access journal of great quality to compete against the too vast number of non-free journals detained by the big publishing houses.

[…]

Of course, such an initiave is a bet, and can only become a success with a common movement of our community towards open access journals. Our contribution to what is a battle against the private publishers is to join our efforts with our northern neighbours to make this struggle develop beyond the boundaries of France, which for the moment acts as a pioneer.

So, that’s another journal added to the list of open access ones.

Embedding a Turing Machine

February 17, 2015

The idea to embed a Turing Machine in famous problems seems to have popped up a lot in recent years, some examples I’ve noticed :

That last paper does mention Bjorn Poonen’s very interesting essay on the topic of undecidability.

Recent efforts to understand Mochizuki’s work

February 15, 2015

As pointed out by Thomas Riepe on Mathoverflow, yesterday Ivan Fesenko released overview notes on Shinichi Mochizuki’s work, with lots of efforts to relate IUT to earlier “mainstream” knowledge. And several workshops are planned in Europe and Japan in the coming years.  To quote Fesenko:

The mathematical vision and perseverance of the author of IUT during 20 years of work on it is admirable.

A valuable addition to this is his investment of time and effort in answering questions about his work and explaining and discussing its parts, via email communication or skype talks and during numerous meetings and seminars at RIMS.

Given the number of upvotes for Minhyong Kim’s two recent open-ended questions What is a Frobenioid? and What is an étale theta function? it seems as though there’s definitely momentum building up.

Auctions of mathematical texts

February 12, 2015

On saturday in Royan there will be on offer (for about €300/350) a handwritten set of lecture notes of Cauchy’s 1824 course on Calcul différentiel et intégral, taken by Lamoricière.  It may well beat that estimate by some margin.

Indeed, last december, the own annotated copy of Hugens’ Horologium Oscillatorium Sive de Motu Pendulorum ad Horologia Aptato Demonstrationes Geometricae (1673), went for \$965,000 after an estimate of 150,000/200,000.

A month earlier, a 1931 offprint of Gödel’s Über Formal Unentscheidbare Sätze der Principia Mathematica und verwandlter Systeme I, achieved £104,500 after an estimate of £12,000/£16,000.

So, there seems to be a market for the rare and historically important items, that’s not too surprising but the level is pretty high (I guess all it takes is just a couple of mathematically enclined high-ranking google executives, or the likes…). And it’s not only auctions, fixed price items exist too, like these 3 manuscript pages by Gauss available for €65,000.

Update: the aforementioned sale has been broadcast online earlier today, and those handwritten notes indeed attracted interest, with a little fight between the floor and online bids ending at €1,300.

A (probably) useless reformulation of the twin prime conjecture

February 8, 2015

In light of the previous post the following is obviously a long stretch, but I can’t resist mentioning it.

The way Jones-Sato-Wada-Wiens construct their polynomial is to start from the characterization of primes by Wilson’s theorem (namely that $k+1$ is prime iff $k+1$ divides $k!+1$), to convert that into a system of exponential equations, and this is itself converted into a larger system of polynomial equations, which are finally assembled into one big polynomial.

Now twin primes can be characterized also by a Wilson-like statement, namely $k+1, k+3$ are twin primes iff $(k+1)(k+3)$ divides $4(k!+1)+k+1$.

So one could construct a polynomial in several integer variables $P_{2}$ along the same lines, and end up with the (difficult to exploit) reformulation of the twin prime conjecture : there are infinitely many twin primes iff $P_{2}$ takes on arbitrarily high positive values on $\mathbb{N}^m$ (where $m$ is the number of variables). Said differently, a disproof of the conjecture would follow from an absolute bound.

By Polymath8b, a similar polynomial $P_{246}$ for the case of pairs $(n, n+246)$ is known to indeed take arbitrarily high positive values, while of course the same is true for the original Jones-Sato-Wada-Wiens polynomial since there are infinitely many primes.  So those two “sandwich” $P_2$ somehow, and there just might be hope to say educated things about that (which I obviously can’t, but the learned reader is encouraged to do so, should that seem to be a good idea).

Update: for sake of completeness, I should have mentioned that the Wilson-like characterization of twin primes goes back to a paper of Clement in the Montly from 1949, and the analog for pairs of primes $n, n+2k$ has been worked out by Lee and Park to be that $n(n+2k)$ divides $2k(2k)!((n-1)!+1)+((2k)!-1)n$.

What do solutions of the Jones-Sato-Wada-Wiens polynomial look like?

February 1, 2015

As soon as one comes across the story of the set of primes being the positive values taken by the Jones-Sato-Wada-Wiens polynomial of degree 25 in 26 variables (see also the senior thesis of Cabusora, or, for french-speaking readers, this RMS paper by Collet and Vidiani), one wants to start playing with it, and see some solutions.

But the aforementioned sources don’t mention any, and neither do more recent books like Denef’s on Hilbert’s tenth problem, the one by de Koninck and Luca on analytic number theory, the one by Crandall and Pomerance on a computational perspective on primes, the one by Everest and Ward on introductory number theory.

