Carry The Two

https://ift.tt/2IW3KiP I was running a bachelor course on representations of finite groups and a master course on simple (mainly sporadic) groups until Corona closed us down. Perhaps these blog-posts can be useful to some. A curious fact, with ripple effect on Mathieu sporadic groups, is that the symmetric group $S_6$ has an automorphism $\phi$, different from an automorphism by conjugation. In the course notes the standard approach was given, based on the $5$-Sylow subgroups of $S_5$. Here’s the idea. Let $S_6$ act by permuting $6$ elements and consider the subgroup $S_5$ fixing say $6$. If such an odd automorphism $\phi$ would exist, then the subgroup $\phi(S_5)$ cannot fix one of the six elements (for then it would be conjugated to $S_5$), so it must act transitively on the six elements. The alternating group $A_5$ is the rotation symmetry group of the icosahedron Any $5$-Sylow subgroup of $A_5$ is the cyclic group $C_5$ generated by a rotation among one of the six body-dia...

https://ift.tt/2YZBu9I If you Googled this number a week ago, all you’d get were links to the paper by Melanie Wood Belyi-extending maps and the Galois action on dessins d’enfants . In this paper she says she can separate two dessins d’enfants (which couldn’t be separated by other Galois invariants) via the order of the monodromy group of the inflated dessins by a certain degree six Belyi-extender. She gets for the inflated $\Delta$ the order 19752284160000 and for inflated $\Omega$ the order 214066877211724763979841536000000000000 (see also this post ). The paper was arXived in 2003, and the Google-test shows that nobody questioned these findings since then. After that post I redid the computations a number of times (as well as for other Belyi-extenders) and always find that these orders are the same for both dessins. And, surprisingly, each time the same numbers keep pupping up. For example, if you take the Belyi-extender $t^6$ (power-map) then it is pretty easy to work out...

https://ift.tt/307peAi On holiday, and re-reading two ‘metabiographies’: Philippe Douroux : Alexandre Grothendieck : Sur les traces du dernier génie des mathématiques and Siobhan Roberts : Genius At Play: The Curious Mind of John Horton Conway . Siobhan Roberts’ book is absolutely brilliant! I’m reading it for the n-th time, first on Kindle, then hardcopy, and now I’m just flicking pages, whenever I need to put a smile on my face. So, here’s today’s gem of a Conway quote (on page 150): Pure mathematicians usually don’t found companies and deal with the world in an aggressive way. We sit in our ivory towers and think. Though Conway complains his words were taken out of context (in an article featuring Stephen Wolfram), he clearly means each one of them. If only university administrations worldwide would take the ‘sitting in an ivory tower and think’-bit as the mission statement, and evaluation criterium, for their pure mathematicians. But then… they obviously prefer man...

https://ift.tt/2Z4Rs1w A Belyi-extender (or dessinflateur ) $\beta$ of degree $d$ is a quotient of two polynomials with rational coefficients \[ \beta(t) = \frac{f(t)}{g(t)} \] with the special properties that for each complex number $c$ the polynomial equation of degree $d$ in $t$ \[ f(t)-c g(t)=0 \] has $d$ distinct solutions, except perhaps for $c=0$ or $c=1$, and, in addition, we have that \[ \beta(0),\beta(1),\beta(\infty) \in \{ 0,1,\infty \} \] Let’s take for instance the power maps $\beta_n(t)=t^n$. For every $c$ the degree $n$ polynomial $t^n – c = 0$ has exactly $n$ distinct solutions, except for $c=0$, when there is just one. And, clearly we have that $0^n=0$, $1^n=1$ and $\infty^n=\infty$. So, $\beta_n$ is a Belyi-extender of degree $n$. A cute observation being that if $\beta$ is a Belyi-extender of degree $d$, and $\beta’$ is an extender of degree $d’$, then $\beta \circ \beta’$ is again a Belyi-extender, this time of degree $d.d’$. That is, Belyi-extenders ...

