Course Syllabus

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.

Important: Due to the course's place in the first-year undergraduate mathematics experience at Harvard and its role in helping our future math concentrators get to know each other and form a community, Math 55 is only open to first-year Harvard College students. 

Lectures: will be held on Mondays, Wednesdays, Fridays from 10:30 to 11:45, location to be confirmed (likely Science Center 507). Students are expected to attend all lectures. Handwritten lecture notes will be provided.

Course staff: 

• Prof. Denis Auroux (auroux@math.harvard.edu)
• Course Assistants to be determined (likely same as in Math 55a)

The CAs will hold regular office hours, as well as various one-time discussion sections in order to go over specific material or skill sets.  Even if you don't need help with specific problems on the current assignment, attending some office hours or sections is recommended -- it gives you an opportunity to review and ask general questions about the course material (or related topics).  It's also a chance to get to know the CAs and your fellow students in smaller groups.

Pre-registration questions: see this FAQ for information about the content of the course, prerequisites, etc.; feel free to e-mail Prof. Auroux with any other inquiries.

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 has recently been reprinted by the American Mathematical Society, with ISBN 978-1-4704-6767-8.
• For real and complex analysis, Prof. McMullen's notes from 2009-10's Math 55b are also a valuable reference.

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 for illness or other serious circumstances (having overcommitted yourself does not count), but should be requested ahead of the deadline; outside of these cases, late submissions will incur a penalty to be computed at the end of the semester once your cumulative lateness across all assignments exceeds 96 hours (4 days).

Exams: there will be a midterm (take-home, mid-February) and a take-home final exam (during late review period / early finals period). 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.  Getting to know the CAs and your fellow students early in the semester, and forming study groups,  is an important part of the experience -- 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.

• Homework 1 (due Wed Feb 5)
• Homework 2 (due Wed Feb 12)
• Homework 3 (due Wed Feb 19)
• Homework 4 (due Wed Feb 26)
• Homework 5 (due Wed March 5)
• Homework 6 (due Wed March 12)
• Homework 7 (due Wed March 26)
• Homework 8 (due Wed April 2)
• Homework 9 (due Wed April 9)
• Homework 10 (due Wed April 16)
• Homework 11 (due Wed April 23)
• Homework 12 (due Wed April 30)

List of Topics and Lecture Notes

Direct links to the lecture notes will be posted here as PDF files. Here is a tentative list of topics / dates; we may deviate from these. 

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