Math 55B
Math 55B is an intensive honors analysis course for first-year Harvard College students, providing a rigorous introduction to real and complex analysis. This course covers the equivalent of Math 25b and Math 113, and prepares students for Math 114 and other advanced courses in analysis.
Course Information
Course Name: Studies in Real and Complex Analysis
Course Number: Math 55B
Instructor: Denis Auroux
Course Description:
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.
Course Content:
This course covers the following main topics:
- Topology: topological spaces, continuity, compactness, connectedness
- Algebraic topology: homotopy, fundamental group, covering spaces
- Real analysis: differentiation and integration in one and several variables, sequences of functions and uniform convergence, differential forms
- Complex analysis: complex functions, analytic functions, Cauchy’s theorem, residue theory, special functions
Textbooks:
Required Texts:
- Topology (2nd edition), James R. Munkres
- Complex Analysis, Lars V. Ahlfors (3rd edition)
Other Recommended References:
- Principles of Mathematical Analysis (3rd edition), Walter Rudin
Course Outline:
- Weeks 1-10: Topology (topological spaces, compactness, connectedness, fundamental group, covering spaces)
- Week 11: Midterm 1
- Weeks 12-20: Real analysis (differentiation and integration in one and several variables, sequences of functions, differential forms)
- Weeks 21-24: Fundamentals of complex analysis (complex functions, analytic functions, Cauchy’s theorem)
- Week 25: Midterm 2
- Weeks 26-37: Advanced complex analysis (residue theory, special functions, Riemann surfaces)
Prerequisites: 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. Students should be familiar with general set-theoretic notation and language, as well as basic knowledge of group theory (groups, subgroups, homomorphisms, cosets and quotient groups). Prior knowledge of basic notions such as open and closed subsets, limits, and continuity in the case of R or R^n is recommended.
Note: This course information is compiled based on Harvard University course content.