# Upper Level Courses

*Eriko Hironaka*

2019 Fall (4 Credits)

**
Schedule: **
TR 01:30 PM - 02:45 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

An introduction to rigorous mathematics, axioms, and proofs, via topics including set theory, symmetry groups, and low-dimensional topology.

- Course Notes:
- Familiarity with algebra, geometry and/or calculus is desirable. Students who have already taken Mathematics 22a,b, 23a,b, 25a,b or 55a,b should not take this course for credit. This course has given fall term and repeated spring term.

- Recommended Prep:
- An interest in mathematical reasoning. Acquaintance with algebra, geometry and/or calculus is desirable. Students who have already taken Math 25a,b or 55a,b should not take this course for credit.

- Requirements:
- Anti-Req: Not to be taken in addition to Mathematics 23a,b or 25a,b or 55a,b.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | For Undergraduate and Graduate Students |

*Peter Kronheimer*

2020 Spring (4 Credits)

**
Schedule: **
MW 03:00 PM - 04:15 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

An introduction to rigorous mathematics, axioms, and proofs, via topics including set theory, symmetry groups, and low-dimensional topology.

- Course Notes:
- Familiarity with algebra, geometry and/or calculus is desirable. Students who have already taken Mathematics 22a,b, 23a,b, 25a,b or 55a,b should not take this course for credit. This course given fall term and repeated spring term.

- Recommended Prep:
- An interest in mathematical reasoning. Acquaintance with algebra, geometry and/or calculus is desirable. Students who have already taken Math 25a,b or 55a,b should not take this course for credit.

- Requirements:
- Anti-Req: Not to be taken in addition to Mathematics 23a,b or 25a,b or 55a,b.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | For Undergraduate and Graduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

FAS: Core Curriculum (old) | Quantitative Reasoning |

All: Cross Reg Availability | Available for Harvard Cross Registration |

*Christian Brennecke*

2020 Spring (4 Credits)

**
Schedule: **
TR 09:00 AM - 10:15 AM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Develops the theory of inner product spaces, both finite-dimensional and infinite-dimensional, and applies it to a variety of ordinary and partial differential equations. Topics: existence and uniqueness theorems, Sturm-Liouville systems, orthogonal polynomials, Fourier series, Fourier and Laplace transforms, eigenvalue problems, and solutions of Laplace’s equation and the wave equation in the various coordinate systems.

- Recommended Prep:
- Mathematics 22a,b, 23a,b or 25a,b or Mathematics 19a,b or 21a,b plus any Mathematics course at the 100 level; or an equivalent background in Mathematics.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

FAS: Course Level | For Undergraduate and Graduate Students |

All: Cross Reg Availability | Available for Harvard Cross Registration |

*W. Hugh Woodin*

2020 Spring (4 Credits)

**
Schedule: **
TR 12:00 PM - 01:15 PM

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Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

An introduction to mathematical analysis and the theory behind calculus. An emphasis on learning to understand and construct proofs. Covers limits and continuity in metric spaces, uniform convergence and spaces of functions, the Riemann integral.

- Recommended Prep:
- Mathematics 19a,b or 21a,b and either an ability to write proofs or concurrent enrollment in Mathematics 101 or 102; or an equivalent background in mathematics.

- Requirements:
- Anti-Req: Not to be taken in addition to Mathematics 23a,b or 25a,b or 55a,b.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | For Undergraduate and Graduate Students |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Alexander Kupers*

2020 Spring (4 Credits)

**
Schedule: **
TR 03:00 PM - 04:15 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Analytic functions of one complex variable: power series expansions, contour integrals, Cauchy’s theorem, Laurent series and the residue theorem. Some applications to real analysis, including the evaluation of indefinite integrals. An introduction to some special functions.

- Recommended Prep:
- Not recommended for most students who took Mathematics 55a and/or Mathematics 55b. Talk to the Director of Undergraduate Studies in Mathematics if you took Mathematics 55a and/or 55b and wish to take this course.

- Requirements:
- Anti-Req: Not to be taken in addition to Mathematics 55b.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | For Undergraduate and Graduate Students |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Christian Brennecke*

2019 Fall (4 Credits)

**
Schedule: **
MW 09:00 AM - 10:15 AM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Lebesgue measure and integration; general topology; introduction to L p spaces, Banach and Hilbert spaces, and duality.

