Course Syllabus

Jump to Today

Quick links:

 

Content: This course provides a rigorous introduction to real and complex analysis and to topology.  The formal prerequisites for Math 55 are minimal, but this class does require a commitment to a demanding course, strong interest in mathematics, and familiarity with proofs and abstract reasoning. 

Shopping week information:

  • See this FAQ for more about the content of the course, the prerequisites, etc. Also make sure to read the important logistics information below.
  • Feel free to e-mail Prof. Auroux with any other inquiries, or to post questions in the Slack workspace.
    Note: Slack access will likely require you to first join the Canvas site by adding the class to your Crimson Cart.
  • Once you have decided to join the class, please post a short message to introduce yourself in Slack. This is especially important if you didn't take Math 55a - it's your chance to meet the other students before the semester starts!

Textbooks: there are two required texts, Munkres' Topology (2nd edition), and Ahlfors' Complex Analysis (3rd edition).  In addition, Rudin's Principles of Mathematical Analysis (3rd edition) is very strongly recommended if you have no prior knowledge of real analysis.  (Munkres and Rudin have cheaper international editions which are nearly identical to the regular text). Finally, a possible alternative reference for complex analysis is Stein and Shakarchi's Complex Analysis.
For real and complex analysis, Prof. McMullen's notes from 2009-10's Math 55b are also a valuable reference.

Course staff:

  • Prof. Denis Auroux (auroux@math.harvard.edu). 
    Office hours: Mondays 12-1pm, Wednesdays 9-10am + 1-2pm (same Zoom as lectures)
  • Avery Parr (Course Assistant)
    OH: Saturdays 12-2pm (Zoom), Sections: Thursday 3-4pm (Zoom)
  • Alfian Tjandra (Course Assistant)
    OH: Fridays 5 -6 pm (Zoom)
  • Cheng Zhou (Course Assistant)
    OH: Sundays 9-10am ET (Zoom), Sections: Saturdays 9-10am ET (Zoom)
  • Richard Xu (Volunteer)
    Combined Section/OH: Mondays 8-10pm (Zoom)
  • Gaurav Goel  (Volunteer)
    Combined Section/OH: Sundays 11:30 am - 1 pm ET (Zoom Link)

Lectures will be held MWF 10:30-11:45am (US Eastern time), on Zoom. Attendance is highly recommended but not mandatory. Handwritten lecture notes will be provided (see in Files) and lectures will be recorded for those who cannot attend live (see in Panopto).

Discussion sections will be held by the CAs weekly at various times, in order to go over material from lecture and homework.  These are highly recommended for everyone; students who cannot attend lecture are expected to attend at least one discussion.  Even if you don't need help with specific problems on the current assignment, section is a chance to review and ask general questions about course material (or related issues that may not be dealt with in the lecture), including any content not thoroughly covered in the live lecture.  It's also a chance to get to know the CAs and your fellow students in smaller groups and learn their perspectives.

Tablets/iPads:  All students in this class should have access to a tablet (with stylus strongly recommended) in order to facilitate mathematical interaction during discussion sections, office hours, and when collaborating on assignments. If you do not have a tablet, this course should be eligible for loaner iPads/styluses; once registered for the class, please fill in this form to request equipment.

Homework will be assigned weekly, and is due on Wednesday each week.  Assignments will be posted on this site, and should be submitted electronically via this website.
Doing the homework in a timely fashion is essential to learning the material properly; given the pace, it is extremely hard to catch up if you fall behind in this class.  Extensions will be granted when circumstances genuinely warrant it (poor time management on your part is not normally a sufficient circumstance), but should be requested ahead of the deadline.

Exams: we will have a take-home midterm (to be taken during the week of February 16) and a take-home final exam (tentatively: to be taken anytime during the period of May 3 to May 12).

Course grades will be based on your homework (65%), the midterm (10%), and the final exam (25%).
One homework score will be dropped for everyone, so you may miss one assignment without penalty, but you are still responsible for working through the material.

Community: Even though online classes can seem impersonal, keep in mind that everyone in this class is part of a community.  Hopefully, you will soon get to know not just the course staff but also your fellow students.  Drop by office hours to introduce yourself and meet others (even if you don't have any specific questions). Participate in the Slack discussions. Form study groups. And please remember: it is up to you to make this community inclusive, welcoming, and supportive of all of its members.