Fortunately, there is an Msc thesis by Gupta which does take the trouble to find the values of all 26 variables for the prime 2, and it is informative. Using his notations, these values are:

$k=0$, $g=0$, $f=17$, $n=2$, $p=3$, $q=16$, $z=9$, $w=1$, $h=2$, $j=5$, $e=32$ (parameter of a Pell equation), $a=7901690358098896161685556879749949186326380713409290912$, $o=8340353015645794683299462704812268882126086134656108363777$ (these two being the smallest solution of that Pell equation), $y=2a$, $x=2a^2-1$, $m=a$, $\ell=1$, $i=0$, $v=2a-3$, $b=0$$s=1$$t=0$, $r\simeq 10^{10^{52}}$, the final $u, c, d$ being larger still.

The basic idea is that the built-in Pell equations inside the polynomial quickly lead to very large values indeed. No hope to play with 26 small integers, say less than $10^6$.

The excellent Audiovisual Mathematics Library of CIRM

January 15, 2015

Last october, the CIRM (which has a gorgeous location in the Calanques) has launched its very nice Audiovisual Mathematics Library, featuring HD videos (many being multi-angle, or post-edited — it does lag quite a bit on my connection, but maybe it’s better with a big university broadband network). So I’ve now added that to my previous list.

It also has a youtube channel, with an especially appealing presentation video:

On a certain 4-chromatic planar graph

January 3, 2015

The Hadwiger-Nelson problem is notoriously difficult. (For context see e.g. this text by Michael Payne.) In fact, it seems that the (weaker?) problem to find a 5-chromatic unit distance graph in the plane is still open (see the interesting 2009 Mathematical Coloring Book by Soifer, I couldn’t find more recent improvements online).

Now recall that the Moser Spindle is a 4-chromatic unit distance graph in the plane, and from this set of open problems we have the coordinates of its vertices:

So the idea would be to build on top of that.  A first step would be to find a 4-chromatic unit distance graph that has necessarily at least two vertices of the 4th color. And then build further on that one.

So I’ve done some trial and error, and found something that is agonisingly close, namely the following graph on 16 vertices has the correct topology (i.e. a Sage computation gives its minimal set of colored vertices as [[13, 6, 8, 1], [11, 10, 15, 5, 2], [4, 3, 14, 12, 9], [7, 0]], so indeed two vertices of the 4th color), but unfortunately it doesn’t quite have distances 1 everywhere.

As it happens, if we enforce unit distances at 6~14 and 10~14 then the abscissa of vertex #14 is not $0.5$ (which would make things work), but instead

$-\frac{895702085}{177666094271969928} \sqrt{514527538436665}+ \frac{415413875483399597}{676176985277553384}$ $\approx 0.49999956608$  (and same problem with vertex #15).

Edit: Some further tinkering didn’t get anywhere. Removing for simplicity vertices #14 and #15 we do get a 4-chromatic unit distance graph such that at least two vertices have the 4th color.

But I then tried to pack several copies together to increase that number, in the hope that eventually some 5th color must be needed, but it is highly nontrivial: gluing two copies (i.e. some graph that contains 4 spindles) does produce four vertices of the 4th color, but gluing a third that has at least one common vertex with both first copies does not necessarily lead to strictly more than 6 vertices of the 4th color, in my example I still get 6 with the partition [[36, 31, 25, 19, 13, 6, 33, 27, 21, 15, 8, 1], [35, 17, 12, 10, 4, 34, 30, 24, 2], [29, 23, 11, 3, 28, 22, 18, 16, 9, 5], [32, 26, 20, 14, 7, 0]].

Here’s the quick Sage code, in case someone wants to test ideas: first the code of the initial idea with vertices #14 and #15:

from sage.graphs.graph_coloring import chromatic_number
from sage.graphs.graph_coloring import vertex_coloring
from sage.graphs.graph_plot import GraphPlot
from math import sqrt

# hack the Moser spindle

d_3={
0:[1,3,2],
1:[0,4,5],
2:[0,3,6],
3:[0,2,6],
4:[1,5,6],
5:[1,4,6],
6:[2,3,4,5],
7:[8,9,10],
8:[7,11,12],
9:[7,10,13],
10:[7,9,13,3],
11:[8,12,13,4],
12:[8,11,13],
13:[9,10,11,12],
14:[6,10,11],
15:[3,4,13]
}

N=Graph(d_3)

# Now let's compute its chromatic number
N.chromatic_number()

# So N is 4-chromatic like the Moser spindle,
# but let's see if it is better in terms of the number of vertices of color 4 (the Spindle has just one)
# by computing its partition into minimal sets of different colors

vertex_coloring(N, value_only=False, k=None)  # or we could similarly do: vertex_coloring(N, value_only=None, k=4)