https://ift.tt/2YGVuxJ I’m trying to get into the latest Manin-Marcolli paper Quantum Statistical Mechanics of the Absolute Galois Group on how to create from Grothendieck’s dessins d’enfant a quantum system, generalising the Bost-Connes system to the non-Abelian part of the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. In doing so they want to extend the action of the multiplicative monoid $\mathbb{N}_{\times}$ by power maps on the roots of unity to the action of a larger monoid on all dessins d’enfants. Here they use an idea, originally due to Jordan Ellenberg, worked out by Melanie Wood in her paper Belyi-extending maps and the Galois action on dessins d’enfants . To grasp this, it’s best to remember what dessins have to do with Belyi maps, which are maps defined over $\overline{\mathbb{Q}}$ \[ \pi : \Sigma \rightarrow \mathbb{P}^1 \] from a Riemann surface $\Sigma$ to the complex projective line (aka the 2-sphere), ramified only in $0,1$ and $\infty$....

https://ift.tt/2YGjobN A long while ago I promised to take you from the action by the modular group $\Gamma=PSL_2(\mathbb{Z})$ on the lattices at hyperdistance $n$ from the standard orthogonal laatice $L_1$ to the corresponding ‘monstrous’ Grothendieck dessin d’enfant. Speaking of dessins d’enfant, let me point you to the latest intriguing paper by Yuri I. Manin and Matilde Marcolli, ArXived a few days ago Quantum Statistical Mechanics of the Absolute Galois Group , on how to build a quantum system for the absolute Galois group from dessins d’enfant (more on this, I promise, later). Where were we? We’ve seen natural one-to-one correspondences between (a) points on the projective line over $\mathbb{Z}/n\mathbb{Z}$, (b) lattices at hyperdistance $n$ from $L_1$, and (c) coset classes of the congruence subgroup $\Gamma_0(n)$ in $\Gamma$. How to get from there to a dessin d’enfant ? The short answer is: it’s all in Ravi S. Kulkarni’s paper, “An arithmetic-geometric method in the stu...

https://ift.tt/2JMoLxU Let’s try to identify the $\Psi(n) = n \prod_{p|n}(1+\frac{1}{p})$ points of $\mathbb{P}^1(\mathbb{Z}/n \mathbb{Z})$ with the lattices $L_{M \frac{g}{h}}$ at hyperdistance $n$ from the standard lattice $L_1$ in Conway’s big picture . Here are all $24=\Psi(12)$ lattices at hyperdistance $12$ from $L_1$ (the boundary lattices): You can also see the $4 = \Psi(3)$ lattices at hyperdistance $3$ (those connected to $1$ with a red arrow) as well as the intermediate $12 = \Psi(6)$ lattices at hyperdistance $6$. The vertices of Conway’s Big Picture are the projective classes of integral sublattices of the standard lattice $\mathbb{Z}^2=\mathbb{Z} e_1 \oplus \mathbb{Z} e_2$. Let’s say our sublattice is generated by the integral vectors $v=(v_1,v_2)$ and $w=(w_1.w_2)$. How do we determine its class $L_{M,\frac{g}{h}}$ where $M \in \mathbb{Q}_+$ is a strictly positive rational number and $0 \leq \frac{g}{h} < 1$? Here’s an example: the sublattice (the thick dots)...

https://ift.tt/32JKkXa Dedekind’s Psi-function $\Psi(n)= n \prod_{p |n}(1 + \frac{1}{p})$ pops up in a number of topics: $\Psi(n)$ is the index of the congruence subgroup $\Gamma_0(n)$ in the modular group $\Gamma=PSL_2(\mathbb{Z})$, $\Psi(n)$ is the number of points in the projective line $\mathbb{P}^1(\mathbb{Z}/n\mathbb{Z})$, $\Psi(n)$ is the number of classes of $2$-dimensional lattices $L_{M \frac{g}{h}}$ at hyperdistance $n$ in Conway’s big picture from the standard lattice $L_1$, $\Psi(n)$ is the number of admissible maximal commuting sets of operators in the Pauli group of a single qudit. The first and third interpretation have obvious connections with Monstrous Moonshine . Conway’s big picture originated from the desire to better understand the Moonshine groups , and Ogg’s Jack Daniels problem asks for a conceptual interpretation of the fact that the prime numbers such that $\Gamma_0(p)^+$ is a genus zero group are exactly the prime divisors of the order of the Mo...