- Recommended Prep:
- Mathematics 22a,b, 23a,b or 25a,b or 55a,b or 112; or an equivalent background in mathematics.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

FAS: Course Level | For Undergraduate and Graduate Students |

*Tianqi Wu*

2019 Fall (4 Credits)

**
Schedule: **
TR 09:00 AM - 10:15 AM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Complex functions; Fourier analysis; Hilbert spaces and operators; Laplace’s equations; Bessel and Legendre functions; symmetries; Sturm-Liouville theory.

- Course Notes:
- Mathematics 115 is especially for students interested in physics.

- Recommended Prep:
- Mathematics 22a,b, 23a,b or 25a,b or 55a,b or 112; or an equivalent background in mathematics.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | For Undergraduate and Graduate Students |

*Marius Lemm*

2019 Fall (4 Credits)

**
Schedule: **
TR 03:00 PM - 04:15 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Develops the theory of convex sets, normed infinite-dimensional vector spaces, and convex functionals and applies it as a unifying principle to a variety of optimization problems such as resource allocation, production planning, and optimal control. Topics include Hilbert space, dual spaces, the Hahn-Banach theorem, the Riesz representation theorem, calculus of variations, and Fenchel duality. Students will be expected to understand and come up with proofs of theorems in real and functional analysis.

- Recommended Prep:
- Mathematics 22a,b, 23a,b or 25a,b or 55a,b; or Mathematics 21a,b plus at least one other more advanced course in mathematics; or an equivalent background in mathematics.

Additional Course Attributes::

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | For Undergraduate and Graduate Students |

*Eriko Hironaka*

2020 Spring (4 Credits)

**
Schedule: **
TR 01:30 PM - 02:45 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Introduction to dynamical systems theory with a view toward applications. Topics include existence and uniqueness theorems for flows, qualitative study of equilibria and attractors, iterated maps, and bifurcation theory.

- Recommended Prep:
- Mathematics 19a,b or 21a,b or Math 22a,b,or Math 23a,b or Math 25a,b or Math 55a,b; or an equivalent background in mathematics.

Additional Course Attributes::

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | For Undergraduate and Graduate Students |

*Horng-Tzer Yau*

2019 Fall (4 Credits)

**
Schedule: **
TR 12:00 PM - 01:15 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

The goal of this class is to provide an introduction to Brownian motion and stochastic calculus. In addition, we will cover basic martingale theory. We will also give an intuitive derivation of Ito’s formula and cover some of its application.

- Course Notes::
- This class assumes familiarity with basic probability theory at the level of Mathematics 154.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | Primarily for Undergraduate Students |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Elden Elmanto*

2019 Fall (4 Credits)

**
Schedule: **
TR 10:30 AM - 11:45 AM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Real and complex vector spaces, linear transformations, determinants, inner products, dual spaces, and eigenvalue problems. Applications to some or all of the following: Ggeometry, systems of linear differential equations, optimization, and Markov processes. This course emphasizes learning to understand and write rigorous mathematics.

- Recommended Prep:
- Mathematics 19b or 21b or an equivalent background in mathematics.

- Requirements:
- Anti-req: Not to be taken in addition to Mathematics 22b, 23a or 25a or 55a.

Additional Course Attributes::

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | For Undergraduate and Graduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Brooke Ullery*

2019 Fall (4 Credits)

**
Schedule: **
MW 12:00 PM - 01:15 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Groups and group actions, vector spaces and their linear transformations, bilinear forms and linear representations of finite groups.

- Recommended Prep:
- Not recommended for most students who took Mathematics 55a and/or Mathematics 55b. Talk to the Director of Undergraduate Studies in Mathematics if you took Mathematics 55a and/or Mathematics 55b and wish to take this course.

- Requirements:
- Anti-Req: Not to be taken in addition to Mathematics 55a.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

FAS: Course Level | For Undergraduate and Graduate Students |

*Sebastien Vasey*

2020 Spring (4 Credits)

**
Schedule: **
WF 09:00 AM - 10:15 AM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Rings and modules. Polynomial rings. Field extensions and the basic theorems of Galois theory. Structure theorems for modules.

