Math 55a: Honors Abstract Algebra
Handout #1 (31 August 2016): About Math 55
Topics We will spend most of the semester studying (bi)linear algebra, and also lay the groundwork for further topics in abstract algebra, culminating with
the main results of either Galois theory or the representation theory of finite groups, if not both. These are fundamental tools for much of modern mathematics. In Math 55b, we will introduce metric topology and combine it with tools from the linear algebra studied in 55a to develop a rigorous treatment of differential and integral calculus, together with an introduction to complex analysis (calculus of differentiable functions of one complex variable). Which math class? Much of the syllabus for Math 25 is similar; Math 55 dif-
fers not so much in the choice of topics as in the level of exposition, and is intended for students with significant experience with and enthusiasm for abstract mathematics. The Mathematics Department offers these courses at separate hours so you can “shop” both, which you are strongly encouraged to do if you are at all unsure which class is better for you — it’s the only way to make an informed informed choice. choice. By special dispensat dispensation ion from the Registrar, you may switch between 25 and 55 without penalty for the first few weeks of the semester. NB: Each year several first-year students are tempted to skip 25/55 altogether and dive right into the upper-level undergraduate or graduate courses; we have found that in nearly all cases this temptation is best resisted. Textbooks We shall base our exposition of linear algebra on Sheldon Axler’s Lin (3rd ed., Springer, 2015), though we will routinely ear Algebra Done Right (3rd
go beyond beyond Axler. Axler. That text should should now be available available for purchase purchase at the 1 Coop. For the rest of 55a we’ll likely follow a (necessarily small) selection from Michael Artin’s Artin’s Algebra. Office Hours, etc. I’ll hold weekly office hours in the Lowell House Dining Hall
overlapping with “Brain Break”; at least for now it will be Tuesdays from 7:30 to 9:00 PM. I’ll also occasionally stop by the newly instituted Math Night in the Leverett House Dining Hall, which meets Mondays from 8:00 to 10:00 PM. You You can also e-mail me to set up an appointment outside of Office Hours, or for any other reason, at
[email protected] . Please use plain text only, no HTML or “smart quotes” or o r other fancy features that will just make it harder for me to properly read and process your e-mail: believe believe it or not, I’d much prefer to see “ xˆ2 can’t can’t *really* work” than 2 2 “x can’t really work” in my inbox. If you can’t say it all in plain text, you 1
I’m told that you can also use your Harvard credentials credentials to access the PDF for free, and even print a paperback copy for personal use at a deep discount. For further information inquire at one of the Harvard Harvard libraries. 2 Yes, I use such features in the Math 55a webpage, but I don’t read my e-mail in a browser! Conversely Conversely,, if you have technological or other issues that make it hard to read my online notes, please let me know so I can try to work out an accommodation.
can add an attachment in PDF (or possibly JPG/PNG etc., but not Micro$**t Word or Excel etc.). The assistant for the course is Wyatt Mackey (wmackey@college ). Section time and place will be determined once the class roster has stabilized and we know what everybody’s schedule is. Most handouts, problem sets, etc. will be posted at www.math.harvard.edu/˜elkies/M55a.16 = the Math 55a Web page. Grading Most of your grade (about 53 ) will be based on weekly problem sets. Do-
ing mathematics is the only reliable way to learn it, and most of the material in 55 cannot reasonably be done in the framework of a few-hour exam. There will likely be one or two in-class quizzes that will mostly test your recollection of basic concepts; such a quiz will count for at most the equivalent of two homework assignments. A final take-home exam will account for most of the remaining 52 − 2q of your grade, with class participation used mostly to decide borderline cases. Math 55 is not “graded on a curve”; I would be most delighted to find that every single student in the class has earned an A. (When I have taught 55 previously, most but not all students did earn A or A−.) Homework and take-home exams may be (and often are) hand-written; for Wyatt’s and my sake, and also for yours when reviewing your corrected homework sets, please write neatly. If your handwriting tends to indecipherability, consider writing in the much more easily legible BLOCK LETTERS , which (perhaps counterintuitively) takes about the same time as scrawling. Honor code “Faculty legislation requires all instructors to include a statement outlining their policies regarding collaboration on their syllabi . . .”:
You are encouraged to discuss the course with other students, your CA, and/or me. It is much easier to learn mathematics if you have other people who will help you test your understanding and overcome problems. By all means discuss homework problems with other students, but you should always write your homework solutions out yourself in your own words. If you work with a group of fellow 55a students, please also indicate the composition of the group; of course you should always follow the usual guidelines for “working with sources”. For the final take-home exam you will be on your honor to work on your own.
