Upper Level Courses

Mathematics concentrators in their final two undergraduate semesters can take this course to work individually on their senior thesis.

Course Notes:

Limited to candidates in Mathematics who obtain the permission of both the faculty member under whom they want to work and the Director of Undergraduate Studies. May not count for concentration in Mathematics without special permission from the Director of Undergraduate Studies. Graded sat/unsat only.

Additional Course Attributes:

Programs of directed study supervised by a person approved by the Department.

Course Notes:

May not ordinarily count for concentration in Mathematics.

Additional Course Attributes:

Programs of directed study supervised by a person approved by the Department.

Course Notes:

May not ordinarily count for concentration in Mathematics.

Additional Course Attributes:

Supervised small group tutorial. Topics to be arranged.

Course Notes:

May be repeated for course credit with permission from the Director of Undergraduate Studies. Only one tutorial may count for concentration credit.

Additional Course Attributes:

Supervised small group tutorial. Topics to be arranged.

Course Notes:

May be repeated for course credit with permission from the Director of Undergraduate Studies. Only one tutorial may count for concentration credit.

Additional Course Attributes:

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:

An introduction to abstract mathematical thought and proof techniques, via topics including set theory, group theory, analysis, and topology.

Course Notes:

Students who have already taken Mathematics 25a,b or 55a,b should not take this course for credit. Ordinarily, students who have already taken Mathematics 22a,b or 23a,b should not take this course for credit, but they may do so with the instructor’s permission. This course is given fall term and repeated spring term.

Recommended Prep:

An interest in mathematical reasoning. Acquaintance with algebra, geometry and/or calculus is desirable.

Requirements:

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

Additional Course Attributes:

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:

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.

Additional Course Attributes:

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:

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:

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.

Additional Course Attributes:

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::

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.

Additional Course Attributes:

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.

Additional Course Attributes:

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:

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.

Additional Course Attributes:

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:

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:

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. (See the course website for plans to accommodate diverse time zones of students in this course.)

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:

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:

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:

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:

An introduction to large cardinals and their inner models, with special emphasis on Woodin’s recent advances toward finding an ultimate version of Godel’s L. Topics include: Weak extender models, the HOD Dichotomy Theorem, and the HOD Conjecture. (After the first lecture, the course will arrange meeting times to accommodate all students.)

Requirements:

Prerequisite: Mathematics 145A

Additional Course Attributes:

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:

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:

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:

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:

Derived categories is the right framework to view derived functors (Ext, Tor, higher direct images, cohomology etc.) in geometry and algebra. A fruitful, modern perspective is to view derived categories themselves as invariants of the objects (varieties, groups, algebras etc.) that one is interested in. In this seminar style class, we will start from the very beginning of the story with basic notions of categories and functors, equipping ourselves with an extensive list of examples. We will then study the notion of additive and abelian categories where one can do homological algebra. We will perform some important calculations of derived functors, including sheaf and group cohomology and equip ourselves with important tools such as spectral sequences. From this we will place everything in context using the language of triangulated and derived categories. In the last part of the class we will explore some “modern classics” from algebra and algebraic geometry using the theory we developed.

Recommended Prep:

A first course in algebraic geometry and algebraic topology.

Additional Course Attributes: