Course Syllabus

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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. 

Lectures: will be held in person Mondays, Wednesdays, Fridays from 10:30 to 11:45, Science Center 507, and simultaneously broadcast over Zoom.

Important: if you are unwell or have been told to isolate/quarantine for any reason, please do not come to class. Instead, you should plan on watching the lecture via the backup Zoom meeting link. (Ideally let Prof. Auroux know ahead of time if possible).

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 joined 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 (except page numbering and index in the case of Munkres).
  • Ahlfors' book was out of print for a while, but it has just been reprinted by the American Mathematical Society, with ISBN 978-1-4704-6767-8; see here if you can't find it elsewhere.
  • 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: Mon & Tue 9-10am in SC 530,  and Mon 1:30-2:30pm in SC 411
  • Oliver Cheng (Course Assistant)
    Office hours: Fri 3-5 PM Zoom: https://harvard.zoom.us/j/2318474747
  • Edis Memis (Course Assistant)
  • Dora Woodruff (Course Assistant)
  • Eric Yan (Course Assistant)

Lectures will be held MWF 10:30-11:45am.  Students are expected to attend all lectures. Handwritten lecture notes will be provided.

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.  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 lecture.  It's also a chance to get to know the CAs and your fellow students in smaller groups. See here a live list of locations and times for office hours and sections.

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: there will be a take-home midterm (February 14-18) and a take-home final exam (May 2-10); you will be given a window of a few days for each exam, for flexibility.

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:  One of the best features of Math 55 is the sense of community that most students get out of it.  Get to know the CAs and your fellow students early in the semester, and form study groups -- they'll be your support network when the math gets rough. Drop by office hours to introduce yourself and meet others (even if you don't have any specific questions). Participate in the Slack discussions. And please remember: it is up to you to make this community inclusive, welcoming, and supportive of all of its members -- and of all the other people around you, including those who aren't in Math 55.

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.

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

Course Summary:

Date Details Due