305 – A potpurri of groups

February 27, 2012

Here are a few examples of groups and links illustrating some of them. I will be adding to this list; if you find additional links that may be useful or interesting, please let me know. A nice general place to look at is the page for the book “Visual group theory.”

  • S_n, A_n, the symmetric and alternating groups in n letters.
  • Abelian groups, such as ({\mathbb Z}/n{\mathbb Z},+), (({\mathbb Z}/n{\mathbb Z})^*,\cdot),{\mathbb Z},{\mathbb Q}.
  • Dihedral groups. Here is a page by Erin Carmody illustrating the symmetries of the square. The Wikipedia page on dihedral groups has additional illustrations and interesting examples.
  • Braid groups. Patrick Dehornoy has done extensive research on braid groups, and his page has many useful surveys and papers on the topic. Again, the Wikipedia page is a useful introduction. The applet we saw in class is here.
  • Matrix groups. For example, GL_n({\mathbb R}), the group of all invertible n\times n matrices with real entries, or SL_n({\mathbb R}), the group of all n\times n matrices with real entries and determinant 1.
  • The plane symmetry (or Wallpaper) groups.
  • Coxeter groups.
  • Crystallographic groups.
  • Any group is (isomorphic to) a group of permutations, but the groups corresponding to permutation puzzles are naturally described this way. For example, Dana Ernst recently gave a talk on this topic.

305 – The Futurama theorem

February 24, 2012

Here is a link to Dana Ernst talk on the Futurama theorem of Ken Keeler. In this version, products are just as we treat them (left to right). Also, in this version, there is the “try it yourself” exercise we did not do, but you may want to practice a few examples on your own anyway to make sure you understand the argument.

Finally, the questions mentioned at the end of Ernst’s talk (see here) are all interesting, and if you figure one or several of them out, please turn them in for extra credit.

(By the way, the same applies to all problems we have been discussing. For example, if through the term you figure a way of producing a really long sequence giving us a better bound for n(3) than what you obtained when the homework was due, please turn it in as well.)

305 – Campanology

February 24, 2012

Campanology, or bell ringing, is an English tradition, where a round of cathedral bells is rung by permuting their order. The book discusses some examples of the possible patterns used in practice (the Plain Lead on four bells, and the Plain Bob Minimus). Additional examples can be found in the Wikipedia link, and links are provided there to a few additional sites, such as bellringing.org.

I strongly recommend that you read Section 3.5 in the book dealing with this topic. It introduces through an example the useful notion of cosets, and also it is quite interesting. For example, it shows how several ideas from group theory were used in practice since the seventeenth century, predating the introduction of the concepts by Galois and his contemporaries, and in a completely different setting.

I did not know about this until I read the book, and of course now I see mentions of bell-ringing everywhere.

The quote above, for example, is from the book Combinatorial Set Theory by Lorenz Halbeisen.

A nice article on the topic (in French) can be found here; coincidentally, the article also talks at the end about juggling. The video of the talk on the mathematics of juggling by Allen Knutson that we saw an excerpt of today is here. A few technical and expository articles on this topic, by renown mathematician (and juggler) Ronald Graham, and his coauthors, can be found in Graham’s page. See for example here, here, here, and here.


February 19, 2012

All blackboards are gone. They were replaced during the break. (So, no chance to use this MathOverflow question on the near future.) Makes me think of T.H. Huxley’s On a Piece of Chalk (see here) and wonder what the equivalent will be in a few decades.

On the plus side, we now have computers and projection equipment on each classroom. I am using this quite a bit in my abstract algebra class. Except that, during the first few weeks, it was more often than not that the keyboard would be locked away.

Annoyed, I called OIT and asked that they please make sure it was unlocked before my class. It worked for a few days. But then, again, I found it locked.

I called again (I was charming, I am sure). So, somebody came to the classroom, looked at me, smiled. And pressed a button, to open the drawer.


(In my defense, the class is at 8:30 in the morning, and I’m supposed to drink less coffee these days. But still.)

(“Thanks. I’m sorry. I’m an idiot.” “Oh, no, no. It is new.” “It is a button.”)

305 – Homework III

February 17, 2012

This homework set is due Wednesday, February 29th, at the beginning of lecture, but feel free to turn it in earlier if possible. Recall that “show” means “prove”.

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305 – A lower bound for n(3)

February 14, 2012

This continues the previous post on Friedman’s theorem.

