## 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.

Topics:

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.

## 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.

## 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 ${\mathbb R}^3:$

## 305 – Solving cubic and quartic equations

January 23, 2012

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

## 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.

January 13, 2012

Syllabus for Math 515: Advanced calculus AKA Analysis II.

Instructor: Andrés E. Caicedo.
Contact Information: See here.
Time: MWF 9:40-10:30 am.
Place: MG 124.
Office Hours: MF 11-12.

Text: An introduction to measure theory“, by Terence Tao. AMS, Graduate studies in mathematics, vol 126, 2011. ISBN-10: 0-8218-6919-1. ISBN-13: 978-0-8218-6919-2. Errata.

Mathematicians find it easier to understand and enjoy ideas which are clever rather than subtle. Measure theory is subtle rather than clever and so requires hard work to master.

Thomas W. Körner, Fourier Analysis, p. 572.

Contents: From the Course Description on the Department’s site:

Introduction to the fundamental elements of real analysis and a foundation for writing graduate level proofs. Topics may include: Banach spaces, Lebesgue measure and integration, metric and topological spaces.

We will emphasize measure theory, paying particular attention to the Lebesgue integral. Additional topics, depending on time, may include the Banach-Tarski paradox, and an introduction to Functional Analysis.

Grading: Based on homework. No late homework is allowed. Collaboration is encouraged, although you must turn in your own version of the solutions, and give credit to books/websites/… you consulted and people you talked/emailed/… to.

There will be no exams in this course. However, an important component of being proficient in mathematics is a certain amount of mental agility in recalling notions and basic arguments. I plan to assess these by requesting oral presentations of solutions to some of the homework problems throughout the term.

## 305 – Abstract Algebra I

January 13, 2012

Syllabus for Mathematics 305: Abstract Algebra I.

Section 1.
Instructor: Andres Caicedo.
Time: MWF 8:40-9:30 am. (Sorry.)
Place: MG 120.
Office Hours: MF 11-12. See here for contact information.

Text:Adventures in Group Theory. Rubik’s Cube, Merlin’s Machine, and Other Mathematical Toys”, 2nd edn. By David Joyner. The Johns Hopkins University Press (2002). ISBN-10:0801869471. ISBN-13: 978-0801869471. Errata.

I will provide additional handouts and references as needed.

It may be a good idea to get a Rubik’s cube, as many examples we will see may be easier to understand with a cube in front of you. There are several online cube solvers (I particularly like this one), and they may be used as well, but I still recommend you get a physical copy.

The book presents many examples using the mathematics software SAGE. SAGE, developed by William Stein, is open source and may be freely downloaded. Consider installing it in your own computers so you can practice on your own. SAGE is very powerful and you will probably find it useful not just for this course.

(It was recently proved that Rubik’s cube can be solved in 20 moves or less, and 19 moves do not suffice in general.)