- Requirements:
- Prerequisite: Mathematics 122 or Mathematics 55a

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | For Undergraduate and Graduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Michael Hopkins*

2019 Fall (4 Credits)

**
Schedule: **
MW 01:30 PM - 02:45 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Factorization and the primes; congruences; quadratic residues and reciprocity; continued fractions and approximations; Pell’s equation; selected Diophantine equations; theory of integral quadratic forms. Also, selected applications to coding, introduction to elliptic curves and introduction to zeta functions if time permits.

- Recommended Prep:
- Mathematics 101 or 122 or 25a or 23a; or 55a which can be taken concurrently; or an equivalent experience and comfort level with abstract mathematics.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | For Undergraduate and Graduate Students |

*Barry Mazur*

2020 Spring (4 Credits)

**
Schedule: **
TR 12:00 PM - 01:15 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Algebraic number theory: number fields, unique factorization of ideals, finiteness of class group, structure of unit group, Frobenius elements, local fields, ramification, weak approximation, adeles, and ideles.

- Recommended Prep:
- Knowledge of the material in Mathematics 123.

- Requirements:
- Prerequisite: Mathematics 123

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | For Undergraduate and Graduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Eylem Yildiz*

2020 Spring (4 Credits)

**
Schedule: **
MW 09:00 AM - 10:15 AM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Presents several classical geometries, these being the affine, projective, Euclidean, spherical and hyperbolic geometries. They are viewed from many different perspectives, some historical and some very topical. Emphasis on reading and writing proofs.

- Recommended Prep:
- Mathematics 19a,b or 21a,b or 22a,b or 23a or 25a or 55a which may be taken concurrently; or an equivalent background in mathematics.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | For Undergraduate and Graduate Students |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Denis Auroux*

2019 Fall (4 Credits)

**
Schedule: **
MW 10:30 AM - 11:45 AM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

First, an introduction to abstract topological spaces, their properties (compactness, connectedness, metrizability) and their corresponding continuous functions and mappings. Then, an introduction to algebraic topology including homotopy theory, fundamental groups and covering spaces.

- Recommended Prep:
- Some acquaintance with metric space topology as taught in Mathematics 22a,b, 23a,b, 25a,b, 55a,b, 101, 102, or 112; and with groups as taught in Mathematics 101, 122 or 55a.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | For Undergraduate and Graduate Students |

*Alexander Kupers*

2020 Spring (4 Credits)

**
Schedule: **
WF 12:00 PM - 01:15 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Differential manifolds, smooth maps and transversality. Winding numbers, vector fields, index and degree. Differential forms, Stokes’ theorem, introduction to cohomology.

- Recommended Prep:
- Mathematics 22a,b, 23a,b or 25a,b or 55a,b or 112; or an equivalent background in mathematics.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | For Undergraduate and Graduate Students |

*Sebastien Picard*

2019 Fall (4 Credits)

**
Schedule: **
TR 01:30 PM - 02:45 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

The course is an introduction to Riemannian geometry with the focus (for the most part) being the Riemannian geometry of curves and surfaces in space where the fundamental notions can be visualized.

- Recommended Prep:
- Mathematics 19a,b or 21a,b or 22a,b or 23a or 25a or 55a (may be taken concurrently); or an equivalent background in mathematics.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

FAS: Course Level | For Undergraduate and Graduate Students |

*Brooke Ullery*

2020 Spring (4 Credits)

**
Schedule: **
MW 01:30 PM - 02:45 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Affine and projective spaces, plane curves, Bezout’s theorem, singularities and genus of a plane curve, Riemann-Roch theorem.

- Recommended Prep:
- Knowledge of the material in Mathematics 123

- Requirements:
- Prerequisite: Mathematics 123

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

FAS: Course Level | For Undergraduate and Graduate Students |

All: Cross Reg Availability | Available for Harvard Cross Registration |

*Sebastien Vasey*

2019 Fall (4 Credits)

**
Schedule: **
TR 12:00 PM - 01:15 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

An introduction to set theory: ordinals, cardinals, transfinite induction, the cumulative hierarchy, ZFC, the theory of the infinite, and the basics of independence.

