UC Santa Barbara General CatalogUniversity of California, Santa Barbara

Mathematics

Division of Mathematical, Life, and Physical Sciences
South Hall 6607
Undergraduate e-mail: ugrad@math.ucsb.edu
Graduate e-mail: math-gradinfo@math.ucsb.edu
Website: www.math.ucsb.edu
Department Chair: David R. Morrison


 

Some courses displayed may not be offered every year.
For actual course offerings by quarter, please consult the Schedule of Classes - Class Search or GOLD (for current students).

Mathematics
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Collapse Courses Lower Division 
MATH 3A. Calculus with Applications, First Course
(4) STAFF
Prerequisite: Algebra Diagnostic Test.
Differential Calculus including analytic geometry, functions and limits, derivatives, techniques and applications of differentiation, logarithmic and trigonometric functions.
MATH 3B. Calculus with Applications, Second Course
(4) STAFF
Prerequisite: Mathematics 3A with a minimum grade of C.
Integral calculus including definite and indefinite integrals, techniques of integration, with applications in mathematics and physics.
MATH 3C. Differential Equations and Linear Algebra, First Course
(4) STAFF
Prerequisite: Mathematics 3B with a minimum grade of C.
First order ODEs including direction fields, separation of variables, first order linear equations, growth and decay, nonlinear models. Linear algebra including systems of linear equations, matrix inverses, determinants, vector spaces and subspaces, basis and dimension.
MATH 3H. Honors Seminar-Calculus
(1) STAFF
Prerequisite: Concurrent enrollment in Mathematics 3A or 3B or 3BI or 3C or 3CI.
Emphasizing fundamental concepts and applications. Intended for highly motivated and well prepared students.
MATH 4A. Linear Algebra with Applications
(4) STAFF
Prerequisite: Math 3B or 3BI with a minimum grade of C.
Systems of linear equations, matrix algebra, determinants, vector spaces and subspaces, basis and dimension, linear transformations, eigenvalues and eigenvectors, diagonalization, and orthogonality.
MATH 4AI. Inquiry Based Linear Algebra
(4) STAFF
Prerequisite: Mathematics 3B or Mathematics 3BI with a minimum grade of C.
Honors version of Mathematics 4A. Mathematical inquiry course is developed through problem solving and discovery.
MATH 4B. Differential Equations
(4) STAFF
Prerequisite: Math 4A or Math 4AI with a minimum grade of C.
First and second order differential equations, separation of variables, linear differential equations, systems of first order equations, nonlinear differential equations and stability.
MATH 4BI. Inquiry Based Differential Equations
(4) STAFF
Prerequisite: Mathematics 3C or 3CI with a minimum grade of C.
Honors version of Mathematics 4B. Mathematical inquiry course is developed through problem solving and discovery.
MATH 5A. Differential Equations and Linear Algebra, Second Course
(4) STAFF
Prerequisite: Mathematics 3C or 3CI with a grade of C or better.
Second order linear ODEs, linear transformations including eigenvalues, eigenvectors and diagonalization. Linear systems of ODEs. Nonlinear systems and linearization.
MATH 5B. Vector Calculus with Applications, First Course
(4) STAFF
Prerequisite: Mathematics 5A or Mathematics 5AI with a grade of C or better.
Calculus of functions of several variables, vector-valued functions of one variable, scalar and vector Fields, integration along paths, double and triple integrals, integration over surfaces, properties, and applications of integrals, classical integration theorems of vector calculus.
MATH 5BI. Inquiry Based Calculus IV
(4) STAFF
Prerequisite: Mathematics 5A or Mathematics 5AI with a minimum grade of C.
Honors version of Mathematics 5B. Mathematical inquiry course is developed through problem solving and discovery.
MATH 5C. Vector calculus with Applications, Second Course
(4) STAFF
Prerequisite: Mathematics 5B or 5BI with a grade of C or better.
Integral theorems of vector calculus (continuation), infinite series, Fourier series, integrals and transforms, partial differential equations.
MATH 6A. Vector Calculus with Applications, First Course
(4) STAFF
Prerequisite: Mathematics 4A (or 4AI) or Mathematics 5A (or 5AI) with a letter grade of C or better.
Calculus of functions of several variables, vector-valued functions of one variable, scalar and vector fields, integration along paths, double and triple integrals, integration over surfaces, properties, and applications of integrals, and classical integration theorems of vector calculus.
MATH 6AI. Inquiry Based Calculus IV
(4) STAFF
Prerequisite: Mathematics 4A (or 4AI) or Mathematics 5A (or 5AI) with a letter grade of C or better.
Honors version of Mathematics 6A. Mathematical inquiry course is developed through problem solving and discovery.
MATH 6B. Vector calculus with Applications, Second Course
(4) STAFF
Prerequisite: Mathematics 5B or 5BI with a minimum grade of C; or, Mathematics 4B (or 4BI) and Mathematics 6A (or 6AI), each with a minimum grade of C.
