Instructor

Laurent Berger. I'm a number theorist. Here is my homepage with more details. My office is 322 in the science center, my phone number is 5-9560, and the best way to get in touch with me is by email, at laurent@math.harvard.edu.

Course Assistants

The course assistants are Benjamin Bakker and Inna Zakharevich. Their job will include: grading your homework, holding office hours and running problem sessions.

CA Website

The course assistants have set up a website of their own, which contains copies of the course notes, assignments, section information, their office hours, etc.

Section times

Homework

There will be weekly homework, which will be an essential part of the course. Late homework will generally not be accepted, but the lowest three scores of the semester will be dropped. The homework will be posted on the web and you need to be able to read files in one of the following three formats: dvi, ps or pdf. Collaboration on assignments is encouraged: this means you can discuss problems you're having trouble with, but you have to write the solutions yourself. Note: ``I do problems 1-10 and you do 11-20'' does not qualify as ``collaboration''.

Date due	Problems (tex, dvi, ps and pdf)	Solutions (dvi, ps and pdf)
Sep 26th		Part 1 Part 2 Part 3
Oct 3rd		Part A Part B Part C
Oct 10th		Part A Part B Part C
Oct 17th		Part A Part B Part C
Oct 24th		Part A Part B Part C
Oct 31st		Part A Part B
Midterm
Nov 7th		Part A Part B
Nov 14th		Part A Part B
Nov 26th		Part A Part B
Dec 5th		Part A Part B
Dec 15th		Part A Part B
Final

Web resources

• Biographies of mathematicians
• MathWorld

What's next?

Here is the menu for the past and future few classes:

• Sets
  • Sets and maps
  • Finite sets
  • Countable sets
  • Cardinals
• Topological spaces
  • Definition, sequences
  • Open and closed sets
  • Continuous functions
  • Connected spaces
  • Complete spaces
  • Compact spaces
• The real numbers
  • Definition and basic properties
  • Series
  • Rational and irrational numbers
  • Functions on the real numbers
• Integration and differentiation
  • Integration of ruled functions
  • Differentiation
  • Higher derivatives and Taylor's formula
  • Application to two series
  • Asymptotic developments
  • Improper integrals
  • Continuity and differentiation under the integral sign
• Algebra
  • Groups
  • Rings and fields
  • Complex numbers
• Linear algebra
  • Vector spaces and matrices
  • Rank
  • Endomorphisms
  • Determinants

Course Description

A rigorous treatment of topics such as topology, some algebra, linear algebra, real analysis and more advanced topics such as differential equations, complex analysis, Fourier analysis and maybe some functional analysis. The goal of the course is to prepare you for advanced mathematics. There will be an emphasis on learning how to understand and write proofs.

Office hours

Thursdays 9-12. Instead of coming to see me it is more efficient to send me an email at laurent@math.harvard.edu. I will then post the answer to your anonymised email (unless you don't want me to) on this virtual office hours webpage. Check that webpage before asking a question since somebody probably asked it already.

Textbook

There is no textbook assigned for 25a. Instead, I will give numerous exercises (see above) and also indicate references of books (see below) that you can take a look at. The Math 23 bibliography is also a useful source of references. Note that the books published by Dover (e.g. the textbooks for 23) are pretty cheap (under $20 usually).

Sets, topology

• Topology: a first course by James Munkres.

Analysis

• Principles of mathematical analysis by Walter Rudin. The first few chapters cover some of what we're doing now. This is the first textbook for 55.

Linear Algebra

• Linear algebra by S. Lang (Springer Verlag UTM) is a good textbook on linear algebra, but is a bit too easy for the course (and only covers the linear algebra part).

• Linear algebra done right by S. Axler. Speaks for itself... This is the second textbook for 55.

Algebra

• Lang's Algebra (now Springer Verlag GTM 211, previously Addison-Wesley) is a good reference for, well, algebra (including the linear kind) but it is not a textbook.

Advanced References

• Most of what we will cover is in S. Lang's Real and functional analysis (Springer Verlag GTM 142) which is not a bad investment but is too advanced for the course. Consider it as good supplementary reading during the second half of the course.

Grading

A combination of homework, a midterm and a final exam, 40%--30%--30%. The final exam should be on January 23rd. Students generally do very well.

Attendance

Attendance is not mandatory, but it is expected of you. You should read pages 79-85 of your Student Handbook for more on that topic.

Special needs

In general, it is your responsibility to let me know well in advance if you have special needs, a conflict because of a religious holiday, etc. If you need special accomodations because of a disability, you need to speak to me and bring a letter from SDR by October 1st.

Last modified: Nov 7th 2003