MATH 10500-10600. Fundamental Mathematics I-II.

Students who place into this course must take it in their first year in the College. Must be taken for a quality grade. Both precalculus courses together count as only one elective. These courses do not meet the general education requirement in mathematical sciences. This two-course sequence covers basic precalculus topics. The Autumn Quarter course is concerned with elements of algebra, coordinate geometry, and elementary functions. The Winter Quarter course continues with algebraic, trigonometric, and exponential functions.

MATH 11200-11300. Studies in Mathematics I-II.

MATH 11200 AND 11300 cover the basic conceptual foundations of mathematics by examining the ideas of number and symmetry. MATH 11200 addresses number theory, including a study of the rules of arithmetic, integral domains, primes and divisibility, congruences, and modular arithmetic. MATH 11300’s main topic is symmetry and geometry, including a study of polygons, Euclidean construction, polyhedra, group theory, and topology. These courses emphasize the understanding of ideas and the ability to express them through rigorous mathematical arguments. While students may take MATH 11300 without having taken MATH 11200, it is recommended that MATH 11200 be taken first. Either course in this sequence meets the general education requirement in mathematical sciences. These courses are at the level of difficulty of the MATH 13100-13200-13300 calculus sequence.

MATH 13100-13200-13300. Elementary Functions and Calculus I-II-III.

This sequence provides the opportunity for students who are somewhat deficient in their precalculus preparation to complete the necessary background and cover basic calculus in three quarters. This is achieved through three regular one-hour class meetings and two mandatory one-and-one-half hour tutorial sessions each week. A class is divided into tutorial groups of about eight students each, and these meet with an undergraduate junior tutor for problem solving related to the course. MATH 13100 component of this sequence gives a careful treatment of limits and the continuity and differentiability of algebraic functions. Topics examined in MATH 13200 include applications of differentiation; exponential, logarithmic, and trigonometric functions; the definite integral and the fundamental theorem, and applications of the integral. In MATH 13300, subjects include more applications of the definite integral, infinite sequences and series, and Taylor expansions. Students are expected to understand the definitions of key concepts (i.e., limit, derivative, integral) and to be able to apply definitions and theorems to solve problems. In particular, all calculus courses require students to do proofs. Students completing MATH 13100-13200-13300 have a command of calculus equivalent to that obtained in 15100-15200-15300. Students may not take the first two quarters of this sequence for P/F grading. MATH 13100-13200 meets the general education requirement in mathematical sciences.

MATH 15100-15200-15300. Calculus I-II-III.

This is the regular calculus sequence in the department. Students entering this sequence are to have mastered appropriate precalculus material and, in many cases, have had some previous experience with calculus in high school or elsewhere. MATH 15100 undertakes a careful treatment of limits, the differentiation of algebraic and transcendental functions, and applications. Work in MATH 15200 is concerned with the mean value theorem, integration, techniques of integration, and applications of the integral. MATH 15300 deals with additional techniques and theoretical considerations of integration, infinite sequences and series, and Taylor expansions. All Autumn Quarter offerings of MATH 15100, 15200, and 15300 begin with a rigorous treatment of limits and limit proofs. Students may not take the first two quarters of this sequence for P/F grading. MATH 15100-15200 meets the general education requirement in mathematical sciences.

MATH 16100-16200-16300. Honors Calculus I-II-III.

Students may not take the first two quarters of this sequence for P/F grading. MATH 16100-16200 meets the general education requirement in mathematical sciences. MATH 16100-16200-16300 is an honors version of MATH 15100-15200-15300. A student with a strong background in the problem-solving aspects of one-variable calculus may, by suitable achievement on the Calculus Accreditation Exam, be permitted to register for MATH 16100-16200-16300. This sequence emphasizes the theoretical aspects of one-variable analysis and, in particular, the consequences of completeness in the real number system. MATH 16300 also includes an introduction to linear algebra. At least one section of this sequence is offered as an inquiry-based learning (IBL) course. Students interested in IBL should have an AP score of 5 on the BC Calculus exam and fluency in spoken English.

MATH 17500. Basic Number Theory. 100 Units.