Integral theorems of vector calculus (continuation), infinite series, Fourier series, integrals and transforms, partial differential equations.
MATH 7H. Honors Seminar-Calculus
(1) STAFF
Prerequisite: Concurrent enrollment in Mathematics 3A or 3B or 3BI or 3C or 4A or 4AI or 4B or 4BI or 5A or 5AI or 6A or 6AI or 6B.
Emphasizing fundamental concepts and applications. Intended for highly motivated and well prepared students.
MATH 8. Transition to Higher Mathematics
(5) STAFF
Prerequisite: Mathematics 3B or 3C or 3CI or 4A or 4AI or 4B or 4BI or 5A or 5AI or 5B or 5BI or 5C or 6A or 6AI or 6B with a grade of B or better.
Introduction to the elements of propositional logic, techniques of mathematical proof, and fundamental mathematical structures, including sets, functions, relations, and other topics as time permits. Mastery of this material is essential for students planning to major in mathematics.
MATH 15. Precalculus
(4) STAFF
Prerequisite: Algebra Diagnostic Test.
A function approach integrating algebra and trigonometry. Topics include: one-on-one and onto functions; inverse functions; properties and graphs of polynomial, rational, exponential, and logarithmic functions; properties and graphs of inverse trigonometric identities and trignometric equations.
MATH 34A. Calculus for Social and Life Sciences
(4) STAFF
Introduction to differential and integral calculus with applications to modeling in the biological sciences.
MATH 34B. Calculus for Social and Life Sciences
(4) STAFF
Prerequisite: Mathematics 3A or 3AS or 34A with a Grade of C or better.
Continued study of differential and integral calculus with differential andintegral calculus with applications. Introduction to mathematical modeling with differential equations. Calculus of several variables including an introduction to partial derivatives.
MATH 94. Group Studies in Mathematics
(1-4) STAFF
Prerequisite: Consent of instructor.
Lectures and discussions on special topics.
Collapse Courses Upper Division 
MATH 100A. Mathematics for Elementary Teaching, I
(3) STAFF
Prerequisite: Upper-division standing.
This class teaches ways to think about and explain elementary school mathematics. Topics include: cultural and base-n number systems, algorithms, elementary number theory, probability, and graphing.
MATH 100B. Mathematics for Elementary Teaching, II
(3) STAFF
Prerequisite: Mathematics 100A.
Completes the explanation of elementary school mathematics by discussing geometry and algebra. Discusses the pedagogy with the California mathematics framework, the NCTM standards, and "replacement units".
MATH 101A. Classical Number Systems
(4) STAFF
Prerequisite: Mathematics 3A; and Mathematics 8.
Especially suitable for prospective teachers. A conceptual rather than an axiomatic development starting with the natural numbers and progressing through the integral, rational, real, and complex number systems. The historical implications of these developments in number systems.
MATH 101B. Mathematical Systems
(4) STAFF
Prerequisite: Mathematics 101A.
Especially suitable for prospective teachers. The theory of operations within rings and fields and the foundations of the real number system. Ideals, quotient rings, and factorization theorems. The history and the historical implications of these developments in mathematical systems.
MATH 102A. Modern Euclidean and Noneuclidean Geometry
(4) STAFF
Prerequisite: Mathematics 3B.
Especially suitable for prospective teachers. Topics in plane and solid geometry. The axioms of pure, euclidean, projective, and noneuclidean geometry. Transformational geometry (isometries, dilitations, involutions, perspectivities, and projectivities). The history and the historical implications of these developments in geometry.
MATH 102B. Modern Euclidean and Noneuclidean Geometry
(4) STAFF
Prerequisite: Mathematics 102A.
Especially suitable for prospective teachers. Topics in plane and solid geometry. The axioms of pure, euclidean, projective, and noneuclidean geometry. Transformational geometry (isometries, dilitations, involutions, perspectivities, and projectivities). The history and the historical implications of these developments in geometry.
MATH 103. Introduction to Group Theory
(4) STAFF
Prerequisite: Mathematics 8.
Intended primarily for prospective teachers. Permutation groups, cyclic groups, theory of finite groups, group homomorphisms and isomorphisms, and Abelian groups. Applications to number theory and geometry.
MATH 104A. Introduction Into Numerical Analysis
(4) STAFF
Prerequisite: Mathematics 4B or 4BI, 6A or 6AI, and 6B, or 5A or 5AI, 5B or 5BI and 5C; and, Computer Science 5AA-ZZ or 10 or 8 or 16 or Engineering 3 with a grade of C or above.
Numerical methods for the solution of nonlinear equations (Newton method), for integration (quadrature formulas and composite integration), and for the initial value problem for ordinary differential equations (Euler and Kutta methods).
MATH 104B. Numerical Analysis
(4) STAFF
Prerequisite: Mathematics 104A.
