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2020-21 UCSC General Catalog
2019-20 UCSC General Catalog

Group theory: 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): MATH 111A and MATH 117 are recommended as preparation.

5

Enrollment is restricted to graduate students.

Yes

Fall

Vector spaces, linear transformations, eigenvalues and eigenvectors, the Jordan canonical form, 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, skewsymmetric forms, tensor products of vector spaces, tensor algebras, symmetric algebras, exterior algebras, Clifford algebras and spin groups.

5

Prerequisite(s): MATH 200 is recommended as preparation. Enrollment is restricted to graduate students.

Winter

Module theory: 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.

5

Prerequisite(s): MATH 201 is recommended as preparation. Enrollment is restricted to graduate students.

Spring

Topics include tensor product of modules over rings, projective modules and injective modules, Jacobson radical, Wedderburns' theorem, category theory, Noetherian rings, Artinian rings, affine varieties, projective varieties, Hilbert's Nullstellensatz, prime spectrum, Zariski topology, discrete valuation rings, and Dedekind domains.

5

Prerequisite(s): MATH 200, MATH 201, and MATH 202. Enrollment is restricted to graduate students.

Fall

Completeness and compactness for real line; sequences and infinite series of functions; Fourier series; calculus on Euclidean space and the implicit function theorem; metric spaces and the contracting mapping theorem; the Arzela-Ascoli theorem; basics of general topological spaces; the Baire category theorem; Urysohn's lemma; and Tychonoff's theorem.

5

Prerequisite(s): MATH 105A and MATH 105B are recommended as preparation.Enrollment is restricted to graduate students.

Fall

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 's theorem. L^{p} spaces, derivative of a measure, the Radon-Nikodym theorem, and the fundamental theorem of calculus.

5

Prerequisite(s): MATH 204. Enrollment is restricted to graduate students.

Winter

Banach spaces, Hahn-Banach theorem, uniform boundedness theorem, the open mapping and closed graph theorems, weak and weak* topology, the Banach-Alaoglu theorem, Hilbert spaces, self-adjoint operators, compact operators, spectral theory, Fredholm operators, spaces of distributions and the Fourier transform, and Sobolev spaces.

5

Prerequisite(s): MATH 204 and MATH 205 recommended as preparation. Enrollment is restricted to graduate students.

Spring

Holomorphic and harmonic functions, Cauchy's integral theorem, the maximum principle and its consequences, conformal mapping, analytic continuation, the Riemann mapping theorem.

5

Prerequisite(s): MATH 103 is recommended as preparation. Enrollment is restricted to graduate students.

Spring

Definition of manifolds; the tangent bundle; the inverse function theorem and the implicit function theorem; transversality; Sard's theorem and the Whitney embedding theorem; vector fields, flows, and the Lie bracket; Frobenius's theorem. MATH 204 recommended for preparation.

5

Enrollment is restricted to graduate students.

Fall

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. The Poincaré lemma and the Mayer-Vietoris sequence. Theorems of deRham and Hodge.

5

Prerequisite(s): MATH 208. MATH 201 is recommended as preparation. Enrollment is restricted to graduate students.

Winter

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, the Lefschetz fixed point theorem, the exact homotopy sequence of a fibration and the Hurewicz isomorphism theorem, and remarks on characteristic classes.

5

Prerequisite(s): MATH 208 and MATH 209 recommended as preparation. Enrollment is restricted to graduate students.

Spring

Continuation of MATH 210. Topics include theory of characteristic classes of vector bundles, cobordism theory, and homotopy theory.

5

Prerequisite(s): MATH 200, MATH 201, and MATH 202 recommended as preparation. Enrollment is restricted to graduate students.

Winter

Principal bundles, associated bundles and vector bundles, connections and curvature on principal and vector bundles. More advanced topics include: introduction to cohomology, the Chern-Weil construction and characteristic classes, the Gauss-Bonnet theorem or Hodge theory, eigenvalue estimates for Beltrami Laplacian, and comparison theorems in Riemannian geometry.

5

Prerequisite(s): MATH 208. Enrollment is restricted to graduate students.

Spring

First of the two PDE courses 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; conservation laws; some methods for solving equations in closed form; and the Cauchy-Kovalevskaya theorem.

5

MATH 106 and MATH 107 are recommended as preparation. Enrollment is restricted to graduate students.

Winter

Second course of the PDE series covering basically most of Part II in Evans' book and some topics in nonlinear PDE including Sobolev spaces, 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.

5

Prerequisite(s): MATH 106, MATH 107, and MATH 213A are recommended as preparation. Enrollment is restricted to graduate students.

