# Upper Level Courses

*Curtis McMullen*

2022 Fall (4 Credits)

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

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Instructor Permissions: **
None

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Enrollment Cap: **
n/a

This course provides an introduction to conceptual and axiomatic mathematics, the writing of proofs, and mathematical culture, with sets, groups and knots as the main topics.

- Course Notes::
- Course Requirements: An interest in mathematical reasoning. Some acquaintance with algebra, geometry and/or calculus is desirable. Students who have already taken Math 23a/b, 25a/b or 55a/b should*not* take this course.

:

*Janet Chen*

2023 Spring (4 Credits)

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

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Instructor Permissions: **
Instructor

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Enrollment Cap: **
n/a

This course provides an introduction to conceptual and axiomatic mathematics, the writing of proofs, and mathematical culture, with sets, groups and real analysis as the main topics.

:

*Niki Myrto Mavraki*

2023 Spring (4 Credits)

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

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Instructor Permissions: **
None

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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.

:

*Ana Balibanu*

2023 Spring (4 Credits)

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

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Instructor Permissions: **
None

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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.

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*Horng-Tzer Yau*

2022 Fall (4 Credits)

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

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Instructor Permissions: **
None

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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.

:

*Paul Bamberg (he)*

2022 Fall (4 Credits)

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

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Instructor Permissions: **
None

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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.

:

*Paul Bamberg (he)*

2023 Spring (4 Credits)

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

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Instructor Permissions: **
None

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Enrollment Cap: **
n/a

A self-contained treatment of the theory of probability and random processes with specific application to the theory of option pricing. Topics: axioms for probability, calculation of expectation by means of Lebesgue integration, conditional probability and conditional expectation, martingales, random walks and Wiener processes, and the Black-Scholes formula for option pricing. Students will work in small groups to investigate applications of the theory and to prove key results.

:

*Assaf Shani*

2023 Spring (4 Credits)

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

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Instructor Permissions: **
None

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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.

:

*Ana Balibanu*

2022 Fall (4 Credits)

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

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Instructor Permissions: **
None

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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: Geometry, 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 22a, 23a or 25a or 55a.

:

*Niki Myrto Mavraki*

2022 Fall (4 Credits)

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

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Instructor Permissions: **
None

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Enrollment Cap: **
n/a

The theory of groups and group actions, rings, ideals and factorization.

- 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.

:

*Curtis McMullen*

2023 Spring (4 Credits)

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

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Instructor Permissions: **
None

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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

:

*Barry Mazur*

2023 Spring (4 Credits)

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

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Instructor Permissions: **
None

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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 22a or 23a or 25a or 101 or 122; or 55a which can be taken concurrently; or an equivalent experience and comfort level with abstract mathematics.

:

*Salim Tayou*

2023 Spring (4 Credits)

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

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Instructor Permissions: **
None

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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.

:

*Paul Bamberg (he)*

2023 Spring (4 Credits)

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

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Instructor Permissions: **
None

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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.

- Class Notes::
- This course presents the classical geometries, the affine, projective, Euclidean, spherical and hyperbolic geometries. These are viewed from many different perspectives, some historical and some very modern (in particular, with regards to recent discoveries in finite affine and projective geometry.) The course will also have an emphasis on analyzing and writing proofs in geometry. Note that this courses will be very different from the Spring 2022 version of Math 1b; in particular, there will be no Greek or R-script in this course. The course will focus solely on the elegant mathematics. (Those interested in a version of Math 130 with an opportunity to read Euclid in the original Greek might consider enrolling in the Harvard Summer School courses S-Math 139 Reading elements of Euclid in Greek.) There is an additional Friday, 1:30-2:45 meeting of the course for students who wish to learn Greek along with mathematics.

- 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.

:

*Laura DeMarco*

2022 Fall (4 Credits)

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

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Instructor Permissions: **
None

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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.

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*Michael Hopkins*

2023 Spring (4 Credits)

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

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Instructor Permissions: **
None

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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.

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*Puskar Mondal*

2022 Fall (4 Credits)

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

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Instructor Permissions: **
None

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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.

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*Mihnea Popa*

2023 Spring (4 Credits)

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

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Instructor Permissions: **
None

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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.

:

*Assaf Shani*

2022 Fall (4 Credits)

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

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Instructor Permissions: **
None

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Enrollment Cap: **
n/a

Introduction to the incompleteness phenomenon, covering the incompleteness theorems and the basic results of recursion theory.

:

*Paul Bamberg (he)*

2022 Fall (4 Credits)

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

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Instructor Permissions: **
Instructor

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Enrollment Cap: **
n/a

An introduction to finite groups, finite fields, quaternions, finite geometry, finite topology, combinatorics, and graph theory. A recurring theme of the course is the symmetry group of the regular icosahedron. 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.

- Recommended Prep:
- Mathematics 19b, 21b, 22a, 23a, or 25a. Previous experience with proofs is recommended but not required.

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

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*Cesar Cuenca*

2023 Spring (4 Credits)

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

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Instructor Permissions: **
None

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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.

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*Cesar Cuenca*

2022 Fall (4 Credits)

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

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Instructor Permissions: **
None

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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.

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*Paul Bamberg (he)*

2022 Fall (4 Credits)

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Schedule: **
F 03:00 PM - 05:00 PM

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Instructor Permission: **
None

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Enrollment Cap: **
n/a

Presents the probability theory and statistical principles which underly the tools that are built into the open- source programming language R. Each class presents the theory behind a statistical tool, then shows how the implementation of that tool in R can be used to analyze real-world data. The emphasis is on modern bootstrapping and resampling techniques, which rely on computational power. Topics include discrete and continuous probability distributions, permutation tests, the central limit theorem, chi-square and Student t tests, linear regression, and Bayesian methods.

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*Joseph D. Harris*

2023 Spring (4 Credits)

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

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Instructor Permissions: **
None

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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.

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*Philip Wood*

2023 Spring (4 Credits)

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

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Instructor Permissions: **
None

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Enrollment Cap: **
n/a

How can a computer check if a mathematical proof is completely and truly correct? This course will be an introduction into the world of formal verification of mathematics, starting with basic examples of sets and natural numbers, and moving on to more advanced mathematics. We will work with Lean, an open-source programming language for formal verification that has been used to verify large portions of mathematics, including a few examples reaching all the way to the forefront of current mathematics research.

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