Numerical methods for the solution of systems of linear equations (direct and iteractive methods), and the finite difference methods for boundary value problems for (ordinary and partial) differential equations.
MATH 104C. Advanced Topics in Numerical Analysis
(4) STAFF
Prerequisite: Mathematics 104B.
Topics in approximation theory; numerical methods for finding eigenvalues of a matrix; and advanced topics in numerical methods for ordinary and partial differential equations.
MATH 108A. Introduction to Linear Algebra
(4) STAFF
Prerequisite: Mathematics 3C or 3CI or 4A or 4AI, 4B or 4BI or 5A or 5AI; and 8.
Abstract vector spaces subspaces. Span and linear independence. Basis and dimension. Linear maps. Eigenvalues and eigenvectors.
MATH 108B. Introduction to Linear Algebra
(4) STAFF
Prerequisite: Mathematics 108A.
Diagonalization, inner product spaces, projections, least-squares approximations, invariant factors and elementary divisors, canonical forms, topics from advanced matrix theory, applied linear algebra, and group representation theory.
MATH 108C. Introduction to Linear Algebra
(4) STAFF
Prerequisite: Mathematics 108B.
Diagonalization, inner product spaces, projections, least-squares approximations, invariant factors and elementary divisors, canonical forms, topics from advanced matrix theory, applied linear algebra, and group representation theory.
MATH 109A. Introduction to Mathematical Logic
(4) STAFF
Prerequisite: Mathematics 8 or Computer Science 40.
An introduction to mathematical logic with applications in computer scienceand mathematics. Topics include propositional and predicate calculi; models; proof systems, decidability and undecidability, automated theorem-proving, unification, logic programming, and program verification.
MATH 111A. Introduction to Abstract Algebra
(4) STAFF
Prerequisite: Mathematics 108A.
An introduction to algebraic structures with an emphasis on groups.
MATH 111B. Abstract Algebra
(4) STAFF
Prerequisite: Mathematics 111A.
Rings, fields, Galois theory.
MATH 111C. Abstract Algebra
(4) STAFF
Prerequisite: Mathematics 111B.
Rings, fields, Galois theory.
MATH 113. Non-euclidean Geometry
(4) STAFF
Prerequisite: Mathematics 8.
An introduction to hyperbolic geometry with some discussion of other non-euclidean systems.
MATH 115A. Introduction to Number Theory
(4) STAFF
Prerequisite: Mathematics 8.
Divisibility, congruences, primitive roots an indices, quadratic residues and the quadratic reciprocity law, number-theoretic functions, Diophantine equations, the distribution of primes, number-theorhetic methods in cryptography, quadratic forms, continued fractions, and the approximation of real numbers, algebraic number theory, partitions.
MATH 115B. Introduction to Number Theory
(4) STAFF
Prerequisite: Mathematics 115A.
Divisibility, congruences, primitive roots and indices, quadratic residues and the quadradic reciprocity law, number-theorhetic functions, Diophantine equations, the distribution of primes, number-theorhetic methods in cryptography, quadratic forms, continued fractions, and the approximation of real numbers, algebraic number theory, partitions.
MATH 116. Combinatorial Analysis
(4) STAFF
Prerequisite: Mathematics 8.
Elementary counting principles, binomial coefficients, generating functions, recurrence relations, the principle of inclusion and exclusion, distributions and partitions, systems of distinct representatives, applications to computation.
MATH 117. Methods of Analysis
(4) STAFF
Prerequisite: Mathematics 8.
Introduction to methods of proof in analysis. topics include limits, sequences and series, continuity, compactness, as well as other topics. This course is intended to follow Mathematics 8 and to introduce students to the level of sophistication of upper-division mathematics.
MATH 118A. Introduction to Real Analysis
(4) STAFF
Prerequisite: Mathematics 5A or 5AI and 5B or 5BI or 4B or 4BI and 6A or 6AI and 108A-B and 117.
The real number system, elements of set theory, continuity, differentiability, Riemann integral, implicit function theorems, convergence processes, and special topics.
MATH 118B. Introduction to Real Analysis
(4) STAFF
Prerequisite: Mathematics 118A.
The real number system, elements of set theory, continuity, differentiability, Riemann integral, implicit function theorems, convergence processes, and special topics.
MATH 118C. Introduction to Real Analysis
(4) STAFF
Prerequisite: Mathematics 118B.
The real number system, elements of set theory, continuity, differentiability, Riemann integral, implicit function theorems, convergence processes, and special topics.
MATH 119A. Ordinary Differential Equations
(4) STAFF
Prerequisite: Mathematics 5A or 5AI and 5B or 5BI or 4B or 4BI and 6A or 6AI.
Existence, uniqueness, and stability; the geometry of phase space; linear systems and hyperbolicity; maps and diffeomorphisms.
MATH 119B. Chaotic Dynamics and Bifurcation Theory
(4) STAFF
Prerequisite: Mathematics 5C or 6B, and 119A or consent of instructor.
Hyperbolic structure and chaos; center manifolds; bifurcation theory; and the Feigenbaum and Ruelle-Takens cascades to strange attractors.
MATH 122A. Introduction to Theory of Complex Variables
(4) STAFF
Prerequisite: Mathematics 5A or 5AI, and 5B or 5BI, or 4B or 4BI, and 6A or 6AI.
Complex numbers, functions, differentiability, series extensions of elementary functions, complex integration, calculus of residues, conformal maps, mapping functions, applications.
MATH 122B. Introduction to Theory of Complex Variables
(4) STAFF
Prerequisite: Mathematics 122A.
Complex numbers, functions, differentiability, series extensions of elementary functions, complex integration, calculus of residues, conformal maps, mapping functions, applications.
MATH 124A. Partial Differential Equations
(4) STAFF
Prerequisite: Mathematics 5A or 5AI, 5B or 5BI, and 5C or 4B or 4BI, and 6A or 6AI, and 6B.
Wave, heat, and potential equations.
MATH 124B. Fourier Series and Numerical Methods
(4) STAFF
Prerequisite: Mathematics 5A or 5AI, and 5B or 5BI, and 5C; or 4B or 4BI, and 6A or 6AI, and 6B; and 124A or consent of instructor.
Fourier series; generalized functions; and numerical methods.
MATH 132A. Introduction to Operations Research
(4) STAFF
Prerequisite: Math 108A.
Linear programming, the simplex method, duality, applications to the transportation and assignment problems, sensitivity analysis, problem formulation.
MATH 132B. Introduction to Operations Research
(4) STAFF
Prerequisite: Mathematics 117 and 132A.
Network analysis: shortest route, minimal spanning tree and maximal flow problems; PERT including the critical path method; dynamic programming; game theory; integer programming, nonlinear programming.
MATH 137A. Graph and Network Theory
(4) STAFF
Prerequisite: Mathematics 5A or 5AI or 4B or 4BI and 8.
Elements of graph and network theory including paths, circuits, trees, coloring, planarity, matching theory, Hall's theorem, applications to scheduling theory, flows in networks, Menger's theorem, and other topics as time permits.
MATH 137B. Graph and Network Theory
(4) STAFF
Prerequisite: Mathematics 137A.
Elements of graph and network theory including paths, circuits, trees, coloring, planarity, matching theory, Hall's theorem, applications to scheduling theory, flows in networks, Menger's theorem, and other topics astime permits.
MATH 145. Introduction to Topology
(4) STAFF
Prerequisite: Mathematics 8.
Metric spaces, continuity, compactness, classification of surfaces, Euler characteristics, and fundamental groups. Additional topics at the discretion of the instructor.
MATH 147A. Introductory Differential Geometry
(4) STAFF
Prerequisite: Mathematics 5B or 5BI or 6A or 6AI; and, Mathematics 108A or 117.
Curves and surfaces in three-dimensional Euclidean space, first and second fundamental forms, Gaussian and mean curvature, geodesics, Gauss-Bonnet theorem, and non-Euclidean geometry.
MATH 147B. Introductory Differential Geometry
(4) STAFF
Prerequisite: Mathematics 147A.
Curves and surfaces in three-dimensional Euclidean space, first and second fundamental forms, Gaussian and mean curvature, geodesics, Gauss-Bonnet theorem, and non-Euclidean geometry.
MATH 170. Introduction to Mathematical Finance
(4) STAFF
Prerequisite: PSTAT 120A-B, 160A.
Describes mathematical methods for estimating and evaluating asset pricing models, equilibrium and derivative pricing, options, bonds, and the term-structure of interest rates. Also introduces finance optimization models for risk management and financial engineering.
MATH 178. Introduction to Cryptography
(4) STAFF
Prerequisite: Computer Science 10; and, PSTAT 120A or 121A or equivalent courses.
An introduction to the basic concepts and techniques of cryptography and cryptanalysis. Topics include: The Shannon Theory, classical systems, the enigma machine, the data encryption standard, public key systems, digital signatures, file security.
MATH 181A. Pedagogical Content Knowledge for Secondary Mathematics Teaching
(4) LAGER, JACOB
Prerequisite: Mathematics 5A and an upper-division mathematics course, or consent of the instructor.
Designed for prospective middle and high school teachers. Focuses on the representations, strategies, and language learners use to conceptualize and develop fundamental ideas of mathematics. Includes advanced problem solving and its implications for teaching and learning at the secondary level.
MATH 181B. Advanced Problem Solving: Mathematical, Historical, and Pedagogical Contexts
(4) STAFF
Prerequisite: Consent of instructor.
Continuation of Math 181A.
MATH 190. Special Topics in Mathematics
(4) STAFF
Prerequisite: Consent of instructor.