Spring

Nilpotent groups, solvable groups, Hall subgroups, the Frattini subgroup, the Fitting subgroup, the Schur-Zassenhaus theorem, fusion in p-subgroups, the transfer map, Frobenius theorem on normal p-complements.

5

Prerequisite(s): MATH 200 and MATH 201 recommended as preparation. Enrollment is restricted to graduate students.

Operators on Banach spaces and Hilbert spaces. The spectral theorem. Compact and Fredholm operators. Other special classes of operators.

5

Prerequisite(s): MATH 204, MATH 205, MATH 206, and MATH 207 are recommended as preparation. Enrollment is restricted to graduate students.

Topics include: the Lebesgue set, the Marcinkiewicz interpolation theorem, singular integrals, the Calderon-Zygmund theorem, Hardy Littlewood-Sobolev theorem, pseudodifferential operators, compensated compactness, concentration compactness, and applications to PDE.

5

Prerequisite(s): MATH 204, MATH 205, and MATH 206 recommended as preparation. Enrollment is restricted to graduate students.

Winter

Topics include elliptic equations, existence of weak solutions, the Lax-Milgram theorem, interior and boundary regularity, maximum principles, the 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.

5

Prerequisite(s): MATH 204, MATH 205, and MATH 206 recommended as preparation. Enrollment is restricted to graduate students.

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.

5

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.

5

Enrollment is restricted to graduate students.

Lie groups and Lie algebras, and their finite dimensional representations.

5

Prerequisite(s): MATH 200, MATH 201, and MATH 202. MATH 225A and MATH 227 recommended as preparation. Enrollment is restricted to graduate students.

Winter

Lie groups and Lie algebras, and their finite dimensional representations.

5

Prerequisite(s): MATH 220A. Enrollment is restricted to graduate students.

Spring

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. MATH 200, MATH 201, and MATH 202 are recommended as preparation.

5

Enrollment is restricted to graduate students.

Fall

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.

5

Enrollment is restricted to graduate students.

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. MATH 200, MATH 201, MATH 202, and MATH 208 are recommended as preparation.

5

Enrollment is restricted to graduate students.

A continuation of course 223A. Topics include theory of schemes and sheaf cohomology, formulation of the Riemann-Roch theorem, birational maps, theory of surfaces. Weekly problem solving. MATH 223A is recommended as preparation.

5

Enrollment is restricted to graduate students.

Winter

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.

5

Prerequisite(s): MATH 201 and MATH 202 recommended as preparation Enrollment is restricted to graduate students.

Fall

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.

5

Prerequisite(s): MATH 225A. Enrollment is restricted to graduate students.

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, the Kac determinant formula, and unitary and discrete series representations.

5

Enrollment is restricted to graduate students.

Continuation of MATH 226A: 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.

5

Enrollment is restricted to graduate students.

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: Schur's Lemma, the Peter-Weyl theorem, roots, weights, classification of Lie groups, the classical groups.

5

Prerequisite(s): MATH 200, MATH 201, MATH 204, and MATH 208. Enrollment is restricted to graduate students.

Fall

Linear incidence geometry is introduced. Linear and classical groups are reviewed, and geometries associated with projective and polar spaces are introduced. Characterizations are obtained.

5

Enrollment is restricted to graduate students.

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

5

Enrollment is restricted to graduate students.

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

5

Prerequisite(s): MATH 208, MATH 209, MATH 210, MATH 211, and MATH 212 recommended as preparation. Enrollment is restricted to graduate students.

Winter

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 the 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.

5

Prerequisite(s): MATH 103, MATH 204, and MATH 205; MATH 117 recommended as preparation. Enrollment is restricted to graduate students.

Riemann surfaces, conformal maps, harmonic forms, holomorphic forms, the Reimann-Roch theorem, the theory of moduli.

5

Enrollment is restricted to graduate students.

Spring

An introduction to the qualitative theory of systems of ordinary differential equations. Structural stability, critical elements, stable manifolds, generic properties, bifurcations of generic arcs.

5

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.

5

Prerequisite(s): MATH 200, MATH 201, MATH 202, and either MATH 207 or MATH 103A are recommended as preparation. Enrollment is restricted to graduate students.

Fall

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. MATH 200 and MATH 202 strongly recommended.

5

Enrollment is restricted to graduate students.

Spring

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 Brauer's induction theorems. Recommended: successful completion of MATH 200-MATH 202.

5

Enrollment is restricted to graduate students.

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 MATH 200-MATH 203 and MATH 240A.

