515 – Advanced Analysis

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.

I will use this website to post additional information, and encourage you to use the comments feature. If you leave a comment, please use your full name, which will simplify my life filtering spam out.


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.)

Read the rest of this entry »


Update

January 13, 2012

There is this book we read at bedtime, All the things I love about you, by LeUyen Pham. Francisco has added to it a few twists of his own:

For example, here the boy is running away from an explosion.

He is learning to spell, but he is stubborn. And sometimes he spells backwards. So Y-R-M-A-C A-T-O-Y-O-T spells car.

His favorite movie right now is Jumanji. During the stampede scenes, the living room gets destroyed. He is thorough.

For christmas he got the Doctor, and for his birthday, a bike. And, of course, a million other things.

All of which must be uniformly spread through the living room while elephants trumpet.