Information about the special topics to be presented may be obtained from the office of the Department of Mathematics.
MATH 193. Internship in Mathematics
(1-4) STAFF
Prerequisite: Consent of instructor and department.
Faculty sponsored academic internship in industrial or research firms.
MATH 194GS. Group Studies for Advanced Students
(1) STAFF
Prerequisite: Consent of instructor.
Participants will select a math-related book or papers, read the section before the next meeting and discuss reading at the meeting. Readings may include biographies of mathematicians, histories or popularizations of mathematics, textbooks, and readings in mathematical physics or biology.
MATH 195A. Internship in Mathematics Teaching
(4) STAFF
Prerequisite: Upper-division standing in the major; two upper-division mathematics courses.
Supervised mathematics internship in local schools and participation in themathematics teaching seminar on mathematics learning and teaching. A paper on mathematics and its teaching required.
MATH 195B. Internship in Mathematics Teaching
(4) STAFF
Prerequisite: Upper-division standing in the major; two upper-division mathematics courses.
Supervised mathematics internship in local schools and participation in themathematics teaching seminar on mathematics learning and teaching. A paper on mathematics and its teaching required.
MATH 197A. Senior Thesis
(1-4) STAFF
Prerequisite: Open to senior majors only; consent of department and instructor.
Independent research under the supervision of a faculty member which will result in a senior thesis. Student will concentrate on reading and gathering material for a thesis.
MATH 197B. Senior Thesis
(1-4) STAFF
Prerequisite: Mathematics 197A. Open to senior majors only; consent of department and instructor.
Independent research under the supervision of a faculty member which will result in a senior thesis. Student will concentrate on writing a thesis.
MATH 199. Independent Studies in Mathematics
(1-5) STAFF
Prerequisite: Upper-division standing; completion of 2 upper-division courses in math; consent of instructor and department.
Coursework shall consist of academic research supervised by a faculty member on a topic not available in established course offerings.
Collapse Courses Graduate 
MATH 201A. Real Analysis
(4) STAFF
Prerequisite: Mathematics 118A-B-C.
Measure theory and integration. Point set topology. Principles of functional analysis. Lp-spaces. The Riesz representation theorem. Topics in real and functional analysis.
MATH 201B. Real Analysis
(4) STAFF
Prerequisite: Mathematics 118A-B-C.
Measure theory and integration. Point set topology. Principles of functional analysis. Lp-spaces. The Riesz representation theorem. Topics in real and functional analysis.
MATH 201C. Real Analysis
(4) STAFF
Prerequisite: Mathematics 118A-B-C.
Measure theory and integration. Point set topology. Principles of functional analysis. Lp-spaces. The Riesz representation theorem. Topics in real and functional analysis.
MATH 202A. Complex Analysis
(4) STAFF
Prerequisite: Mathematics 118A-B-C or 122A.
Analytic functions. Complex integration. Cauchy's theorem. Series and product developments. Entire functions. Conformal mappings. Topics in complex analysis.
MATH 202B. Complex Analysis
(4) STAFF
Prerequisite: Mathematics 118A-B-C or 122A.
Analytic functions. Complex integration. Cauchy's theorem. Series and product developments. Entire functions. Conformal mappings. Topics in complex analysis.
MATH 202C. Complex Analysis
(4) STAFF
Prerequisite: Mathematics 118A-B-C or 122A.
Analytic functions. Complex integration. Cauchy's theorem. Series and product developments. Entire functions. Conformal mappings. Topics in complex analysis.
MATH 206A. Matrix Analysis and Computation
(4) STAFF
Prerequisite: Consent of instructor.
Graduate level-matrix theory with introduction to matrix computations. SVD's, pseudoinverses, variational characterization of eigenvalues, perturbation theory, direct and iterative methods for matrix computations.
MATH 206B. Numerical Simulation
(4) STAFF
Prerequisite: Consent of instructor.
Linear multistep methods and Runge-Kutta methods for ordinary differential equations: stability, order and convergence. Stiffness. Differential algebraic equations. Numerical solution of boundary value problems.
MATH 206C. Numerical Solution of Partial Differential Equations--Finite Difference Methods
(4) STAFF
Prerequisite: Consent of instructor.
Finite difference methods for hyperbolic, parabolic and elliptic PDE's, with application to problems in science and engineering. Convergence, consistency, order and stability of finite difference methods. Dissipation and dispersion. Finite volume methods. Software design and adaptivity.
MATH 206D. Numerical Solution of Partial Differential Equations - Finite Element Methods
(4) STAFF
Prerequisite: Consent of instructor.
Weighted residual and finite element methods for the solution of hyperbolic, parabolic and elliptic partial differential equations, with application to problems in science and engineering. Error estimates. Standard and discontinuous Galerkin methods.
MATH 214A. Ordinary Differential Equations
(4) STAFF
Prerequisite: Not open to mathematics majors.
