Mathematics

194 Baskin Engineering
(831) 459-2969
http://www.math.ucsc.edu

Lower-Division Courses

2. College Algebra for Calculus. F,W
Operations on real numbers, complex numbers, polynomials, and rational expressions; exponents and radicals; solving linear and quadratic equations and inequalities; functions, algebra of functions, graphs; conic sections; mathematical models; sequences and series. Prerequisite(s): placement exam score of 12 or higher. The Staff

3. Precalculus. F,W,S
Inverse functions and graphs; exponential and logorithmic functions, their graphs, and use in mathematical models of the real world; rates of change; trigonometry, trigonometric functions, and their graphs; and geometric series. Students cannot receive credit for both course 3 and Applied Mathematics and Statistics 3. Applied Mathematics and Statistics 3 can substitute for course 3. Prerequisite(s): course 2 or placement exam score of 20 or higher. (General Education Code(s): Q.) The Staff

11A. Calculus with Applications. F,W,S
A modern course stressing conceptual understanding, relevance, and problem solving. The derivative of polynomial, exponential, and trigonometric functions of a single variable is developed and applied to a wide range of problems involving graphing, approximation, and optimization. Students cannot receive credit for both this course and course 19A or Applied Mathematics and Statistics 11A or Economics 11A. Prerequisite(s): course 3 or Applied Mathematics and Statistics 3; or placement exam score of 31 or higher; or AP Calculus AB exam score of 3 or higher. (General Education Code(s): IN, Q.) The Staff

11B. Calculus with Applications. F,W,S
Starting with the fundamental theorem of calculus and related techniques, the integral of functions of a single variable is developed and applied to problems in geometry, probability, physics, and differential equations. Polynomial approximations, Taylor series, and their applications conclude the course. Students cannot receive credit for this course and course 19B or Economics 11B. Prerequisite(s): course 11A or AP Calculus AB exam score of 4 or 5, or BC exam score of 3 or higher, or IB Mathematics Higher Level exam score of 5 or higher. (General Education Code(s): IN, Q.) The Staff

19A. Calculus for Science, Engineering, and Mathematics. F,W,S
The limit of a function, calculating limits, continuity, tangents, velocities, and other instantaneous rates of change. Derivatives, the chain rule, implicit differentiation, higher derivatives. Exponential functions, inverse functions, and their derivatives. The mean value theorem, monotonic functions, concavity, and points of inflection. Applied maximum and minimum problems. Students cannot receive credit for both this course and course 11A or Applied Mathematics and Statistics 11A or Economics 11A. Prerequisite(s): course 3 or Applied Mathematics and Statistics 3 or placement exam score of 40 or higher or AP Calculus AB exam score of 3 or higher. (General Education Code(s): IN, Q.) The Staff

19B. Calculus for Science, Engineering, and Mathematics. F,W,S
The definite integral and the fundamental theorem of calculus. Areas, volumes. Integration by parts, trigonometric substitution, and partial fractions methods. Improper integrals. Sequences, series, absolute convergence and convergence tests. Power series, Taylor and Maclaurin series. Students cannot receive credit for both this course and course 11B, Applied Math and Statistics 11B, or Economics 11B. Prerequisite(s): course 19A or AP Calculus AB exam score of 4 or 5, or BC exam score of 3 or higher, or IB Mathematics Higher Level exam score of 5 of higher. (General Education Code(s): IN, Q.) The Staff

20A. Honors Calculus. F
Challenging course designed to approach single-variable calculus from the perspective of modern mathematics. Emphasis is on the evolution and historical development of core concepts underlying calculus and analysis. Prerequisite(s): placement exam score of 46 or higher; or AP Calculus AB exam score of 4 or 5; or BC exam of 3 or higher; or IB Mathematics Higher Level exam score of 5 or higher. Enrollment limited to 30. (General Education Code(s): IN, Q.) The Staff

20B. Honors Calculus. W
Challenging course designed to approach single-variable calculus from the perspective of modern mathematics. Emphasis is on the evolution and historical development of core concepts underlying calculus and analysis. Prerequisite(s): course 20A. Enrollment limited to 60. (General Education Code(s): IN, Q.) The Staff

21. Linear Algebra. F,W,S
Systems of linear equations, matrices, determinants. Introduction to abstract vector spaces, linear transformation, inner products, geometry of Euclidean space, and eigenvalues. One quarter of college mathematics is recommended as preparation. (General Education Code(s): Q.) The Staff

22. Introduction to Calculus of Several Variables. F,W,S
Functions of several variables. Continuity and partial derivatives. The chain rule, gradient and directional derivative. Maxima and minima, including Lagrange multipliers. The double and triple integral and change of variables. Surface area and volumes. Applications from biology, chemistry, earth sciences, engineering, and physics. Students cannot receive credit for this course and course 23A. Prerequisite(s): course 11B or 19B or 20B. The Staff

