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Freshman Research Initiative (FRI): Research Methods in Mathematics

University of Texas at Austin, Fall semester, 2010

  • Course number: M 310T. Unique identifier: 55205.

  • Tuesday, Thursday 9:30-11:00 a.m. RLM 9.166

  • Instructor: Tim Perutz (Assistant Professor)

  • Office hours: Tuesday, Wednesday 4-5 p.m., RLM 10.136.

  • Email: perutz AT math DOT utexas DOT edu

  • TA: Michael Kelly.

  • Textbooks: No required texts. For recommended texts, see below.

Start here

For a taste of what this course is about, click here for an in illustrated preview (PDF format). The official syllabus (which consists of the preview plus information from this page) is available here.

Outline

This course aims to help you make the transition from high-school-style mathematics to advanced mathematics. Making this transition is not about handling more complicated formulae but about learning mathematical modes of thought. Advanced mathematics asks for crystal-clear arguments. It is also a creative subject with wide horizons. The mathematics covered in this course may be the most challenging you have seen so far, but it might just be the most satisfying. It will give you a head-start in topics studied systematically by math majors (analysis, linear algebra, geometry).

The course will be divided into three quite different parts:

I. From counting to calculus.

We will begin by re-examining familiar number systems, and will go on to see how to solve a problem that baffled mathematicians for centuries: how to make calculus make logical sense.

II. Algebra and geometry of linear maps.

In this part we will discuss linear maps and matrices, focusing on examples such as rotation about an axis in 3-dimensional space or reflection in a mirror. This material is applicable to computer graphics, mechanical engineering, quantum physics, chemistry, and to any part of science that uses differential equations.

III. The symmetries of plane patterns.

The third part will be about the mathematics of symmetry, especially repeating patterns like those found on wallpaper, tiles or brick walls. This is visual mathematics. You will learn how to identify a pattern by its symmetries. The course will culminate in a modern, "topological" proof that there are precisely 17 kinds of repeating pattern. Examples of most of them can be found on the walls of the Alhambra Palace in Spain, built in the 14th Century.

Textbooks

The course will be self-contained, so textbooks are not absolutely required. For each part of the course, there is a recommended text which will help you. I suggest that you buy at least one of them if you can afford it. The text for the first part is pricy but will serve you well in other courses. The text for the second part is slim and cheap. The one for the third part is an expensive but gorgeously illustrated book. I'll ask the Kuehne library to reserve copies. Here they are:
  • For part 1: Michael Spivak, Calculus, 4th edition. Publish or Perish, 2008.

    A brilliantly written text on analysis - a different kettle of fish from other books with the same title. A solution manual is also available. The third edition is out of print, but the fourth edition is available direct from the publisher's website, currently for $85. It's a nicely produced, weighty hardback. This book will make an excellent supplement to any courses in calculus or analysis you may take in the future.

  • For part 2: Alexander Givental, Linear algebra and differential equations. Berkeley Mathematics Lecture Notes Vol 11. American Mathematical Society, 2001.

    Tersely written - read it slowly! - this book goes straight to what's important. We will cover a fraction of what's here. The American Mathematical Society currently sells this for $21.

  • For part 3: J. Conway, H. Burgiel and C. Goodman-Strauss, The Symmetries of Things. A. K. Peters, 2008.

    Wonderful pictures. Much of this material isn't published anywhere else, except in research papers. Here it is at Amazon for a list price of $75, but as I write it's available for $66.15.

For further reading about mathematics generally, I recommend the following.
  • T. Gowers, Mathematics: A Very Short Introduction. Oxford University Press, 2002.

    Brevity and price are virtues of this book, but so is its quality. To get an impression of what mathematicians actually do, this is an excellent place to start. Currently available for less than $10!

  • T. Koerner, The Pleasures of Counting. Cambridge University Press, 1996. %!TEX TS-program =

Syllabus

Date

Notices

Lecture plan

From counting to calculus
Aug. 26 Introduction
Aug. 31 Counting and induction
Sept. 2 HW 1 dueAddition and multiplication
Sept. 7 Inequality
Sept. 9 HW 2 due Rational numbers and upper bounds
Sept. 14 The real numbers
Sept. 16 HW 3 dueFunctions and limits
Sept. 21 Continuity
Sept. 23 HW 4 dueThe intermediate value theorem
Sept. 28 Differentiation
Sept. 30 HW 5 due Differentiation rules
Oct. 5 The "plus a constant" theorem

Algebra and geometry of linear maps
Oct. 7 Extended assignment 1 dueVectors and matrices in 2 dimensions.
Oct. 12 Linear maps and matrices.
Oct. 14 HW 6 dueOrthogonal transformations. Rotations and reflections.
Oct. 19 Classifying orthogonal transformations of the plane.
Oct. 21 HW 7 dueVectors and matrices in three dimensions.
Oct. 26 Writing down reflections and rotations in three dimensions.
Oct. 28 HW 8 dueEigenvectors.
Nov. 2 Classifying orthogonal transformations in three dimensions.