# So this topology is better in the sense that it must have at least two vertices of color 4

#Let's see now the metric aspect, i.e. whether or not all the length of the edges can be equal 1
vertices_pos_3={}
vertices_pos_3[0]=[0,0]
vertices_pos_3[1]=[1,0]
vertices_pos_3[2]=[(3-sqrt(33))/12,(3*sqrt(11)+sqrt(3))/12]
vertices_pos_3[3]=[(3+sqrt(33))/12,(3*sqrt(11)-sqrt(3))/12]
vertices_pos_3[4]=[(9-sqrt(33))/12,(3*sqrt(11)-sqrt(3))/12]
vertices_pos_3[5]=[(9+sqrt(33))/12,(3*sqrt(11)+sqrt(3))/12]
vertices_pos_3[6]=[0.5,sqrt(11)/2]
vertices_pos_3[7]=[0,(3*sqrt(11)-sqrt(3)-6)/6]
vertices_pos_3[8]=[1,(3*sqrt(11)-sqrt(3)-6)/6]
vertices_pos_3[9]=[(3-sqrt(33))/12,2*((3*sqrt(11)-sqrt(3)-6)/12)-(3*sqrt(11)+sqrt(3))/12]
vertices_pos_3[10]=[(3+sqrt(33))/12,(3*sqrt(11)-sqrt(3)-12)/12]
vertices_pos_3[11]=[(9-sqrt(33))/12,(3*sqrt(11)-sqrt(3)-12)/12]
vertices_pos_3[12]=[(9+sqrt(33))/12,2*((3*sqrt(11)-sqrt(3)-6)/12)-(3*sqrt(11)+sqrt(3))/12]
vertices_pos_3[13]=[0.5,2*((3*sqrt(11)-sqrt(3)-6)/12)-sqrt(11)/2]
vertices_pos_3[14]=[-(895702085/177666094271969928)*sqrt(514527538436665) + (415413875483399597/676176985277553384),
-(19/32519092464)*sqrt(178940893)*sqrt(1427)*sqrt(31)*sqrt(13)*sqrt(5) + (15303925/22788432)]
vertices_pos_3[15]=[-(895702085/177666094271969928)*sqrt(514527538436665) + (415413875483399597/676176985277553384),
2*((3*sqrt(11)-sqrt(3)-6)/12)
+(19/32519092464)*sqrt(178940893)*sqrt(1427)*sqrt(31)*sqrt(13)*sqrt(5) - (15303925/22788432)]

# Explanations:
# 10 is simply a shift of 3 by -1 along y-axis, same with 11 from 4 (i.e. those two edges 10~3 and 11~4 have length 1 by design)

# coordinates of 14 come from the fact that it is located at the intersection of two circles of radius 1 centered at 10 and 6
# so we get them from (we pick the first solution, with lowest y):
#x, y = var('x, y')
#solve([(x-0.5)^2+(y-sqrt(11)/2)^2==1, (x-(3+sqrt(33))/12)^2+(y-(3*sqrt(11)-sqrt(3)-12)/12)^2==1 ], x, y)

# finally the axis of up-down symmetry is y=(3*sqrt(11)-sqrt(3)-6)/12 from which we deduce coordinates of
# 7 from 0, 8 from 1, 9 from 2, 12 from 5, 15 from 14

# So the excruciating problem is that 14 and 15 don't have x=0.5 but only barely so,
# i.e. that topology just barely cannot have all unit distances...

pl = N.graphplot(pos=vertices_pos_3)
pl.show()


And now the topology of the three copies:

from sage.graphs.graph_coloring import chromatic_number
from sage.graphs.graph_coloring import vertex_coloring
from sage.graphs.graph_plot import GraphPlot
from math import sqrt

# hack the Moser spindle

d_3={
0:[1,3,2],
1:[0,4,5],
2:[0,3,6],
3:[0,2,6],
4:[1,5,6,14,16,19],
5:[1,4,6],
6:[2,3,4,5],
7:[8,9,10],
8:[7,11,12],
9:[7,10,13],
10:[7,9,13,3],
11:[8,12,13,4],
12:[8,11,13,20,22,25],
13:[9,10,11,12],
14:[15,16,4],
15:[14,17,18],
16:[14,4,19],
17:[15,18,19],
18:[15,17,19],
19:[4,16,17,18],
20:[21,22,12],
21:[20,23,24],
22:[20,12,25],
23:[21,24,25,17],
24:[21,23,25],
25:[22,12,23,24],
26:[27,4,28],
27:[26,29,30],
28:[26,4,31],
29:[27,30,31],
30:[27,29,31],
31:[28,4,29,30],
32:[33,34,11],
33:[32,35,23],
34:[32,11,36],
35:[33,23,36,29],
36:[34,11,35,23],
}

N=Graph(d_3)

# let's compute its chromatic number
N.chromatic_number()

# let's compute its partition into minimal sets of different colors
vertex_coloring(N, value_only=False, k=None)  # or we could similarly do: vertex_coloring(N, value_only=None, k=4)