23A. Multivariable Calculus. F,W,S
Vectors in n-dimensional Euclidean space. The inner and cross products. The derivative of functions from n-dimensional to m-dimensional Euclidean space is studied as a linear transformation having matrix representation. Paths in 3-dimensions, arc length, vector differential calculus. Taylor's theorem in several variables, extrema of real-valued functions, constrained extrema and Lagrange multipliers, the implicit function theorem, some applications. Students cannot receive credit for this course and course 22. Prerequisite(s): course 19B or 20B. The Staff

23B. Multivariable Calculus. F,W,S
Double integral, changing the order of integration. Triple integrals, maps of the plane, change of variables theorem, improper double integrals. Path integrals, line integrals, parametrized surfaces, area of a surface, surface integrals. Green's theorem, Stokes theorem, conservative fields, Gauss' theorem. Applications to physics and differential equations, differential forms. Prerequisite(s): course 23A. The Staff

24. Ordinary Differential Equations. S
First and second order ordinary differential equations, with emphasis on the linear case. Methods of integrating factors, undetermined coefficients, variation of parameters, power series, numerical computation. Students cannot receive credit for this course and Applied Mathematics and Statistics 27. Prerequisite(s): course 22 or 23A; course 21 is recommended as preparation. The Staff

30. Mathematical Problem Solving. F
Students learn techniques of problem solving such as induction, contradiction, exhaustion, dissection, analogy, generalization, specialization, and others in the context of solving problems drawn from number theory, probability, combinatorics, graph theory, geometry, and logic. Prerequisite(s): course 11A or 19A or 20A or Math Placement Exam score of 40 or higher. B. Cooperstein

99. Tutorial. F,W,S
The Staff

Upper-Division Courses

100. Introduction to Proof and Problem Solving. F,W,S
Students learn the basic concepts and ideas necessary for upper-division mathematics and techniques of mathematical proof in the context of specific topics. Introduction to sets, relations, elementary mathematical logic, proof by contradiction, mathematical induction, and counting arguments. Prerequisite(s): courses 11A and 11B or 19A and 19B or 20A and 20B. Enrollment limited to 40. The Staff

103. Complex Analysis. F,S
Complex numbers, analytic and harmonic functions, complex integration, the Cauchy integral formula, Laurent series, singularities and residues, conformal mappings. Prerequisite(s): course 23B; and either course 100 or Computer Science 101. The Staff

105A. Real Analysis. W,S
The basic concepts of one-variable calculus are treated carefully and rigorously. Set theory, the real number system, numerical sequences and series, continuity, differentiation. Prerequisite(s): course 23B and either course 100 or Computer Science 101. The Staff

105B. Real Analysis. S
Metric spaces, differentiation and integration of functions. The Riemann-Stieltjes integral. Sequences and series of functions. Prerequisite(s): course 105A. The Staff

105C. Real Analysis. *
The Stone-Weierstrass theorem, Fourier series, differentiation and integration of functions of several variables. Prerequisite(s): course 105B. The Staff

106A. Systems of Ordinary Differential Equations. F
Linear systems, exponentials of operators, existence and uniqueness, stability of equilibria, periodic attractors, and applications. Prerequisite(s): either Applied Mathematics and Statistics 27 or preferably courses 21 and 24; and either course 100 or Computer Science 101. The Staff

106B. Partial Differential Equations. S
Topics covered include first and second order linear partial differential equations, the heat equation, the wave equation, Laplace's equation, separation of variables, eigenvalue problems, Green's functions, Fourier series. Prerequisite(s): either courses 21 and 24 or Applied Mathematics and Statistics 27; and either course 100 or Computer Science 101; course 106A is recommended as preparation. The Staff

110. Introduction to Number Theory. F
Prime numbers, unique factorization, congruences with applications (e.g., to magic squares). Rational and irrational numbers. Continued fractions. Introduction to Diophantine equations. No calculus required. An introduction to some of the ideas and outstanding problems of modern mathematics. Prerequisite(s): course 100 or Computer Science 101. (General Education Code(s): Q.) The Staff

111A. Algebra. F,W
Group theory including the Sylow theorem, the structure of abelian groups, permutation groups. Introduction to rings and fields including polynomial rings, factorization, the classical geometric constructions, and Galois theory. Prerequisite(s): course 21 or Applied Mathematics and Statistics 27 and either course 100 or Computer Science 101. The Staff

111B. Algebra. S
Group theory including the Sylow theorem, the structure of abelian groups, permutation groups. Introduction to rings and fields including polynomial rings, factorization, the classical geometric constructions, and Galois theory. Prerequisite(s): course 111A. The Staff

112. Mathematical Probability Theory. F
Introductory probability course for mathematicians, designed as a prerequisite for advanced probability courses at the graduate level. Moving from elementary topics of probability spaces and random variables, independent identical trials, the law of large numbers, the Demoivre-Laplace central limit theorem, also includes basic Martingale theory, finite Markov chains, percolations, and branching processes. Prerequisite(s): courses 21 and 23B. The Staff