Existence, uniqueness, and stability; the geometry of phase space; linear systems and hyperbolicity; maps and diffeomorphisms.
MATH 214B. Chaotic Dynamics and Bifurcation Theory
(4) STAFF
Prerequisite: Not open to mathematics majors.
Hyberbolic structure and chaos; center manifolds; bifurcation theory; and the Feigenbaum and Ruelle-Takens cascades to strange attractors.
MATH 215A. Partial Differential Equations
(4) STAFF
Prerequisite: Not open to mathematics majors.
Wave, heat, and potential equations.
MATH 215B. Fourier Series and Numerical Methods
(4) STAFF
Prerequisite: Not open to mathematics majors.
Fourier series; generalized functions; and numerical methods.
MATH 220A. Modern Algebra
(4) STAFF
Prerequisite: Mathematics 108A-B and 111A-B.
Group theory, ring and module theory, field theory, Galois theory, other topics.
MATH 220B. Modern Algebra
(4) STAFF
Prerequisite: Mathematics 108A-B and 111A-B.
Group theory, ring and module theory, field theory, Galois theory, other topics.
MATH 220C. Modern Algebra
(4) STAFF
Prerequisite: Mathematics 108A-B and 111A-B.
Group theory, ring and module theory, field theory, Galois theory, other topics.
MATH 221A. Foundations of Topology
(4) STAFF
Prerequisite: Mathematics 118A or equivalent.
Metric spaces, topological spaces, continuity, Hausdorff condition, compactness, connectedness, product spaces, quotient spaces. Other topics as time allows.
MATH 221B. Homotopy Theory
(4) STAFF
Prerequisite: Mathematics 221A.
Homotopy groups, exact sequences, fiber spaces, covering spaces, van Kampen Theorem.
MATH 221C. Differential Topology
(4) STAFF
Prerequisite: Mathematics 221A.
Topological manifolds, differentiable manifolds, transversality, tangent bundles, Borsuk-Ulam theorem, orientation and intersection number, Lefschetz fixed point theorem, vector fields.
MATH 225A. Topics in Number Theory
(4) STAFF
Prerequisite: Mathematics 220A-B-C.
Selected topics in number theory.
MATH 225B. Topics in Number Theory
(4) STAFF
Prerequisite: Mathematics 220A-B-C.
Selected topics in number theory.
MATH 227A. Advanced Topics in Geometric and Algebraic Topology
(4) STAFF
Prerequisite: Consent of instructor.
Topics, varying from year to year, include piecewise linear and differential topology, manifolds, fiber bundles and fiber spaces, homotopy theory, and spectral sequences.
MATH 227B. Advanced Topics in Geometric and Algebraic Topology
(4) STAFF
Prerequisite: Consent of instructor.
Topics, varying from year to year, include piecewise linear and differential topology, manifolds, fiber bundles and fiber spaces, homotopy theory, and spectral sequences.
MATH 227C. Advanced Topics in Geometric and Algebraic Topology
(4) STAFF
Prerequisite: Consent of instructor.
Topics, varying from year to year, include piecewise linear and differential topology, manifolds, fiber bundles and fiber spaces, homotopy theory, and spectral sequences.
MATH 228A. Functional Analysis
(4) STAFF
Prerequisite: Mathematics 201A-B-C.
Topics in functional analysis such as operators on Hilbert space, convex analysis, fixed point theorems, distribution theory, unbounded operators.
MATH 228B. Functional Analysis
(4) STAFF
Prerequisite: Mathematics 201A-B-C.
Topics in functional analysis such as operators on Hilbert space, convex analysis, fixed point theorems, distribution theory, unbounded operators.
MATH 231A. Lie Groups and Lie Algebras
(4) STAFF
Prerequisite: Consent of instructor.
Differentiable manifolds, definition and examples of lie groups, lie group-lie algebra correspondence, nilpotent and solvable lie algebras, classification of semi-simple lie algebras over the complexes, representations of lie groups and lie algebras, special topics.
MATH 231B. Lie Groups and Lie Algebras
(4) STAFF
Prerequisite: Consent of instructor.
Differentiable manifolds, definition and examples of lie groups, lie group-lie algebra correspondence, nilpotent and solvable lie algebras, classification of semi-simple lie algebras over the complexes, representations of lie groups and lie algebras, special topics.
MATH 232A. Algebraic Topology
(4) STAFF
Prerequisite: Mathematics 108A-B and 145.
Singular homology and cohomology, exact sequences, Hurewicz theorem, Poincare duality.
MATH 232B. Algebraic Topology
(4) STAFF
Prerequisite: Mathematics 108A-B and 145.
Singular homology and cohomology, exact sequences, Hurewicz theorem, Poincare duality.
MATH 232C. Algebraic Topology
(4) STAFF
Prerequisite: Mathematics 108A-B and 145.