https://ift.tt/2JDJMuy Fortunately, there are a few certainties left in life: In spring, you might expect the next instalment of Connes’ and Consani’s quest for Gabriel’s topos . Here’s the latest: $\overline{\mathbf{Spec}(\mathbb{Z})}$ and the Gromov norm . Every six months or so, Mochizuki’s circle-of-friends tries to create some buzz announcing the next IUTeich-workshop. I’ll spare you the link, if you are still interested in all of that, follow math_jin or IUTT_bot_math_jin on Twitter. And then, there’s the never-ending story of Grothendieck’s griboullis , kept alive by the French journalist and author Philippe Douroux. Here are some recent links: Alexandre Grothendieck : une mathématique en cathédrale gothique , an article (in French) by Philippe Douroux in Le Monde, May 6th (behind paywall). L’histoire étonnante des archives du mathématicien Alexandre Grothendieck , an article (in French) on France Inter by Mathieu Vidar, based on info from Philippe Douroux. Les archi...

https://ift.tt/2NUC5Ew Last time we revisited Robin’s theorem saying that 5040 being the largest counterexample to the bound \[ \frac{\sigma(n)}{n~log(log(n))} < e^{\gamma} = 1.78107... \] is equivalent to the Riemann hypothesis. There’s an industry of similar results using other arithmetic functions. Today, we’ll focus on Dedekind’s Psi function \[ \Psi(n) = n \prod_{p | n}(1 + \frac{1}{p}) \] where $p$ runs over the prime divisors of $n$. It is series A001615 in the online encyclopedia of integer sequences and it starts off with 1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, 14, 24, 24, 24, 18, 36, 20, 36, 32, 36, 24, 48, 30, 42, 36, 48, 30, 72, 32, 48, 48, 54, 48, … and here’s a plot of its first 1000 values To understand this behaviour it is best to focus on the ‘slopes’ $\frac{\Psi(n)}{n}=\prod_{p|n}(1+\frac{1}{p})$. So, the red dots of minimal ‘slope’ $\approx 1$ correspond to the prime numbers, and the ‘outliers’ have a maximal number of distinct small prime divisor...

https://ift.tt/2L9IyJs Yesterday, there was an interesting post by John Baez at the n-category cafe: The Riemann Hypothesis Says 5040 is the Last . The 5040 in the title refers to the largest known counterexample to a bound for the sum-of-divisors function \[ \sigma(n) = \sum_{d | n} d = n \sum_{d | n} \frac{1}{n} \] In 1983, the french mathematician Guy Robin proved that the Riemann hypothesis is equivalent to \[ \frac{\sigma(n)}{n~log(log(n))} < e^{\gamma} = 1.78107... \] when $n > 5040$. The other known counterexamples to this bound are the numbers 3,4,5,6,8,9,10,12,16,18,20,24,30,36,48,60,72,84,120,180,240,360,720,840,2520. In Baez’ post there is a nice graph of this function made by Nicolas Tessore, with 5040 indicated with a grey line towards the right and the other counterexamples jumping over the bound 1.78107… Robin’s theorem has a remarkable history, starting in 1915 with good old Ramanujan writing a part of this thesis on “highly composite numbers” (nu...

http://bit.ly/2HUlE81 In April my Google+ account will disappear. Here I collect some G+ posts, in chronological order, having a common theme. Today, math-history (jokes and puns included). September 20th, 2011 Was looking up pictures of mathematicians from the past and couldn’t help thinking ‘Hey, I’ve seen this face before…’ Leopold Kronecker = DSK Adolf Hurwitz = Groucho Marx June 2nd, 2012 The ‘Noether boys’ (Noether-Knaben in German) were the group of (then) young algebra students around Emmy Noether in the early 1930’s. Actually two of them were girls (Grete Hermann and Olga Taussky). The picture is taken from a talk Peter Roquette gave in Heidelberg. Slides of this talk are now available from his website . In 1931 Jacques Herbrand (one of the ‘Noether boys’) fell to his death while mountain-climbing in the Massif des Écrins (France). He was just 23, but already considered one of the greatest minds of his generation. He introduced the notion of recursive functions ...