Academic integrity policy: You are encouraged to discuss and collaborate with each other on the homework assignments. However,  make sure that you can work through the problems yourself, and write up your answers on your own. This is not only a matter of academic integrity, but also crucial for properly learning the material and the problem-solving skills that this course aims to cover.  For exams, collaboration or consultation of sources other than those explicitly permitted is not allowed.

 

Homework assignments

Direct links to the homework assignments will be posted here as PDF files. Go to "Assignments" to see the LaTeX source, and to submit your solutions.

Lecture notes

Direct links to the lecture notes will be given here. Go to "Files" to see them all in a folder. Videos will be posted in Panopto (restricted access).

  • Lecture 1 (Mon Jan 25: Course logistics; metric spaces: open sets, continuity, limits) (handout)
  • Lecture 2 (Wed Jan 27: Topological spaces, bases, subspaces and products; interior and closure)
  • Lecture 3 (Fri Jan 29: interior and closure, limit points; Hausdorff spaces; product topology)
  • Lecture 4 (Mon Feb 1: product and uniform topologies; connected spaces)
  • Lecture 5 (Wed Feb 3: connected and path-connected spaces; compact spaces)
  • Lecture 6 (Mon Feb 8: compact spaces; compactness in metric spaces; sequential compactness)
  • Lecture 7 (Wed Feb 10: sequential compactness; completeness; compactifications)
  • Lecture 8 (Fri Feb 12: compactifications; countability and separation axioms, Urysohn's theorem)
  • Lecture 9 (Wed Feb 17: Urysohn's theorem; quotient topology and gluing)
  • Lecture 10 (Fri Feb 19: quotient spaces; homotopy; homotopy equivalence; composition of paths)
  • Lecture 11 (Mon Feb 22: the fundamental group; homotopy invariance; covering maps)
  • Lecture 12 (Wed Feb 24: covering spaces, path-lifting; π₁(S¹))
  • Lecture 13 (Fri Feb 26: Brouwer fixed point theorem and other applications; calculations of π₁)
  • Lecture 14 (Wed Mar 3: more about covering spaces; classification)
  • Lecture 15 (Fri Mar 5: universal covering space; Seifert-Van Kampen theorem; π₁ of surfaces)
  • Lecture 16 (Mon Mar 8: continuity of real functions; sequences and series, power series; derivatives)
  • Lecture 17 (Wed Mar 10: differentiation and integration in one variable)
  • Lecture 18 (Fri Mar 12: the Riemann integral; L^p norms; equicontinuity and Arzela-Ascoli)
  • Lecture 19 (Mon Mar 15: Stone-Weierstrass theorem, Fourier series)
  • Lecture 20 (Wed Mar 17: Fourier series; differentiation in several variables)
  • Lecture 21 (Fri Mar 19: differentiation in several variables, inverse function theorem)
  • Lecture 22 (Mon Mar 22: integration; differential forms; Stokes' theorem)
  • Lecture 23 (Wed Mar 24: integration of differential forms; Stokes' theorem; complex functions)
  • Lecture 24 (Fri Mar 26: complex derivative, analytic functions; rational functions; the Riemann sphere)
  • Lecture 25 (Mon Mar 29: power series; exp and log; Cauchy's theorem and integral formula)
  • Lecture 26 (Fri Apr 2: Cauchy's integral formula, derivatives)
  • Lecture 27 (Mon Apr 5: Taylor series; zeros of analytic functions; further consequences of Cauchy)
  • Lecture 28 (Wed Apr 7: equicontinuity; antiderivatives; Laurent series; poles and singularities)
  • Lecture 29 (Fri Apr 9: poles and singularities; meromorphic functions; maximum principle)
  • Lecture 30 (Mon Apr 12: maximum principle; harmonic functions; open mapping; argument principle)
  • Lecture 31 (Wed Apr 14: the argument principle; residue calculus: residues and definite integrals)
  • Lecture 32 (Fri Apr 16: more definite integrals via residues; partial fractions)
  • Lecture 33 (Mon Apr 19: partial fractions and infinite sum expansions)
  • Lecture 34 (Wed Apr 21: infinite product expansions; partitions)
  • Lecture 35 (Fri Apr 23: end-of-semester announcements; Gamma and zeta functions)
  • Lecture 36-37 (Apr 26-28: abelian integrals, elliptic functions, Riemann surfaces, Weierstrass P-function)
  • Review, part 1: Topology
  • Review, part 2: Real Analysis
  • Review, part 3: Complex Analysis

 

Course Summary:

Date Details Due