Singular homology and cohomology, exact sequences, Hurewicz theorem, Poincare duality.
MATH 236A. Homological Algebra
(4) STAFF
Prerequisite: Mathematics 220A-B-C.
Algebraic construction of homology and cohomology theories, aimed at applications to topology, geometry, groups and rings. Special emphasis on hom and tensor functors; projective, injective and flat modules; exact sequences; chain complexes; derived functors, in particular, ext and tor.
MATH 236B. Homological Algebra
(4) STAFF
Prerequisite: Mathematics 220A-B-C.
Algebraic construction of homology and cohomology theories, aimed at applications to topology, geometry, groups and rings. Special emphasis on hom and tensor functors; projective, injective and flat modules; exact sequences; chain complexes; derived functors, in particular, ext and tor.
MATH 237A. Algebraic Geometry
(4) STAFF
Prerequisite: Mathematics 220A-B-C.
Affine/projective varieties, Hilbert's Nullstellensatz, morphisms of varieties, rational maps, dimension, singular/nonsingular points, blowing up of varieties, tanent spaces, divisors, differentials, Riemann-Roch theorem. Special topics may include: elliptic curves, intersection numbers, Bezout's theorem, Max Noether's theorem.
MATH 237B. Algebraic Geometry
(4) STAFF
Prerequisite: Mathematics 220A-B-C.
Affine/projective varieties, Hilbert's Nullstellensatz, morphisms of varieties, rational maps, dimension, singular/nonsingular points, blowing up of varieties, tangent spaces, divisors, differentials, Riemann-Roch theorem. Special topics may include: elliptic curves, intersection numbers,Bezout's theorem, Max Noether's theorem.
MATH 240A. Introduction to Differential Geometry and Riemannian Geometry
(4) STAFF
Topics include geometry of surfaces, manifolds, differential forms, Lie groups, Riemannian manifolds, Levi-Civita connection and curvature, curvature and topology, Hodge theory. Additional topics such as bundles and characteristic classes, spin structures, and Dirac operator, comparison theorems in Riemannian geometry.
MATH 240B. Introduction to Differential Geometry and Riemannian Geometry
(4) STAFF
Topics include geometry of surfaces, manifolds, differential forms, Lie groups, Riemannian manifolds, Levi-Civita connection and curvature, curvature and topology, Hodge theory. Additional topics such as bundles and characteristic classes, spin structures, and Dirac operator, comparison theorems in Riemannian geometry.
MATH 240C. Introduction to Differential Geometry and Riemannian Geometry
(4) STAFF
Topics include geometry of surfaces, manifolds, differential forms, Lie groups, Riemannian manifolds, Levi-Civita connection and curvature, curvature and topology, Hodge theory. Additional topics such as bundles and characteristic classes, spin structures, and Dirac operator, comparison theorems in Riemannian geometry.
MATH 241A. Topics in Differential Geometry
(4) STAFF
Prerequisite: Mathematics 240A-B-C..
Various topics are covered including sectional curvature and Ricci curvature, minimal submanifolds, Atiyah-Singer index theorem and eta invariant, Einstein manifold, symplectic geometry, geometry of gauge theories, geometric PDE, Morse theory and Floer theory.
MATH 241B. Topics in Differential Geometry
(4) STAFF
Prerequisite: Mathematics 240A-B-C.
Various topics are covered including sectional curvature and Ricci curvature, minimal submanifolds, Atiyah-Singer index theorem and eta invariant, Einstein manifold, symplectic geometry, geometry of gauge theories, geometric PDE, Morse theory and Floer theory.
MATH 241C. Topics in Differential Geometry
(4) STAFF
Prerequisite: Mathematics 240A-B-C.
Various topics are covered including sectional curvature and Ricci curvature, minimal submanifolds, Atiyah-Singer index theorem and eta invariant, Einstein manifold, symplectic geometry, geometry of gauge theories, geometric PDE, Morse theory and Floer theory.
MATH 243A. Ordinary Differential Equations
(4) STAFF
Prerequisite: Mathematics 118A-B-C.
Existence and stability of solutions, Floquet theory, Poincare-Bendixson theorem, invariant manifolds, existence and stability of periodic solutions, Bifurcation theory and normal forms, hyperbolic structure and chaos, Feigenbaum period-doubling cascade, Ruelle-Takens cascade.
MATH 243B. Ordinary Differential Equations
(4) STAFF
Prerequisite: Mathematics 118A-B-C.
Existance and stability of solutions, Floq uet theory, Poincare-Bendixson theorem, invariant manifolds, existence and stability of periodic solutions, Bifurcation theory and normal forms, hyperbolic structure and chaos, Feigenbaum period-doubling cascade, Ruelle-Takens cascade.
MATH 243C. Ordinary Differential Equations
(4) STAFF
Prerequisite: Mathematics 118A-B-C.