113. Discrete Mathematics. *
Basic course in theorems and applications of discrete mathematics. Sequences and series, matrix operations, recursion relations, discrete probability, algorithms, finite state machines, boolean functions, trees, elementary number theory, generating functions, graph theory. Particular emphasis on combinatorics. Applications dealing with searching and sorting, cryptography, coding, quantum mechanics, and Markov processes. Prerequisite(s): courses 19A-B, 21, or equivalent. The Staff

114. Introduction to Financial Mathematics. W
Financial derivatives: contracts and options. Hedging and risk managment. Arbitrage, interest rate, and discounted value. Geometric random walk and Brownian motion as models of risky assets. Ito's formula. Initial boundary value problems for the heat and related partial differential equations. Self-financing replicating portfolio; Black-Scholes pricing of European options. Dividends. Implied volatility. American options as free boundary problems. Prerequisite(s): course 24 or Applied Mathematics and Statistics 27. Corequisite(s): course 112 or Applied Mathematics and Statistics 131 or Computer Engineering 107. The Staff

115. Graph Theory. *
Graph theory, trees, vertex and edge colorings, Hamilton cycles, Eulerian circuits, decompositions into isomorphic subgraphs, extremal problems, cages, Ramsey theory, Cayley's spanning tree formula, planar graphs, Euler's formula, crossing numbers, thickness, splitting numbers, magic graphs, graceful trees, rotations, and genus of graphs. Prerequisite(s): course 21 or Applied Mathematics and Statistics 27 and either course 100 or Computer Science 101. The Staff

117. Advanced Linear Algebra. W
Review of abstract vector spaces. Dual spaces, bilinear forms, and the associated geometry. Normal forms of linear mappings. Introduction to tensor products and exterior algebra. Prerequisite(s): course 21 or Applied Mathematics and Statistics 27 and either course 100 or Computer Science 101. The Staff

118. Advanced Number Theory. F
Topics include divisibility and congruences, arithmetical functions, quadratic residues and quadratic reciprocity, quadratic forms and representations of numbers as sums of squares, Diophantine approximation and transcendence theory, quadratic fields. Additional topics as time permits. Prerequisite(s): course 110 or 111A. The Staff

120. Coding Theory. *
An introduction to mathematical theory of coding. Construction and properties of various codes, such as cyclic, quadratic residue, linear, Hamming, and Golay codes; weight enumerators; connections with modern algebra and combinatorics. Prerequisite(s): course 21. The Staff

121A. Differential Geometry. S
Topics include Euclidean space, tangent vectors, directional derivatives, curves and differential forms in space, mappings. Curves, the Frenet formulas, covariant derivatives, frame fields, the structural equations. The classification of space curves up to rigid motions. Vector fields and differentiable forms on surfaces; the shape operator. Gaussian and mean curvature. The theorem Egregium; global classification of surfaces in three space by curvature. Prerequisite(s): courses 21 and 23B and either course 100 or Computer Science 101. Course 105A strongly recommended. The Staff

121B. Differential Geometry and Topology. *
Examples of surfaces of constant Gauss curvature, surfaces of revolutions, minimal surfaces. Abstract manifolds; integration theory; Riemannian manifolds. Total curvature and geodesics; the Euler characteristic, the theorem of Gauss-Bonnet. Length-minimizing properties of geodesics, complete surfaces, curvature and conjugate points covering surfaces. Surfaces of constant curvature; the theorems of Bonnet and Hadamard. Prerequisite(s): course 121A. The Staff

124. Introduction to Topology. W
Topics include introduction to point set topology (topological spaces, continuous maps, connectedness, compactness), homotopy relation, definition and calculation of fundamental groups and homology groups, Euler characteristic, classification of orientable and nonorientable surfaces, degree of maps, and Lefschetz finex point theorem. Prerequisite(s): course 100; course 111A recommended. The Staff

126. Mathematical Control Theory. *
Control theory concerns steering and stabilizing systems by means of tunable parameters. Examples are flight controllers, CD players, and biological or robotic locomotion. Studies the mathematical foundations, tools, and basic theorems of linear and nonlinear deterministic control. Prerequisite(s): courses 23B and 24 or Applied Mathematics and Statistics 27, and either course 100 or Computer Science 101. The Staff

128A. Classical Geometry: Euclidean and Non-Euclidean. F
Rigorous foundations for Euclidean and non-Euclidean geometries. History of attempts to prove the parallel postulate and of the simultaneous discovery by Gauss, J. Bolyai, and Lobachevsky of hyperbolic geometry. Consistency proved by Euclidean models. Classification of rigid motions in both geometries. Prerequisite(s): either course 100 or Computer Science 101. The Staff

