Upper Level Courses

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

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.

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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.

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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 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.

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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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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.

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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 class is an advanced undergraduate course and it is an introduction to algebraic number theory. Number fields are fundamental objects of study in number theory and algebraic geometry. They are ubiquitous in several areas of mathematics.We will first start by studying unique factorization of ideals in number fields, we will define the Picard group of the ring of integers of a number field and prove that it is a finite group. Our next object of study will be the structure of the units group, the structure of the Galois group, the Frobenius elements, and ramification theory. The final topic will be an introduction to analytic number theory: after introducing the Dedekind Zeta function, we will prove the class number formula. If time permits, we will introduce the adeles and the ideles.

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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Introduction to mathematical logic focusing on the fundamentals of first-order logic (language, axioms, completeness theorem, etc.) and the basic results of model theory (compactness), Lowenheim-Skolem, omitting typesetc.

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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.)

Prerequisite:

Mathematics 145A

Additional Course Attributes:

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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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:

In combinatorics and elsewhere one often encounters a “design”,or a collection of subsets of some finite set S whose elements areevenly distributed in a suitable sense; for instance the collection ofedges of a regular graph (each of whose vertices is contained in the samenumber of edges) or the collection of lines of a finite projective plane(any two of whose points are contained in a unique line).Of particular interest are designs symmetric under a large group ofpermutations of S. The consideration of specific classical designs andtheir symmetries will lead us to the general study of designs andpermutation groups. We conclude with the construction and detailedanalysis of the remarkable designs associated with Mathieu’s sporadicgroups of permutations of 12- and 24- element sets.

Recommended Prep:

The ability to write proofs and some knowledge of linear algebra will be needed.

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