The symmetries of plane patterns
Nov. 4 Extended assignment 2 dueClassifying rigid transformations of the plane
Nov. 9 Rosette patterns. The seven frieze patterns. Their signatures.
Nov. 11 HW 9 dueSignatures and orbifolds.
Nov. 16 Wallpaper patterns. The kaleidoscopic patterns *632, *333, *442 and *2222. The gyroscopic patterns 632, 333, 442 and 2222.
Nov. 18 HW 10 due The mixed patterns 3*3, 4*2, 2*22 and 22*. The remaining wallpaper patterns **, *x, xx, 22x and o. Their orbifolds. Recognizing them.
Nov. 23 The classification theorem via Conway's magic theorem.
Nov. 30 HW 11 dueEuler characteristics.
Dec. 2 Extended assignment 3 dueProving the magic theorem.

Assessment

  • Homework (45%).

    Weekly homework problems (except when an extended assignment is due), including questions of various sorts. Some will be routine problems to help learn the material. Others will give you practice in writing proofs. A few will be tough nuts which I hope you will enjoy solving. Homework will be due in class on Thursdays. Your lowest homework grade is dropped.

  • Extended assignments (45%).

    There will be three of these - one for each part of the course - each worth 15%. You will choose from a short list of titles. In each of them, I will ask you to explain a topic going slightly beyond what I covered in lectures (so you will need to use the library). I might ask you to explain a particular theorem and its proof, or an application of the theory. In addition, I may ask you to solve a small number of problems relevant to that material. I'll be looking for mathematical accuracy, relevance and clarity.

  • Participation (10%).

    By participation, I mean attendance in class, and asking, discussing or answering questions in class and/or at my office hours. (Any constructive discussion in counts: not just correct answers.)

Policies

This will be quite a small class, and I hope it will be one with plenty of participation from all the students in it. To make this work, I expect you to adhere to some common-sense rules: don't skip class (illness excepted); arrive on time; put your phone and other gadgets away (definitely no texting, instant messaging, etc.); take notes in class.

You must adhere to the academic standards described here.

You can discuss your homework with others, but to avoid any hint of plagiarism, you should write it up alone.

If you have (or think you might have) a disability relevant to this class, you can request accommodations from the Division of Diversity and Community, Services for Students with Disabilities. You need to do so early in the semester. If there's something I can do differently that will help, I'll gladly discuss it with you.

If you need to miss class for a religious holiday, university policy asks that you let me know two weeks in advance.

Class notes

Class notes will appear here. Beware: these will usually only be outline notes, so skip class at your peril!

Help with finding errors is appreciated.

Lecture 1 notes, slides.

Lecture 2 notes (revised), slides.

Lecture 3 notes (revised).

Lecture 4 notes.

Lecture 5 notes.

Lecture 6 notes.

Lecture 7 notes.

Lecture 8 notes.

Lecture 9 notes.

Lecture 10 notes.

Lecture 11 notes.

Lecture 12 notes.

Lecture 13 notes.

Homework

Homework 1. Due at the beginning of class, Thursday 2 Sept. Solutions.

Homework 2. Due at the beginning of class, Thursday 9 Sept. Solutions.

Homework 3. Due at the beginning of class, Thursday 16 Sept. Solutions.

Homework 4. Due at the beginning of class, Thursday 23 Sept. Solutions.

Homework 5. Due at the beginning of class, Thursday 30 Sept. Solutions.

Extended assignment 1. Due at the beginning of class, Thursday 7 Oct. Solutions to the "fields" option. Solutions to the "circle problem" option.

Homework 6. Due at the beginning of class, Thursday 14 Oct. Solutions.

Homework 7. Due at the beginning of class, Thursday 21 Oct. Solutions.

Homework 8. Due at the beginning of class, Thursday 28 Oct.

Extended assignment 2. Due at the beginning of class, Thursday 4 Nov.

Homework 9. Due at the beginning of class, Thursday 11 Nov.

Homework 10. Due at the beginning of class, Thursday 18 Nov.

Extended Assignment 3. Due by noon, Tuesday 14 Dec. Turn in to my office, or by email, or to my mailbox in the math office on the 8th floor of RLM.