This course covers basic properties of the integers following from the division algorithm, primes and their distribution, and congruences. Additional topics include existence of primitive roots, arithmetic functions, quadratic reciprocity, and transcendental numbers. The subject is developed in a leisurely fashion, with many explicit examples.

Terms Offered: Autumn
Prerequisite(s): MATH 16300 or 19900

MATH 17600. Basic Geometry. 100 Units.

This course covers advanced topics in geometry, including Euclidean geometry, spherical geometry, and hyperbolic geometry. We emphasize rigorous development from axiomatic systems, including the approach of Hilbert. Additional topics include lattice point geometry, projective geometry, and symmetry.

Terms Offered: Winter
Prerequisite(s): MATH 16300 or 19900

MATH 19520. Mathematical Methods for Social Sciences. 100 Units.

This course takes a concrete approach to the basic topics of multivariable calculus. Topics include a brief review of one-variable calculus, parametric equations, alternate coordinate systems, vectors and vector functions, partial derivatives, multiple integrals, and Lagrange multipliers.

Terms Offered: Autumn, Winter, Spring
Prerequisite(s): MATH 15300 or equivalent

MATH 19620. Linear Algebra. 100 Units.

This course takes a concrete approach to the basic topics of linear algebra. Topics include vector geometry, systems of linear equations, vector spaces, matrices and determinants, and eigenvalue problems.

Terms Offered: Autumn, Winter, Spring
Prerequisite(s): MATH 13300 or 15200

MATH 19900. Introduction to Analysis and Linear Algebra. 100 Units.

This course is intended for students who are making the transition from MATH 15300 to 20300, or for students who need more preparation in learning to read and write proofs. This course covers the fundamentals of theoretical mathematics and prepares students for upper-level mathematics courses beginning with MATH 20300. Topics include the construction of the real numbers, completeness and the least upper bound property, the topology of the real line, the structure of finite-dimensional vector spaces over the real and complex numbers. Students who are majoring or minoring in mathematics may not use both MATH 16300 and 19900 to meet program requirements.

Terms Offered: Autumn, Winter, Spring
Prerequisite(s): Superior performance on the Calculus Accreditation Exam or MATH 15300 or 13300

MATH 20000-20100. Mathematical Methods for Physical Sciences I-II.

This sequence is intended for students who are majoring in a department in the Physical Sciences Collegiate Division other than mathematics. MATH 20000 covers multivariable calculus, including the algebra and geometry of Euclidean space, differentiation and integration of functions of several variables, vector valued functions and the classical theorems of vector analysis (i.e., theorems of Green, Gauss, and Stokes). MATH 20100 introduces ordinary differential equations (e.g., first and second order linear differential equations, series solutions, and the Laplace transform) and complex analysis (i.e., basic properties of the complex plane and analytic functions through Cauchy’s theorem).

MATH 20300-20400-20500. Analysis in Rn I-II-III.

This three-course sequence is intended for students who plan to major in mathematics or who require a rigorous treatment of analysis in several dimensions. Both theoretical and problem solving aspects of multivariable calculus are treated carefully. Topics in MATH 20300 include metric spaces, the topology of Rn, compact sets, the geometry of Euclidean space, and limits and continuous mappings. MATH 20400 deals with partial differentiation, vector-valued functions, extrema, and the inverse and implicit function theorems. MATH 20500 is concerned with multiple integrals, line and surface integrals, and the theorems of Green, Gauss, and Stokes. This sequence is the basis for all advanced courses in analysis and topology.

MATH 20700-20800-20900. Honors Analysis in Rn I-II-III.

This highly theoretical sequence in analysis is intended for the most able students. Topics include the real number system, metric spaces, basic functional analysis, and the Lebesgue integral.

MATH 21100. Basic Numerical Analysis. 100 Units.

This course covers direct and iterative methods of solution of linear algebraic equations and eigenvalue problems. Topics include numerical differentiation and quadrature for functions of a single variable, approximation by polynomials and piece-wise polynomial functions, approximate solution of ordinary differential equations, and solution of nonlinear equations.

Terms Offered: Spring
Prerequisite(s): MATH 20000 or 20300

MATH 21200. Advanced Numerical Analysis. 100 Units.

