Upper Level Courses

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

Course Notes:

Course Note: 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.

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

Course Notes:

May not ordinarily count for concentration in Mathematics.

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

Course Notes:

May not ordinarily count for concentration in Mathematics.

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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.

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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.

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This course is an introduction to abstract mathematical thought and proof techniques, via topics including set theory, group theory, and real analysis.

Course Notes::

Problem sessions (optional but highly encouraged) Mondays 12 - 1 pm. 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 offered in the Fall and Spring terms. Anti-Req: Not to be taken in addition to Mathematics 25a,b or 55a,b.

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.

Course Notes::

Problem sessions (optional but highly encouraged) Mondays 12 - 1 pm.

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This course is an introduction to abstract mathematical thought and proof techniques, via topics including set theory, group theory, and real analysis.

Course Notes:

Problem sessions (optional but highly encouraged) Mondays 12 - 1 pm. 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 offered in the Fall and Spring terms. Anti-Req: Not to be taken in addition to Mathematics 25a,b or 55a,b.

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.

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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 an equivalent background in mathematics.

Requirements:

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

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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.

Course Notes:

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

Requires:

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

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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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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.

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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.

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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.

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This is a second course in linear algebra, with an emphasis on understanding linear algebra at a more abstract level and learning to read and write proofs. Topics include real and complex vector spaces, linear transformations, and eigenvalues and eigenvectors.

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.

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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.

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

Course Notes:

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.

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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.

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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.

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This course is an introduction to algebraic number theory. Topics will include unique factorization in rings of integers, finiteness of the class group and the Dirichlet unit theorem. There will also be applications of these results to solve Diophantine equations. We will also study p-adic fields, and if time permits, adeles.

Recommended Prep:

Knowledge of the material in Mathematics 123.

Requirements:

Prerequisite: Mathematics 123.

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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.

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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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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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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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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.

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An introduction to first-order logic (the basis of mathematical reasoning) from a meta-mathematical point of view. Topics include: The axiomatic method, the principles of first-order logic, the meta-mathematical point of view, the central theorems (soundness and completeness), and the expressive limitations (Löwenheim-Skolem).

Course notes::

An additional hour of lecture will be scheduled independently.

Recommended Prep::

Familiarity with proof based mathematical reasoning at the level of Math 22a, 25a, 55a or Math 101

Course Requirements::

Anti-Requisite: Cannot be taken for credit if MATH 141A or PHIL 143 already complete or in progress.

Jointly Offered with::

Faculty of Arts & Sciences as PHIL 143

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Course Notes::

Familiarity with the material in either Math 143 or Math 141

Course Requirements::

Anti-requisite: Cannot be taken for credit if PHIL 144 already complete or in progress.

Jointly Offered with::

Faculty of Arts & Sciences as PHIL 144

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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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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:

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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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:

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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