128B. Classical Geometry: Projective. *
Theorems of Desargue, Pascal, and Pappus; projectivities; homogeneous and affine coordinates; conics; relation to perspective drawing and some history. Prerequisite(s): course 21. The Staff

130. Celestial Mechanics. W
Solves the two-body (or Kepler) problem, then moves onto the N-body problem where there are many open problems. Includes central force laws; orbital elements; conservation of linear momentum, energy, and angular momentum; the Lagrange-Jacobi formula; Sundman's theorem for total collision; virial theorem; the three-body problem; Jacobi coordinates; solutions of Euler and of Lagrange; and restricted three-body problem. Will be offered in 06-07 academic year. Prerequisite(s): courses 19A-B and course 23A or Physics 5A or 6A; courses 21 and 24 strongly recommended. Enrollment limited to 35. The Staff

134. Cryptography. S
Introduces different methods in cryptography (shift cipher, affine cipher, Vigenere cipher, Hill cipher, RSA cipher, ElGamal cipher, knapsack cipher). The necessary material from number theory and probability theory is developed in the course. Common methods to attack ciphers discussed. Will be offered in 06-07 academic year. Prerequisite(s): course 100; course 110 recommended as preparation. R. Boltje

141. Introduction to Nonlinear Mathematics. *
Modeling problems involving nonlinear differential equations. Applications to chemical reactions, electrical circuits, shock waves, ecosystems, microeconomics, stochastic processes. Exact solutions, intuitive and pictorial methods of analysis. Prerequisite(s): courses 21 and 24 or Applied Mathematics and Statistics 27; course 100 or Computer Science 101; 106A recommended. The Staff

145. Introductory Chaos Theory. *
The Lorenz and Rossler attractors, measures of chaos, attractor reconstruction, applications from the sciences. Concurrent enrollment in course 145L is required. Students cannot receive credit for this course and Applied Mathematics and Statistics 146. Prerequisite(s): course 22 or 23A; course 21; course 100 or Computer Science 101. Concurrent enrollment in course 145L is required. The Staff

145L. Introductory Chaos Laboratory (1 credit). *
Laboratory sequence illustrating topics covered in course 145. One three-hour session per week in microcomputer laboratory. Concurrent enrollment in course 145 is required. The Staff

148. Numerical Analysis. *
The theory of constructive methods in mathematical analysis and its application with scientific computation. Some typical topics are difference equations, linear algebra, iteration, Bernoulli's method, quotient difference algorithm, the interpolating polynomial, numerical differentiation and integration, numerical solution of differential equations, finite Fourier series. Prerequisite(s): course 22 or 23A; course 21 and 24 or Applied Mathematics and Statistics 27; course 100 or Computer Science 101. Concurrent enrollment in course 148L is required. The Staff

148L. Numerical Analysis Laboratory (1 credit). *
Laboratory sequence illustrating topics covered in course 148. One three-hour session per week in microcomputer laboratory. Concurrent enrollment in course 148 is required. The Staff

160. Mathematical Logic I. S
Propositional and predicate calculus. Resolution, completeness, compactness, and Löwenheim-Skolem theorem. Recursive functions, Gödel incompleteness theorem. Undecidable theories. Hilbert's 10th problem. Prerequisite(s): course 100 or Computer Science 101. The Staff

161. Mathematical Logic II. *
Continuation of course 160: arithmetization of syntax, Tarski's theorem on the undefinability of truth, Gödel's first incompleteness theorem, naive set theory and its limitations (Russell's paradox), cardinal numbers, cardinal arithmetic, Axiom of Choice, finite, countable and uncountable sets, and Continuum Hypothesis. Prerequisite(s): course 160. Enrollment limited to 45. The Staff

181. History of Mathematics. W
A survey from a historical point of view of various developments in mathematics. Specific topics and periods to vary yearly. The Staff

188. Supervised Teaching. F,W,S
Supervised tutoring in self-paced courses. May not be repeated for credit. Students submit petition to sponsoring agency. The Staff

194. Senior Seminar. W,S
Designed to expose the student to topics not normally covered in the standard courses. The format varies from year to year. In recent years each student has written a paper and presented a lecture on it to the class. Prerequisite(s): course 103 or 105A or 111A. Enrollment priority given to seniors. The Staff

195. Senior Thesis. F,W,S
Students research a mathematical topic under the guidance of a faculty sponsor and write a senior thesis demonstrating knowledge of the material. Students submit petition to sponsoring agency. May be repeated for credit. The Staff

199. Tutorial. F,W,S
Students submit petition to sponsoring agency. May be repeated for credit. The Staff

Graduate Courses

200. Algebra I. F
Subgroups, cosets, normal subgroups, homomorphisms, isomorphisms, quotient groups, free groups, generators and relations, group actions on a set. Sylow theorems, semidirect products, simple groups, nilpotent groups, and solvable groups. Ring theory: Chinese remainder theorem, prime ideals, localization. Euclidean domains, PIDs, UFDs, polynomial rings. Prerequisite(s): courses 111A and 117 are recommended as preparation. May be repeated for credit. The Staff