This course covers topics similar to those of Math 21100 but at a more rigorous level. The emphasis is on proving all of the results. Previous knowledge of numerical analysis is not required. Programming is also not required. The course makes extensive use of the material developed in the analysis sequence (ending in Math 20500 or Math 20900) and provides an introduction to other areas of analysis such as functional analysis and operator theory.

Terms Offered: Autumn
Prerequisite(s): MATH 20500 or 20900

MATH 22000. Introduction to Mathematical Methods in Physics. 100 Units.

This course, with concurrent enrollment in PHYS 13300, is required of students who plan to major in physics. Topics include infinite series and power series, complex numbers, linear equations and matrices, partial differentiation, multiple integrals, vector analysis, and Fourier series. Applications of these methods include Maxwell's equations, wave packets, and coupled oscillators.

Terms Offered: Spring
Prerequisite(s): MATH 15200 or 16200, and PHYS 13200

MATH 24100. Topics in Geometry. 100 Units.

This course focuses on the interplay between abstract algebra (group theory, linear algebra, and the like) and geometry. Several of the following topics are covered: affine geometry, projective geometry, bilinear forms, orthogonal geometry, and symplectic geometry.

Terms Offered: Spring
Prerequisite(s): MATH 25500 or 25800
Note(s): This course is offered in alternate years.

MATH 24200. Algebraic Number Theory. 100 Units.

Topics include factorization in Dedekind domains, integers in a number field, prime factorization, basic properties of ramification, and local degree.

Terms Offered: Spring
Prerequisite(s): MATH 25500 or 25800

MATH 24300. Introduction to Algebraic Curves. 100 Units.

This course covers the projective line and plane curves, both affine and projective. We also study conics and cubics, as well as the group law on the cubic. Abstract curves associated to function fields of one variable are discussed, along with the genus of a curve and the Riemann-Roch theorem. Curves of low genus are emphasized. Although the formal prerequisite is MATH 25500 or 25800, MATH 25600 or 25900 is strongly recommended.

Terms Offered: Spring
Prerequisite(s): MATH 25500 or 25800, or consent of instructor
Note(s): This course is offered in alternate years.

MATH 25400-25500-25600. Basic Algebra I-II-III.

This sequence covers groups, subgroups, and permutation groups; rings and ideals; fields; vector spaces, linear transformations and matrices, and modules; and canonical forms of matrices, quadratic forms, and multilinear algebra.

MATH 25700-25800-25900. Honors Basic Algebra I-II-III.

This sequence is an accelerated version of MATH 25400-25500-25600 that is open only to students who have achieved a B- or better in prior mathematics courses. Topics include the theory of finite groups, commutative and noncommutative ring theory, modules, linear and multilinear algebra, and quadratic forms. We also cover basic field theory, the structure of p-adic fields, and Galois theory.

MATH 26200. Point-Set Topology. 100 Units.

This course examines topology on the real line, topological spaces, connected spaces and compact spaces, identification spaces and cell complexes, and projective and other spaces. With MATH 27400, it forms a foundation for all advanced courses in analysis, geometry, and topology.

Terms Offered: Winter
Prerequisite(s): MATH 20300 or 20700, and 25400 or 25700

MATH 26300. Introduction to Algebraic Topology. 100 Units.

Topics include the fundamental group of a space; Van Kampen's theorem; covering spaces and groups of covering transformation; existence of universal covering spaces built up out of cells; and theorems of Gauss, Brouwer, and Borsuk-Ulam.

Terms Offered: Spring
Prerequisite(s): MATH 26200

MATH 26700. Introduction to Representation Theory of Finite Groups. 100 Units.

Topics include group algebras and modules, semisimple algebras and the theorem of Maschke; characters, character tables, orthogonality relations and calculation; and induced representations and characters. Applications to permutation groups and solvability of groups are also included.

Terms Offered: Autumn
Prerequisite(s): MATH 25900 or 25600

MATH 26800. Introduction to Commutative Algebra. 100 Units.