Existance and stability of solutions, Floquet theory, Poincare-Bendixson theorem, invariant manifolds, existence and stability of periodic solutions, Bifurcation theory and normal forms, hyperbolic structure and chaos, Feigenbaum period-doubling cascade, Ruelle-Takens cascade.
MATH 246A. Partial Differential Equations
(4) STAFF
Prerequisite: Mathematics 201A-B-C.
First-order nonlinear equations; the Cauchy problem, elements of distribution theory an Sobolev spaces; the heat, wave, and Laplace equations; additional topics such as quasilinear symmetric hyperbolic systems, elliptic regularity theory.
MATH 246B. Partial Differential Equations
(4) STAFF
Prerequisite: Mathematics 201A-B-C.
First-order nonlinear equations; the Cauchy problem, elements of distribution theory and Sobolev spaces; the heat, wave, and Laplace equations; additional topics such as quasilinear symmetric hyperbolic systems, elliptic regularity theory.
MATH 246C. Partial Differential Equations
(4) STAFF
Prerequisite: Mathematics 201A-B-C.
First-order nonlinear equations; the Cauchy problem, elements of distribution theory and Sobolev spaces; the heat, wave, and Laplace equations; additional topics such as quasilinear symmetric hyperbolic systems, ellipitic regularity theory.
MATH 260AAZZ. Seminars in Mathematics
(1-6) STAFF
Prerequisite: Consent of instructor.
Topics in algebra, analysis, applied mathematics, combinatorial mathematics, functional analysis, geometry, statistics, topology, by means of lectures and informal conferences with members of staff.
MATH 260A. Seminars in Mathematics
MATH 260AA. Seminars in Mathematics
MATH 260B. Seminars in Mathematics
MATH 260BB. Seminars in Mathematics
MATH 260C. Seminars in Mathematics
MATH 260CC. Seminars in Mathematics
MATH 260D. Seminars in Mathematics
MATH 260DD. Seminars in Mathematics
MATH 260E. Seminars in Mathematics
MATH 260EE. Seminars in Mathematics
MATH 260ES. Seminars in Mathematics
MATH 260F. Seminars in Mathematics
MATH 260G. Seminars in Mathematics
MATH 260GG. Seminars in Mathematics
MATH 260H. Seminars in Mathematics
MATH 260HH. Seminars in Mathematics
MATH 260I. Seminars in Mathematics
MATH 260II. Seminars in Mathematics
MATH 260J. Seminars in Mathematics
MATH 260JJ. Seminars in Mathematics
MATH 260K. Seminars in Mathematics
MATH 260KK. Seminars in Mathematics
MATH 260L. Seminars in Mathematics
MATH 260LL. Seminars in Mathematics
MATH 260M. Seminars in Mathematics
MATH 260MM. Seminars in Mathematics
MATH 260N. Seminars in Mathematics
MATH 260NN. Seminars in Mathematics
MATH 260O. Seminars in Mathematics
MATH 260OO. Seminars in Mathematics
MATH 260P. Seminars in Mathematics
MATH 260Q. Seminars in Mathematics
MATH 260QQ. Seminars in Mathematics
MATH 260R. Seminars in Mathematics
MATH 260S. Seminars in Mathematics
MATH 260SS. Seminars in Mathematics
MATH 260T. Seminars in Mathematics
MATH 260TT. Seminars in Mathematics
MATH 260U. Foundations in Mathematics
MATH 260UU. Seminars in Mathematics
MATH 260V. Seminars in Mathematics
MATH 260W. Seminars in Mathematics
MATH 260X. Seminars in Mathematics
MATH 260Y. Seminars in Mathematics
MATH 260Z. Seminars in Mathematics
MATH 500. Teaching Assistant Practicum
(1-4) STAFF
Prerequisite: Appointment as teaching assistant and departmental approval.
Supervised teaching of undergraduate mathematics courses.
MATH 501. Teaching Assistant Training
(1-2) STAFF
Prerequisite: Departmental and instructor approval.
Consideration of ideas about the process of learning mathematics and discussion of approaches to teaching.
MATH 502. Teaching Associate Practicum
(1-5) STAFF
Prerequisite: Appointment as associate; departmental approval.
Supervised teaching of undergraduate courses.
MATH 510. Reading for Area Examinations
(2-6) STAFF
Prerequisite: Enrollment in M.A. or Ph.D. program. Consent of instructor.
Reading for area examinations.
MATH 596. Directed Reading and Research
(1-6) STAFF
Prerequisite: Graduate standing and consent of instructor.
Directed reading and research.
MATH 596AA. Directed Reading and Research
MATH 598. Master's Thesis Research and Preparation
(1-6) STAFF
Prerequisite: Graduate standing and consent of instructor.
Master's thesis research and preparation.
MATH 598AA. Master's Thesis Research and Preparation
MATH 599. Dissertation Preparation
(1-6) STAFF
Prerequisite: Consent of instructor.
Dissertation preparation.