201. Algebra II. W
Vector spaces, linear transformations, eigenvalues and eigenvectors, Jordan canonical forms, bilinear forms, quadratic forms, real symmetric forms and real symmetric matrices, orthogonal transformations and orthogonal matrices, Euclidean space, Hermitian forms and Hermitian matrices, Hermitian spaces, unitary transformations and unitary matrices, skew-symmetric forms, tensor products of vector spaces, tensor algebras, symmetric algebras, exterior algebras, Clifford algebras and spin groups. Course 200 is recommended as preparation. The Staff

202. Algebra III. S
Submodules, quotient modules, module homomorphisms, generators of modules, direct sums, free modules, torsion modules, modules over PIDs, and applications to rational and Jordan canonical forms. Field theory: field extensions, algebraic and transcendental extensions, splitting fields, algebraic closures, separable and normal extensions, the Galois theory, finite fields, Galois theory of polynomials. Course 201 is recommended as preparation. The Staff

203. Algebra IV. F
Topics include tensor product of modules over rings, projective modules and injective modules, Jacobson radical, Weederburns' theorem, category theory, Noetherian rings, Artinian rings, affine varieties, projective varieties, Hilbert's Nullstellensatz, prime spectrum, Zariski topology, discrete valuation rings, and Dedekind domains. (Formerly course 208.) Prerequisite(s): courses 200, 201, and 202. The Staff

204. Analysis I. F
Completeness and compactness for real line, sequences and infinite series of functions, Fourier series, calculus on Euclidean space and implicit function theorem, metric spaces and contracting mapping theorem, Arzela-Ascoli theorem, basics of general topological spaces, Baire category theorem, Urysohn lemma, and Tychnoff theorem. (Formerly course 203.) Prerequisite(s): course 105A or equivalent; course 105B is recommended as preparation. The Staff

205. Analysis II. W
Lebesgue measure theory, abstract measure theory, measurable functions, integration, space of absolutely integrable functions, dominated convergence theorem, convergence in measure, Riesz representation theorem, product measure and Fubini theorem, Lp spaces, derivative of a measure and Radon-Nikodym theorem, fundamental theorem of calculus. (Formerly course 204.) Prerequisite(s): course 204. The Staff

206. Analysis III. S
Banach space, Hahn-Banach theorem, uniform boundedness theorem, open mapping theorem and closed graph theorem, weak and weak* topology and Banach-Alaoglu theorem, Hilbert space, self-adjoint operators, compact operators, spectral theory, Fredholm operators, space of distributions and Fourier transform, Sobolev spaces. Courses 204 and 205 recommended as preparation. (Formerly course 205.) The Staff

207. Complex Analysis. W
Holomorphic and harmonic functions, the Cauchy integral theorem, the maximum principle and its consequences, conformal mapping, analytic continuation. The Riemann mapping theorem. (Formerly course 209.) Course 103 is recommended as preparation. The Staff

208. Manifolds I. F
Definition of manifolds, tangent bundle, inverse and implicit function theorems, transversality, Sard's theorem and the Whitney embedding theorem, vector fields, flows, and Lie bracket, Frobenius's theorem. Course 204 recommended for preparation. (Formerly course 234A, Calculus on Manifolds.) The Staff

209. Manifolds II. W
Tensor algebra. Differential forms and associated formalism of pullback, wedge product, exterior derivative, Stokes theorem, integration. Cartan's formula for Lie derivative. Cohomology via differential forms. Poincaré lemma and the Mayer-Vietoris sequence. Theorems of deRham and Hodge. (Formerly course 234B, Differential Geometry I.) Prerequisite(s): course 208. Course 201 is recommended as preparation. The Staff

210. Manifolds III. S
The fundamental group, covering space theory and Van Kampen's theorem (with a discussion of free and amalgamated products of groups), CW complexes, higher homotopy groups, cellular and singular cohomology, the Eilenberg-Steenrod axioms, computational tools including Mayer-Vietoris, cup products, Poincaré duality, and the Lefschetz fixed point theorem, homotopy exact sequence of a fibration and the Hurewicz isomorphism theorem, remarks on characteristic classes. (Formerly course 210A, Algebraic Topology.) Courses 208 and 209 recommended as preparation. The Staff

211. Algebraic Topology. *
Continuation of course 210. Topics include theory of characteristic classes of vector bundles, cobordism theory, and homotopy theory. (Formerly course 210B.) Courses 200 and 210 recommended as preparation. The Staff

212. Differential Geometry. S
Principle bundles, associated bundles and vector bundles, connections and curvature on principle and vector bundles. More advanced topics: introduction to cohomology, the Chern-Weil construction and characteristic classes, the Gauss-Bonnet theorem or Hodge theory, eigenvalue estimates for Beltrami Laplacian, comparison theorems in Riemannian geometry. (Formerly course 234C, Differential Geometry II.) Prerequisite(s): course 208. The Staff