Topics include basic definitions and properties of commutative rings and modules, Noetherian and Artinian modules, exact sequences, Hilbert basis theorem, tensor products, localizations of rings and modules, associated primes and primary decomposition, Artin-Rees Lemma, Krull intersection theorem, completions, dimension theory of Noetherian rings, integral extensions, normal domains, Dedekind domains, going up and going down theorems, dimension of finitely generated algebras over a field, Affine varieties, Hilbert Nullstellensatz, dimension of affine varieties, product of affine varieties, and the dimension of intersection of subvarieties.

Terms Offered: Winter
Prerequisite(s): MATH 25900 or 25600

MATH 27000. Basic Complex Variables. 100 Units.

Topics include complex numbers, elementary functions of a complex variable, complex integration, power series, residues, and conformal mapping.

Terms Offered: Autumn, Spring
Prerequisite(s): MATH 20500 or 20900

MATH 27200. Basic Functional Analysis. 100 Units.

Topics include Banach spaces, bounded linear operators, Hilbert spaces, construction of the Lebesgue integral, Lp-spaces, Fourier transforms, Plancherel's theorem for Rn, and spectral properties of bounded linear operators.

Terms Offered: Winter
Prerequisite(s): MATH 20900 or 27000

MATH 27300. Basic Theory of Ordinary Differential Equations. 100 Units.

This course covers first-order equations and inequalities, Lipschitz condition and uniqueness, properties of linear equations, linear independence, Wronskians, variation-of-constants formula, equations with constant coefficients and Laplace transforms, analytic coefficients, solutions in series, regular singular points, existence theorems, theory of two-point value problem, and Green's functions.

Terms Offered: Winter
Prerequisite(s): MATH 27000 or PHYS 22100

MATH 27400. Introduction to Differentiable Manifolds and Integration on Manifolds. 100 Units.

Topics include exterior algebra; differentiable manifolds and their basic properties; differential forms; integration on manifolds; and the theorems of Stokes, DeRham, and Sard. With MATH 26200, this course forms a foundation for all advanced courses in analysis, geometry, and topology.

Terms Offered: Spring
Prerequisite(s): MATH 26200

MATH 27500. Basic Theory of Partial Differential Equations. 100 Units.

This course covers classification of second-order equations in two variables, wave motion and Fourier series, heat flow and Fourier integral, Laplace's equation and complex variables, second-order equations in more than two variables, Laplace operators, spherical harmonics, and associated special functions of mathematical physics.

Terms Offered: Spring
Prerequisite(s): MATH 27300

MATH 27700-27800. Mathematical Logic I-II.

Mathematical Logic I-II

MATH 28000. Introduction to Formal Languages. 100 Units.

This course is a basic introduction to computability theory and formal languages. Topics include automata theory, regular languages, context-free languages, and Turing machines.

Terms Offered: Autumn
Prerequisite(s): CMSC 15300, or MATH 19900 or 25500
Equivalent Course(s): CMSC 28000

MATH 28100. Introduction to Complexity Theory. 100 Units.

Computability topics are discussed (e.g., the s-m-n theorem and the recursion theorem, resource-bounded computation). This course introduces complexity theory. Relationships between space and time, determinism and non-determinism, NP-completeness, and the P versus NP question are investigated.

Terms Offered: Spring
Prerequisite(s): CMSC 27100, or MATH 19900 or 25500; and experience with mathematical proofs
Equivalent Course(s): CMSC 28100

MATH 28410. Honors Combinatorics. 100 Units.

Experience with mathematical proofs. Methods of enumeration, construction, and proof of existence of discrete structures are discussed in conjunction with the basic concepts of probability theory over a finite sample space. Enumeration techniques are applied to the calculation of probabilities, and, conversely, probabilistic arguments are used in the analysis of combinatorial structures. Other topics include basic counting, linear recurrences, generating functions, Latin squares, finite projective planes, graph theory, Ramsey theory, coloring graphs and set systems, random variables, independence, expected value, standard deviation, and Chebyshev's and Chernoff's inequalities.

Instructor(s): L. Babai     Terms Offered: Winter
Prerequisite(s): MATH 19900 or 25400, or CMSC 27100, or consent of instructor
Note(s): This course is offered in alternate years.

MATH 29200. Chaos, Complexity, and Computers. 100 Units.

This course presents the mathematical bases for the complex, scale-independent behavior seen in chaotic dynamics and fractal patterns.  It illustrates these principles from physical and biological phenomena.  It explores these behaviors concretely using extensive computer simulation exercises, thus developing simulation and data analysis skills. L.

