305 – Projects

January 26, 2012

[Edited, April 3, 2012.]

As mentioned on the syllabus,

I expect groups of two or three per project. The deadline for submission is the scheduled time of the final exam. This will constitute 20% or your total grade. Your final project must be typeset; I encourage you to consult with me through the semester in terms of how it looks and its contents.

What I expect is a paper where you explain the topic, and present its history and a few results on it with complete proofs. Work out a few examples. If relevant, do some numerical simulations. List all the references you consult. (Of course, do not plagiarize.) Some of the topics may end up being too ambitious, and if that occurs, let me know. In that case, it would be better to restrict your presentation (to some aspects of the topic at hand) rather than trying to be comprehensive.

I’ll give you a list of references you may find useful once you pick your topic, but of course if you find additional references, use those instead.


  1. The Banach-Tarski paradox. Chosen by two groups:
    • K. Williams.
    • J. Giudicelli, Ch. Kelly, and J. Kunz.
  2. The 17 plane symmetry groups. Chosen by two groups:
    • S. Burns, C. Fletcher, and A. Zell.
    • A. Nelson, H. Newman, and M. Shipley.
  3. Quaternions and Octonions. Chosen by:
    • K. Mcallister.
  4. The Gordon game (See 5.5.2 on the book.) Chosen by:
    • J. Thompson
  5. The 2\times2\times2 Rubik cube. Chosen by:
    • M. Mesenbrink, and N. Stevenson.

515 – Homework 1

January 25, 2012

This set is due Feb. 8 at the beginning of lecture. Of course, let me know if more time is needed or anything like that.

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A letter

January 24, 2012

I don’t know how I didn’t know about this.


The above is the letter presenting the resignation of the editorial board of Topology, an Elsevier journal. The journal has been discontinued as of this year.

[…] As you are well aware, the Editors have been concerned about the price of Topology since Elsevier gained control of the journal in 1994. […] The journal Topology has an illustrious history with which we, on becoming editors, were extremely proud to be associated. […] However, we feel that Elsevier’s policies towards the publication of mathematics research have undermined this legacy.

Therefore, with great reluctance and sadness, we have made the difficult decision to resign. […]

On Google+, David Roberts gave a link to the journal’s site, with some highlights: As you can see here, the last published issue (vol. 48, 2-4) was June-December 2009. The previous issue was 40 pages and consisted of 2 papers (that you can purchase access to, at $31.50 each. Plus tax.) And there is also a supplement, published on December 2011. Only $31.50 (plus tax) for a 4 page correction.

For some context, see this blog post by Tim Gowers and this page.

305 – Homework I

January 23, 2012

This homework set is due Wednesday, February 1st, at the beginning of lecture, but feel free to turn it in earlier if possible.

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David Hilbert

January 23, 2012

Today is the 150th anniversary of Hilbert’s birthday.

Hilbert is one of the mathematicians I admire the most. I believe I first learned about him while in High School, through “Higher Geometry” by N.V. Efimov. My copy of Reid’s “Hilbert” is one of the first books I bought when I arrived in the States (in 1997).

Here is a link to the reprint in the Bulletin of the AMS of Mathematical Problems.

I would love to have a poster sized copy of the picture above. Springer sold them years ago, but now they seem impossible to find.

515 – The Dehn-Sydler theorem

January 23, 2012

As mentioned in lecture, Hilbert’s third problem was an attempt to understand whether the Bolyai-Gerwien theorem could generalize to {\mathbb R}^3:

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305 – Solving cubic and quartic equations

January 23, 2012

Ars Magna, “The Great Art”, by Gerolamo Cardano.

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414/514 – The Schoenberg functions

January 23, 2012

Here is Jeremy Ryder’s project from last term, on the Schoenberg functions. Here we have a space-filling continuous map f:x\mapsto(\phi_s(x),\psi_s(x)) whose coordinate functions \phi_s and \psi_s are nowhere differentiable.

The proof that \phi_s,\psi_s are continuous uses the usual strategy, as the functions are given by a series to which Weierstrass M-test applies.

The proof that f is space filling is nice and short. The original argument can be downloaded here. A nice graph of the first few stages of the infinite fractal-like process that leads to the graph of f can be seen in page 49 of Thim’s master thesis.

414/514 – Faber functions

January 17, 2012

Here is Shehzad Ahmed’s project from last term, on the Faber functions, a family of examples of continuous nowhere differentiable functions. Ahmed’s project centers on one of them, but the argument can be easily adapted to all the functions in the family.

As usual, the function is given as a series F=\sum_n f_n where the functions f_n are continuous, and we can find bounds M_n with \|f_n\|\le M_n and \sum_n M_n<+\infty. By the Weierstrass M-test, F is continuous. Also, the strategy for nowhere differentiability is typical: We associate with each point x a pair of sequences (a_n)_{n\ge0} and (b_n)_{n\ge0} with a_n strictly decreasing to x and b_n strictly increasing to x. The key lemma (shown, for example, in Johan Thim’s Master thesis available here) is that, if a continuous function f is differentiable at x, then we have

\displaystyle f'(x)=\lim_{n\to\infty}\frac{f(a_n)-f(b_n)}{a_n-b_n}.

In the case of the Faber functions, the functions f_n add `peaks’ in the neighborhood of any point, and the locations of these peaks can be used as the points a_n and b_n; moreover, the slopes accumulate at these peaks, and so the limit on the right hand side of the displayed equation above does not converge and, in fact, diverges to +\infty or -\infty.

Faber’s original paper, Einfaches Beispiel einer stetigen nirgends differentiierbaren Funktion, Jahresbericht der Deutschen Mathematiker-Vereinigung, (1892) 538-540, is nice to read as well. It is available through the wonderful GDZ, the Göttinger Digitalisierungszentrum.

414/514 – Katsuura function

January 17, 2012

Here is Erron Kearns’s project from last term, on the Katsuura function, an example of a continuous nowhere differentiable function.

The presentation is nice: As usual with these functions, this one is defined as the limit of an iterative process, but the presentation makes it very clear the function is a uniform limit of continuous (piecewise linear) functions, and also provides us with a clear strategy to establish nowhere differentiability.

Actually, the function is presented in a similar spirit to many fractal constructions, where we start with a compact set K and some continuous transformations T_1,\dots,T_n. This provides us with a sequence of compact sets, where we set K_0=K and K_{m+1}=\bigcup_{i=1}^n T_i(K_m). Under reasonable conditions, there are several natural ways of making sense of the limit of this sequence, which is again a compact set, call it C, and satisfies C=\bigcup_{i=1}^n T_i(C), i.e., C is a fixed point of a natural “continuous” operation on compact sets.

This same idea is used here, to define the Katsuura function, and its fractal-like properties can then be seen as the reason why it is nowhere differentiable.