213A. Partial Differential Equations I. S
First of the two PDE series covering basically Part I in Evans' book, Partial Differential Equations, which includes transport equations, Laplace equations, heat equations, wave equations, characteristics of nonlinear first order PDE, Hamilton-Jacobi equations, equations of conservation laws, some methods to solve equations with close forms, and Cauchy-Kovalevskaya theorem. Courses 106A and 106B are recommended as preparation. The Staff

213B. Partial Differential Equations II. *
Second of the PDE series covering basically most of Part II in Evans' book and some topics in nonlinear PDE including Sobolev space, Sobolev inequalities, existence, regularity and a priori estimates of solutions to second order elliptic PDE, parabolic equations, hyperbolic equations and systems of conservation laws, and calculus of variations and its applications to PDE. Courses 106A, 106B, and 213A are recommended as preparation. The Staff

214. Theory of Finite Groups. *
Nilpotent groups, solvable groups, Hall subgroups, the Frattini subgroup, the Fitting subgroup, Schur-Zassenhaus theorem, fusion in p-subgroups, the transfer map, Frobenius theorem on normal p-complements. Courses 200 and 201 are recommended as preparation. The Staff

215. Operator Theory. *
Operators on Banach and Hilbert spaces. The spectral theorem. Compact and Fredholm operators. Other special classes of operators. Courses 204, 205, 206, and 207 are recommended as preparation. The Staff

216. Advanced Analysis. *
Topics include maximal function, the Lebesgue set, the Marcinkiewicz interpolation theorem, singular integrals, Calderon-Zygmund theorem, Hardy Littlewood-Sobolev theorem, pseudodifferential operators, compensated compactness, concentration compactness, and applications to PDE. Course 204 is recommended as preparation. The Staff

217. Advanced Elliptic Partial Differential Equations. *
Topics include elliptic equations, existence of weak solutions, Lax-Milgram theorem, interior and boundary regularity, maximum principles, Harnack inequality, eigenvalues for symmetric and non-symmetric elliptic operators, calculus of variations (first variation: Euler-Lagrange equations, second variation: existence of minimizers). Other topics covered as time permits. Courses 204 and 205 are recommended as preparation. The Staff

218. Advanced Parabolic and Hyperbolic Partial Differential Equations. *
Topics include linear evolution equations, second order parabolic equations, maximum principles, second order hyperbolic equations, propagation of singularities, hyperbolic systems of first order, semigroup theory, systems of conservation laws, Riemann problem, simple waves, rarefaction waves, shock waves, Riemann invariants, and entropy criteria. Other topics covered as time permits. The Staff

219. Nonlinear Functional Analysis. *
Topological methods in nonlinear partial differential equations, including degree theory, bifurcation theory, and monotonicity. Topics also include variational methods in the solution of nonlinear partial differential equations. Enrollment restricted to graduate students. The Staff

220A. Representation Theory I. *
Lie groups and Lie algebras, and their finite dimensional representations. Prerequisite(s): courses 200 and 204. The Staff

220B. Representation Theory II. *
Lie groups and Lie algebras, and their finite dimensional representations. Prerequisite(s): course 220A. The Staff

222A. Algebraic Number Theory. F
Topics include algebraic integers, completions, different and discriminant, cyclotomic fields, parallelotopes, the ideal function, ideles and adeles, elementary properties of zeta functions and L-series, local class field theory, global class field theory. Courses 200, 201, and 202 are recommended as preparation. The Staff

222B. Algebraic Number Theory. *
Topics include geometric methods in number theory, finiteness theorems, analogues of Riemann-Roch for algebraic fields (after A. Weil), inverse Galois problem (Belyi theorem) and consequences. The Staff

223A. Algebraic Geometry I. *
Topics include examples of algebraic varieties, elements of commutative algebra, local properties of algebraic varieties, line bundles and sheaf cohomology, theory of algebraic curves. Weekly problem solving. Courses 200, 201, 202, and 208 are recommended as preparation. The Staff

223B. Algebraic Geometry II. *
A continuation of course 223A. Topics include theory of schemes and sheaf cohomology, formulation of Riemann-Roch theorem, birational maps, theory of surfaces. Weekly problem solving. Course 223A is recommended as preparation. The Staff

225A. Lie Algebras. F
Basic concepts of Lie algebras, Engel's theorem, Lie's theorem, Weyl's theorem are proved. Root space decomposition for semi-simple algebras, root systems and the classification theorem for semi-simple algebras over the complex numbers. Isomorphism and conjugacy theorems. Course 202 is recommended as preparation. The Staff