Instructor(s): Staff     Terms Offered: Autumn
Prerequisite(s): PHYS 13300 or 14300; PHYS 25000 or prior programming experience.
Equivalent Course(s): CMSC 27900,PHYS 25100

MATH 29700. Proseminar in Mathematics. 100 Units.

Consent of instructor and departmental counselor. Students are required to submit the College Reading and Research Course Form. Must be taken for a quality grade.

Terms Offered: Autumn, Winter, Spring
Prerequisite(s): Completion of general education mathematics sequence

MATH 30200-30300. Computability Theory I-II.

The courses in this sequence are offered in alternate years.

MATH 30500. Computability and Complexity Theory. 100 Units.

Part one of this course consists of models for defining computable functions: primitive recursive functions, (general) recursive functions, and Turing machines; the Church-Turing Thesis; unsolvable problems; diagonalization; and properties of computably enumerable sets. Part two of this course deals with Kolmogorov (resource bounded) complexity: the quantity of information in individual objects. Part three of this course covers functions computable with time and space bounds of the Turing machine: polynomial time computability, the classes P and NP, NP-complete problems, polynomial time hierarchy, and P-space complete problems.

Instructor(s): A. Razborov     Terms Offered: Winter
Prerequisite(s): Consent of department counselor and instructor
Equivalent Course(s): CMSC 38500

MATH 30900-31000. Model Theory I-II.

MATH 30900 covers completeness and compactness; elimination of quantifiers; omission of types; elementary chains and homogeneous models; two cardinal theorems by Vaught, Chang, and Keisler; categories and functors; inverse systems of compact Hausdorf spaces; and applications of model theory to algebra. In MATH 31000, we study saturated models; categoricity in power; the Cantor-Bendixson and Morley derivatives; the Morley theorem and the Baldwin-Lachlan theorem on categoricity; rank in model theory; uniqueness of prime models and existence of saturated models; indiscernibles; ultraproducts; and differential fields of characteristic zero.

MATH 31200-31300-31400. Analysis I-II-III.

Analysis I-II-III

MATH 31700-31800-31900. Topology and Geometry I-II-III.

Topology and Geometry I-II-III

MATH 32500-32600-32700. Algebra I-II-III.

Algebra I-II-III

MATH 37500. Algorithms in Finite Groups. 100 Units.

We consider the asymptotic complexity of some of the basic problems of computational group theory. The course demonstrates the relevance of a mix of mathematical techniques, ranging from combinatorial ideas, the elements of probability theory, and elementary group theory, to the theories of rapidly mixing Markov chains, applications of simply stated consequences of the Classification of Finite Simple Groups (CFSG), and, occasionally, detailed information about finite simple groups. No programming problems are assigned.

Instructor(s): L. Babai     Terms Offered: Spring
Prerequisite(s): Consent of department counselor. Linear algebra, finite fields, and a first course in group theory (Jordan-Holder and Sylow theorems) required; prior knowledge of algorithms not required
Note(s): This course is offered in alternate years.
Equivalent Course(s): CMSC 36500

MATH 38300. Numerical Solutions to Partial Differential Equations. 100 Units.

This course covers the basic mathematical theory behind numerical solution of partial differential equations. We investigate the convergence properties of finite element, finite difference and other discretization methods for solving partial differential equations, introducing Sobolev spaces and polynomial approximation theory. We emphasize error estimators, adaptivity, and optimal-order solvers for linear systems arising from PDEs. Special topics include PDEs of fluid mechanics, max-norm error estimates, and Banach-space operator-interpolation techniques.

Instructor(s): L. R. Scott     Terms Offered: Spring. This course is offered in alternate years.
Prerequisite(s): Consent of department counselor and instructor
Equivalent Course(s): CMSC 38300

MATH 38509. Advanced Topics: Probability. 100 Units.

This course will include the following topics: continuous-time martingales, Brownian motion, Levy processes, Ito integral and stochastic calculus, and stochastic differential equations and diffusions. Topics may vary.

Terms Offered: Spring
Equivalent Course(s): STAT 38500