## 305 – Homework IV

March 7, 2012

This homework set is due Wednesday, March 21, at the beginning of lecture.

## 197 – Introduction to mathematical thought. Announcement and syllabus

March 5, 2012

This Fall I will be teaching a course in the Honors College, Math 197: Introduction to mathematical thought.

The goal of the course is to present an introduction to the mathematical method, the way mathematics is reasoned, discovered, and advanced. This will be accomplished through a presentation of selected (real world) examples, and an emphasis on the key notion of mathematical proof. Particular attention is paid to aesthetic, historical, and philosophical aspects of mathematics.

Pre-requisites: Instructor’s approval.

Text: We will use several texts and articles. Particularly recommended are:

• T.W. Körner. The pleasures of counting, Cambridge University Press (1996). ISBN: 0-521-56087-X (hardback), 0-521-56823-4 (paperback).

(This will be our official textbook, but we will draw material from all three.)

• Sh. Stein. How the other half thinks. Adventures in mathematical reasoning, McGraw-Hill (2001). ISBN-10: 0071407987, ISBN-13: 978-0071407984.
• W. Dunham. Journey through genius. The great theorems of mathematics, John Wiley & Sons, Inc (1991). ISBN-10: 014014739X, ISBN-13: 978-0140147391.

I will provide additional handouts and references as needed. Supplementary recommendations include:

• I. Lakatos. Proofs and refutations. The logic of mathematical discovery, Cambridge University Press (1976). ISBN-10: 052121078X, ISBN-13: 978-0521210782.
• Ph.J. Davies and R. Hersh. The mathematical experience, Mariner Books (1999). ISBN-10: 0395929687, ISBN-13: 978-0395929681.
• S.G. Krantz. The proof is in the pudding. The changing nature of mathematical proof, Springer (2011). ISBN-10: 0387489088, ISBN-13: 978-0387489087.

Contents: We will present a series of examples illustrating how mathematics is used in the real world, and how it is conceived. We will discuss the nature of mathematical proof, and some of the philosophical issues surrounding it, as well as how it has evolved through the ages. The goal is to see how mathematicians actually reason, and how mathematical ideas are a natural part of the cultural legacy of humankind.

Speciﬁc examples may be chosen depending on the audience background and motivation. Particular examples I would like to include are the mathematics of
codes, how populations evolve through time, the mathematics of inﬁnity, and computer generated proofs.

Grading: Grades will be determined based on homework (60%), a written project (20%), and class participation (20%). There will be no exams.

Attendance to lecture is not required but highly recommended.

## 580 – Topics in Set Theory. ANNOUNCEMENT

March 5, 2012

This Fall I will be teaching Topics in set theory. The unofficial name of the course is Combinatorial Set Theory.

We will cover diverse topics in combinatorial set theory, depending on time and the interests of the audience, with emphasis on three topics: Choice-free combinatorics, cardinal arithmetic, and partition calculus (a generalization of Ramsey theory).

Time permitting, we can also cover large cardinals, determinacy and infinite games, or cardinal invariants (the study of sizes of sets of reals), among others. I’m open to suggestions for topics, so feel free to email me or to post a comment.

Pre-requisites: Permission by instructor. The recommended background is knowledge of cardinals and ordinals. A basic course on set theory (like 502: Logic and Set Theory) would be ideal but is not required.

Textbook: Combinatorial set theory, by Neil H. Williams. Elsevier Science (1977). ISBN-10: 0720407222, ISBN-13: 978-0720407228. The book seems to be out of print.

We will also use:

• Combinatorial Set Theory: Partition Relations for Cardinals, by Paul Erdös, András Hajnal, Attila Máté, and Richard Rado. Elsevier Science (1984). ISBN-10: 0444861572,  ISBN-13: 978-0444861573. Apparently, this is also out of print.

I will distribute notes on the material of these books, on additional topics, and some papers that we will follow, particularly:

• András Hajnal and Jean A. Larson. “Partition relations”, in Handbook of set theory, 129–213, Springer, 2010.
• Jean A. Larson. “Infinite combinatorics”, in Handbook of the history of science, vol. 6, 145-357, Elsevier, 2012.

## 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 Erin Carmody‘s page 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.

Another discussion of the Futurama theorem, by Samuel Coskey (who will be joining our math department starting this Fall) can be found here.

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.

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.

## 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”.