225B. Infinite Dimensional Lie Algebra. S
Finite dimensional semi-simple Lie algebras: PBW theorem, generators and relations, highest weight representations, Weyl character formula. Infinite dimensional Lie algebras: Heisenberg algebras, Virasoro algebras, loop algebras, affine Kac-Moody algebras, vertex operator representations. The Staff

226A. Infinite Dimensional Lie Algebras and Quantum Field Theory I. *
Introduction to the infinite-dimensional Lie algebras that arise in modern mathematics and mathematical physics: Heisenberg and Virasoro algebras, representations of the Heisenberg algebra, Verma modules over the Virasoro algebra, Kac determinant formula, and unitary and discrete series representations. Enrollment restricted to graduate students. The Staff

226B. Infinite Dimensional Lie Algebras and Quantum Field Theory II. *
Continuation of I: Kac-Moody and affine Lie algebras and their representations, integrable modules, representations via vertex operators, modular invariance of characters, and introduction to vertex operator algebras. Enrollment restricted to graduate students. The Staff

227. Lie Groups. W
Lie groups and algebras, the exponential map, the adjoint action, Lie's three theorems, Lie subgroups, the maximal torus theorem, the Weyl group, some topology of Lie groups, some representation theory: Shur's lemma, Peter-Weyl theorem, roots, weights, classification of Lie groups, the classical groups. Prerequisite(s): courses 203 and 208. The Staff

228. Lie Incidence Geometries. *
Linear incidence geometry is introduced. Linear and classical groups are reviewed, and geometries associated with projective and polar spaces are introduced. Characterizations are obtained. The Staff

229. Kac-Moody Algebras. W
Theory of Kac-Moody algebras and their representations. Weil-Kac character formula. Emphasis on representations of affine superalgebras by vertex operators. Connections to combinatorics, PDE, the monster group. The Virasoro algebra. The Staff

232. Morse Theory. S
Classical Morse Theory. The fundamental theorems relating to critical points to the topology of a manifold are treated in detail. The Bolt Periodicity Theorem. A specialized course offered every few years. The Staff

233. Random Matrix Theory. W
Classical matrix ensembles; Wigner semi-circle law; method of moments. Gaussian ensembles. Method of orthogonal polynomials; Gaudin lemma. Distribution functions for spacings and largest eigenvalue. Asymptotics and Riemann-Hilbert problem. Painleve theory and Tracy-Widom distribution. Selberg's Integral. Matrix ensembles related to classical groups; symmetric functions theory. Averages of characteristic polynomials. Fundamentals of free probability theory. Overview of connections with physics, combinatorics, and number theory. Prerequisite(s): courses 103, 117, and 205. The Staff

234. Riemann Surfaces. *
Riemann surfaces, conformal maps, harmonic forms, holomorphic forms, the theorem of Reimann-Roch, the theory of moduli. (Formerly course 211.) The Staff

235. Dynamical Systems Theory. W
An introduction to the qualitative theory of systems of ordinary differential equations. Structural stability, critical elements, stable manifolds, generic properties, bifurcations of generic arcs. Prerequisite(s): courses 106A and 203. The Staff

236. Probability Theory. *
Probability theory taught at the graduate level. Topics covered are weak convergence of probability measures, law of large numbers, central limit theorums, infinitely divisible distributions, dependent random variables, conditional expectation and conditioned probability, Markov chains, basic martingale theory, and ergodic theorems. The Staff

237. Stochastic Calculus. *
Introduces Ito's stochastic calculus. Topics covered include Brownian motion, stochastic integration, exit times, elliptic and parabolic partial differential equations, stochastic differential equations, one dimensional diffusions and functional integration. The Staff

238. Elliptic Functions and Modular Forms. S
The course, aimed at second-year graduate students, will cover the basic facts about elliptic functions and modular forms. The goal is to provide the student with foundations suitable for further work in advanced number theory, in conformal field theory, and in the theory of Riemann surfaces. Successful completion of graduate sequence (courses 200-202) and either 207 or 103 are recommended as preparation. The Staff

239. Homological Algebra. *
Homology and cohomology theories have proven to be powerful tools in many fields (topology, geometry, number theory, algebra). Independent of the field, these theories use the common language of homological algebra. The aim of this course is to acquaint the participants with basic concepts of category theory and homological algebra, as follows: chain complexes, homology, homotopy, several (co)homology theories (topological spaces, manifolds, groups, algebras, Lie groups), projective and injective resolutions, derived functors (Ext and Tor). Depending on time, spectral sequences or derived categories may also be treated. Courses 200 and 202 strongly recommended. The Staff

240A. Representations of Finite Groups I. *
Introduces ordinary representation theory of finite groups (over the complex numbers). Main topics are characters, orthogonality relations, character tables, induction and restriction, Frobenius reciprocity, Mackey's formula, Clifford theory, Schur indicator, Schur index, Artin's and Braver's induction theorems. Recommended: successful completion of courses 200-202. The Staff