I want to give a brief idea of the size of Friedman’s n(3), as defined on homework 2. The reference for what follows is

  • Harvey Friedman, Long finite sequences,  J. Combin. Theory Ser. A, 95 (1) (2001), 102–144.

Recall that n(3) is the largest possible size n of a sequence x_1 x_2 x_3\dots x_{n-1} x_n with property *, all of whose terms are 1, 2, or 3. Here, to have property * means that if i<j\le n/2, then the sequence x_i x_{i+1}\dots x_{2i} is not a subsequence of the sequence x_j x_{j+1}\dots x_{2j}.

In lecture we gave an example of such a sequence, of length 216, namely


where we use exponential notation to indicate the number of times a symbol appears consecutively, so the sequence begins


Actually, n(3) is much larger than 216. We know it is finite but, as Friedman mentions in his paper, it is an incomprehensibly large number. To explain this, we should mention the Ackermann hierarchy of functions. These are functions A_1,A_2,\dots, with each A_i:{\mathbb Z}^+\to{\mathbb Z}^+ a strictly increasing function.

Define A_1(n)=2n. Also, set


where there are n occurrences of A_k.

This is perhaps best understood by computing a few examples. We have A_1(1)=2, A_1(A_1(1))=A_1(2)=4=2^2, A_1(A_1(A_1(1)))=A_1(4)=8=2^3 and, by an easy induction, A_2(n)=2^n.

Now, A_2(1)=2, A_2(A_2(1))=A_2(2)=2^2, A_2(A_2(A_2(1)))=A_2(2^2)=2^{2^2} and, in general, A_3(n) is an exponential tower of 2s of height n.

The descriptions become more complicated now. For example, A_4(n) is an exponential tower of 2s, but its height is itself an exponential tower of 2s (of height n). For example,

\displaystyle A_4(4)=2^{\vdots ^2},

where the tower has height \displaystyle 2^{2^{2^2}}=65536.

Let me now quote from Friedman’s paper:

I submit that A_4 ( 4) is a ridiculously large number, but it is not an incomprehensibly large number. One can imagine a tower of 2’s of a large height, where that height is 65,536, and 65,536 is not ridiculously large.

However, if we go much further, then a profound level of incomprehensibility emerges. The definitions are not incomprehensible, but the largeness is incomprehensible. These higher levels of largeness blur, where one is unable to sense one level of largeness from another.

For instance, A_4( 5 ) is an exponential tower of 2’s of height A_4( 4 ).

It seems safe to assert that, say, A_5( 5 ) is incomprehensibly large. We propose this number as a sort of benchmark. In Section 4 we prove that n( 3 ) >A_7( 184 ), which is considerably larger.

Friedman actually goes much further than this. Using computer data provided by Randall Dougherty, Friedman proves in the last section of the paper that


For all I know, this could still be very far from the actual value of n(3). And I do not know a reasonable description of an upper bound.

In technical terms, Friedman proves that the function n(k) is not primitive recursive, and actually grows much faster than the usual examples of recursive functions that are not primitive recursive. He also describes a function H (a member of the so-called Hardy hierarchy) that grows faster than n(k). This means that n(k)<H(k) for all sufficiently large k, but the argument does not give a specific bound when k=3.

515 – Homework 2

February 14, 2012

This set is due Feb. 29 at the beginning of lecture. Let me know if more time is needed or anything like that. Problem 4 was incorrect as stated; I have fixed it now. Thanks to Tara Sheehan for bringing the problem to my attention.

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515 – Plotting Féjer’s kernels

February 6, 2012

Féjer’s kernels are the functions that play the role of “approximations to the Dirac delta” in the computations we will use to obtain Weierstrass approximation theorem. The nth approximation is given by

\displaystyle K_n(s)= \frac1{n+1}\left(\frac{\sin\frac{(n+1)s}2}{\sin\frac s2}\right)^2 for s\ne0, K_n(0)=n+1.

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515 – The fundamental theorem of calculus

February 6, 2012

Suppose that f\in{\mathcal R}[a,b] and f has an antiderivative G. Then

\displaystyle G(b)-G(a)=\int_a^b f(t)dt.

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Secret history

February 5, 2012

One of the deepest results to emerge from Hall’s school was […] This was obtained by Hall’s student Peter Stroud, who was tragically killed in an accident shortly after completing his doctorate; as a result, the proof was never published, and remains quite difficult to get at (Stroud’s thesis may be consulted in the Cambridge University Library, but photocopying is not allowed).

Dan Segal, Words. Notes on verbal width in groups, Cambridge University Press, 2009.