- Recommended Prep:
- Familiarity with proofs. A previous mathematics course at the level of Mathematics 23ab, 25ab, 55ab, 101, 102, or 112 would be enough.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS Divisional Distribution | Science & Engineering & Applied Science |

FAS: Course Level | For Undergraduate and Graduate Students |

All: Cross Reg Availability | Available for Harvard Cross Registration |

*Paul Bamberg*

2020 Spring (4 Credits)

**
Schedule: **
TR 10:30 AM - 11:45 AM

**
Instructor Permissions: **
Instructor

**
Enrollment Cap: **
14

An introduction to finite groups, finite fields, finite geometry, finite topology, combinatorics, graph theory, and (for section 2 only) elementary algebraic topology. A recurring theme of the course is the symmetry group of the regular icosahedron. Elementary category theory will be introduced as a unifying principle. Taught in a seminar format: students will gain experience in presenting proofs at the blackboard.

- Course Notes:
- Covers material used in Computer Science 121 and Computer Science 124. Enrollment limited to16.

- Recommended Prep:
- For section 1: Mathematics 19b or 21b. Previous experience with proofs is not required. For section 2: Mathematics 23a or 25a or an equivalent background in mathematics that includes experience with proofs.

- Requirements:
- Not to be taken in addition to Computer Science 20, Mathematics 55a/b or Mathematics 122.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | For Undergraduate and Graduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Martin Nowak*

2019 Fall (4 Credits)

**
Schedule: **
TR 03:00 PM - 04:15 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

Introduces basic concepts of mathematical biology and evolutionary dynamics: evolution of genomes, quasi-species, finite and infinite population dynamics, chaos, game dynamics, evolution of cooperation and language, spatial models, evolutionary graph theory, infection dynamics, somatic evolution of cancer.

- Recommended Prep::
- Mathematics 19a,b or 21a,b or 22a,b or 23a,b or 25a,b or 55a,b; or an equivalent background in mathematics.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |

FAS: Course Level | For Undergraduate and Graduate Students |

*Sebastien Vasey*

2019 Fall (4 Credits)

**
Schedule: **
TR 10:30 AM - 11:45 AM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

An introduction to counting techniques and other methods in finite mathematics. Possible topics include: the inclusion-exclusion principle and Mobius inversion, graph theory, generating functions, Ramsey’s theorem and its variants, probabilistic methods.

- Recommended Prep:
- Prerequisites: familiarity with proofs. A previous mathematics course at the level of Mathematics 23ab, 25ab, 55ab, 101, 102, or 112 would be enough.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | For Undergraduate and Graduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

All: Cross Reg Availability | Available for Harvard Cross Registration |

*Christian Brennecke*

2020 Spring (4 Credits)

**
Schedule: **
MW 09:00 AM - 10:15 AM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

An introduction to probability theory. Discrete and continuous random variables; distribution and density functions for one and two random variables; conditional probability. Generating functions, weak and strong laws of large numbers, and the central limit theorem. Geometrical probability, random walks, and Markov processes.

- Recommended Prep:
- A previous mathematics course at the level of Mathematics 19ab, 21ab, or a higher number. For students from 19ab or 21ab, previous or concurrent enrollment in Math 101 or 102 or 112 may be helpful. Freshmen who did well in Math 22a, 23a, 25a or 55a fall term are also welcome to take the course.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS: Course Level | For Undergraduate and Graduate Students |

FAS Divisional Distribution | Science & Engineering & Applied Science |

*Joseph D. Harris*

2020 Spring (4 Credits)

**
Schedule: **
TR 01:30 PM - 02:45 PM

**
Instructor Permissions: **
None

**
Enrollment Cap: **
n/a

An interactive introduction to problem-solving with an emphasis on subjects with comprehensive applications. Each class will be focused around a group of questions with a common topic: logic, information, number theory, probability, and algorithms.

- Recommended Prep:
- Mathematics 19b or 21b or 22a,b or 23a; or an equivalent background in mathematics. More importantly, students should have a broad mathematical curiosity and be eager to brainstorm during in-class problem-solving sessions.

Additional Course Attributes:

Attribute | Value(s) |
---|---|

FAS: Course Level | Primarily for Undergraduate Students |

All: Cross Reg Availability | Available for Harvard Cross Registration |

FAS Divisional Distribution | Science & Engineering & Applied Science |