240B. Representations of Finite Groups II. W
Introduces modular representation theory of finite groups (over a field of positive characteristic). Main topics are Grothendieck groups, Brauer characters, Brauer character table, projective covers, Brauer-Cartan triangle, relative projectivity, vertices, sources, Green correspondence, Green's indecomposability theorem. Recommended completion of courses 200-202 and 240A. The Staff

246. Representations of Algebras. *
Material includes associative algebras and their modules; projective and injective modules; projective covers; injective hulls; Krull-Schmidt Theorem; Cartan matrix; semisimple algebras and modules; radical, simple algebras; symmetric algebras; quivers and their representations; Morita Theory; and basic algebras. Prerequisite(s): courses 200, 201, and 202. The Staff

248. Symplectic Geometry. *
Basic definitions. Darboux theorem. Basic examples: cotangent bundles, Kahler manifolds and co-adjoint orbits. Normal form theorems. Hamiltonian group actions, momentum maps. Reduction by symmetry groups. Atiyah-Guillemin-Sternberg convexity. Introduction to Floer homological methods. Relations with other geometries including contact, Poisson, and Kahler. Prerequisite(s): course 204; course 280 is recommended as preparation. The Staff

249A. Mechanics I. *
Covers symplectic geometry and classical Hamiltonian dynamics. Some of the key subjects are the Darboux theorem, Poisson brackets, Hamiltonian and Langrangian systems, the Legendre transformation, variational principles, Hamilton-Jacobi theory, godesic equations, and an introduction to Poisson geometry. Course 208 is recommended as preparation. The Staff

249B. Mechanics II. *
Hamiltonian dynamics with symmetry. Key topics center around the momentum map and the theory of reduction in both the symplectic and Poisson context. Applications are taken from geometry, rigid body dynamics, and continuum mechanics. Course 249A is recommended as preparation. The Staff

249C. Mechanics III. *
Introduces students to active research topics tailored according to the interests of the students. Possible subjects are complete integrability and Kac-Moody Lie algebras; Smale's topological program and bifurcation theory; KAM theory, stability and chaos; relativity; quantization. Course 249B is recommended as preparation. Offered in alternate academic years. The Staff

252. Fluid Mechanics. *
First covers a basic introduction to fluid dynamics equations and then focuses on different aspects of the solutions to the Navier-Stokes equations. Prerequisite(s): courses 106A and 106B are recommended. Enrollment restricted to graduate students. The Staff

254. Geometric Analysis. F
Introduction to some basics in geometric analysis through the discussions of two fundamental problems in geometry: the resolution of the Yamabe problem and the study of harmonic maps. The analytic aspects of these problems include Sobolev spaces, best constants in Sobolev inequalities, and regularity and a priori estimates of systems of elliptic PDE. Prerequisite(s): courses 208, 212, and 213 are recommended as preparation. The Staff

256. Algebraic Curves. S
Introduction to compact Riemann surfaces and algebraic geometry via an in-depth study of complex algebraic curves. Courses 200, 201, 202, 203, 204, and 207 are recommended as preparation. Enrollment restricted to graduate mathematics and physics majors. The Staff

260. Combinatorics. *
Combinatorial mathematics, including summation methods, binomial coefficients, combinatorial sequences (Fibonacci, Stirling, Eulerian, harmonic, Bernoulli numbers), generating functions and their uses, Bernoulli processes and other topics in discrete probability. Oriented toward problem solving applications. Applications to statistical physics and computer science. The Staff

280. Topics in Analysis. *
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281. Topics in Algebra. *
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282. Topics in Geometry. *
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283. Topics in Combinatorial Theory. *
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284. Topics in Dynamics. *
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285. Topics in Partial Differential Equations. *
Topics such as derivation of the Navier-Stokes equations. Examples of flows including water waves, vortex motion, and boundary layers. Introductory functional analysis of the Navier-Stokes equation. The Staff

286. Topics in Number Theory. *
Topics in number theory, selected by instructor. Possibilities include modular and automorphic forms, elliptic curves, algebraic number theory, local fields, the trace formula. May also cover related areas of arithmetic algebraic geometry, harmonic analysis, and representation theory. Courses 200, 201, 202, and 205 are recommended as preparation. The Staff

287. Topics in Topology. *
Topics in topology, selected by the instructor. Possibilities include generalized (co)homology theory including K-theory, group actions on manifolds, equivariant and orbifold cohomology theory. May be repeated for credit. The Staff

292. Seminar (no credit). F,W,S
A weekly seminar attended by faculty, graduate students, and upper-division undergraduate students. All graduate students are expected to attend. The Staff

296. Special Student Seminar. S
Students and staff studying in an area where there is no specific course offering at that time. The Staff

297. Independent Study. F,W,S
Either study related to a course being taken or a totally independent study. The Staff

298. Master's Thesis Research. F,W,S
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299. Thesis Research. F,W,S
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*Not offered in 2006-07