5

Prerequisite(s): MATH 200, MATH 201, MATH 202, MATH 203, and MATH 240A recommended. Enrollment is restricted to graduate students.

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.

5

Prerequisite(s): MATH 200, MATH 201, and MATH 202. Enrollment is restricted to graduate students.

Basic definitions. Darboux theorem. Basic examples: cotangent bundles, Kähler manifolds and co-adjoint orbits. Normal form theorems. Hamiltonian group actions, moment maps. Reduction by symmetry groups. Atiyah-Guillemin-Sternberg convexity. Introduction to Floer homological methods. Relations with other geometries including contact, Poisson, and Kähler geometry.

5

Prerequisite(s): MATH 204; MATH 208 and MATH 209 are recommended as preparation. Enrollment is restricted to graduate students.

Covers symplectic geometry and classical Hamiltonian dynamics. Some of the key subjects are the Darboux theorem, Poisson brackets, Hamiltonian and Langrangian systems, Legendre transformations, variational principles, Hamilton-Jacobi theory, geodesic equations, and an introduction to Poisson geometry. MATH 208 and MATH 209 are recommended as preparation.

5

MATH 208 and MATH 209 recommended as preparation. Enrollment is restricted to graduate students.

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. MATH 249A is recommended as preparation.

5

Enrollment is restricted to graduate students.

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. MATH 249B is recommended as preparation.

5

Enrollment is restricted to graduate students.

First covers a basic introduction to fluid dynamics equations and then focuses on different aspects of the solutions to the Navier-Stokes equations.

5

Prerequisite(s): MATH 106 and MATH 107 are recommended as preparation. Enrollment is restricted to graduate students.

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.

5

MATH 204, MATH 205, MATH 209, MATH 212, and MATH 213A recommended as preparation. Enrollment is restricted to graduate students.

Spring

Introduction to compact Riemann surfaces and algebraic geometry via an in-depth study of complex algebraic curves.

5

MATH 200, MATH 201, MATH 202, MATH 203, MATH 204, and MATH 207 are recommended as preparation. Enrollment is restricted to graduate mathematics and physics students.

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.

5

Enrollment is restricted to graduate students.

Concepts of inverse problem and ill-posedness on the Hilbert scale. Approaches to inversion, regularization and implementation. In Euclidean geometry: Radon transform; X-ray transform; attenuated X-ray transform (Novikov's inversion formula); weighted transforms. Same topics in different geometric contexts: homogeneous spaces, manifolds with boundary. Non-linear problems: boundary rigidity, lens rigidity, inverse problems for connections. MATH 148, MATH 204, MATH 205, MATH 206, and MATH 208, are recommended for preparation.

5

The Staff

Enrollment is restricted to graduate students.

Fall

5

Enrollment is restricted to graduate students.

Yes

5

Enrollment is restricted to graduate students.

Yes

5

Enrollment is restricted to graduate students.

Yes

5

Enrollment is restricted to graduate students.

Yes

5

Enrollment is restricted to graduate students.

Yes

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.

5

Enrollment is restricted to graduate students.

Yes

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.

5

Enrollment is restricted to graduate students.

Yes

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.

5

Enrollment is restricted to graduate students.

Yes

Prepares graduate students to become successful Teaching Assistants in mathematics courses. Topics include class management, assessment creation, evaluation and grading, student interaction, introduction to teaching and learning strategies, innovation in education, use of technology, and best practices that promote diversity and inclusion.

2

Pedro Morales

Enrollment is restricted to graduate students.

Fall

Prepares graduate students to become successful Graduate Student Instructors in mathematics. Topics include class management, assessment creation, evaluation and grading, student interaction, introduction to teaching and learning strategies, innovation in education, use of technology, and best practices that promote diversity and inclusion.

2

Pedro Morales

Enrollment is restricted to graduate students.

Winter

A weekly seminar attended by faculty, graduate students, and upper-division undergraduate students. All graduate students are expected to attend.

0

Enrollment is restricted to graduate students.

Fall, Winter, Spring

Students and staff studying in an area where there is no specific course offering at that time.

5

Enrollment is restricted to graduate students.

Fall, Winter, Spring

Either study related to a course being taken or a totally independent study. Enrollment restricted to graduate students.

5

Yes

Either study related to a course being taken or a totally independent study. Enrollment restricted to graduate students.

10

Yes

Either study related to a course being taken or a totally independent study. Enrollment restricted to graduate students.

15

Yes

Enrollment restricted to graduate students.

5

Fall, Winter, Spring

Enrollment restricted to graduate students.

5

Yes

Enrollment restricted to graduate students.

10

Yes

Enrollment restricted to graduate students.

15

Yes