# MATH - Mathematics

## MATH2 College Algebra for Calculus

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.

### Credits

5

## MATH2S College Algebra for Calculus

This two-credit, stretch course offers students two quarters to master material covered in course 2: 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. After successful completion of this course in the first quarter, students enroll in course 2 the following quarter to complete the sequence and earn an additional 5 credits.

### Credits

2

## MATH2T Preparatory Math: Tutorial

Independent study of algebra and modern mathematics using adaptive learning software. Instruction emphasizes clear mathematical communication and reasoning when working with sets, equations, functions, and graphs. Drop in labs, online forums, and readings provide opportunities for further learning and exploration.

### Credits

2

## MATH3 Precalculus

Inverse functions and graphs; exponential and logarithmic 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 MATH 3 and AM 3. AM 3 can substitute for MATH 3.

### Credits

5

## MATH4 Mathematics of Choice and Argument

Techniques of analyzing and creating quantitative arguments. Application of probability theory to questions in justice, medicine, and economics. Analysis and avoidance of statistical bias. Understanding the application and limitations of quantitative techniques.

### Credits

5

## MATH11A Calculus with Applications

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 MATH 19A, or AM 11A, or AM 15A, or ECON 11A.

### Credits

5

## MATH11B Calculus with Applications

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 MATH 19B, or AM 11B, or AM 15B, or ECON 11B.

### Credits

5

## MATH19A Calculus for Science, Engineering, and Mathematics

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 MATH 11A, or AM 11A, or AM 15A, or ECON 11A.

### Credits

5

## MATH19B Calculus for Science, Engineering, and Mathematics

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 MATH 11B, or AM 11B, or AM 15B, or ECON 11B.

### Credits

5

## MATH20A Honors Calculus

Methods of proof, number systems, binomial and geometric sums. Sequences, limits, continuity, and the definite integral. The derivatives of the elementary functions, the fundamental theorem of calculus, and the main theorems of differential calculus.

### Credits

5

## MATH20B Honors Calculus

Orbital mechanics, techniques of integration, and separable differential equations. Taylor expansions and error estimates, the Gaussian integral, Gamma function and Stirling's formula. Series and power series, numerous applications to physics.

### Credits

5

## MATH21 Linear Algebra

Systems of linear equations matrices, determinants. Introduces abstract vector spaces, linear transformation, inner products, the geometry of Euclidean space, and eigenvalues. Students cannot receive credit for this course and AM 10.

### Credits

5

## MATH22 Introduction to Calculus of Several Variables

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.

### Credits

5

## MATH23A Vector Calculus

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. (Formerly Multivariable Calculus.)

### Credits

5

## MATH23B Vector Calculus

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. (Formerly Multivariable Calculus.)

### Credits

5

## MATH24 Ordinary Differential Equations

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 AM 20.

### Credits

5

## MATH99 Tutorial

### Credits

5

## MATH99F Tutorial

### Credits

2

## MATH100 Introduction to Proof and Problem Solving

Students learn the basic concepts and ideas necessary for upper-division mathematics and techniques of mathematical proof. Introduction to sets, relations, elementary mathematical logic, proof by contradiction, mathematical induction, and counting arguments.

### Credits

5

## MATH101 Mathematical Problem Solving

Students learn the strategies, tactics, skills and tools that mathematicians use when faced with a novel (new) problem. These include generalization, specialization, the optimization, invariance, symmetry, Dirichlet's box principle among others in the context of solving problems from number theory, geometry, calculus, combinatorics, probability, algebra, analysis, and graph theory. (Formerly, course 30.)

### Credits

5

## MATH103A Complex Analysis

Complex numbers, analytic and harmonic functions, complex integration, the Cauchy integral formula, Laurent series, singularities and residues, conformal mappings. (Formerly course 103.)

### Credits

5

## MATH103B Complex Analysis II

Conformal mappings, the Riemann mapping theorem, Mobius transformations, Fourier series, Fourier and Laplace transforms, applications, and other topics as time permits.

### Credits

2

## MATH105A Real Analysis

The basic concepts of one-variable calculus are treated rigorously. Set theory, the real number system, numerical sequences and series, continuity, differentiation.

### Credits

5

## MATH105B Real Analysis

Metric spaces, differentiation and integration of functions. The Riemann-Stieltjes integral. Sequences and series of functions.

### Credits

5

## MATH105C Real Analysis

The Stone-Weierstrass theorem, Fourier series, differentiation and integration of functions of several variables.

### Credits

5

## MATH106 Systems of Ordinary Differential Equations

Linear systems, exponentials of operators, existence and uniqueness, stability of equilibria, periodic attractors, and applications.

### Credits

5

## MATH107 Partial Differential Equations

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, special functions including Bessel and Legendre functions, distributions and transforms.

### Credits

5

## MATH110 Introduction to Number Theory

Prime numbers, unique factorization, congruences with applications (e.g., to magic squares). Rational and irrational numbers. Continued fractions. Introduction to Diophantine equations. An introduction to some of the ideas and outstanding problems of modern mathematics.

### Credits

5

## MATH111A Algebra

Group theory including the Sylow theorem, the structure of abelian groups, and permutation groups. Students cannot receive credit for this course and course 111T.

### Credits

5

## MATH111B Algebra

Introduction to rings and fields including polynomial rings, factorization, the classical geometric constructions, and Galois theory.

### Credits

5

## MATH111T Algebra

Introduction to groups, rings and fields; integers and polynomial rings; divisibility and factorization; homomorphisms and quotients; roots and permutation groups; and plane symmetry groups. Also includes an introduction to algebraic numbers, constructible numbers, and Galois theory. Focuses on topics most relevant to future K-12 teachers. Students cannot receive credit for this course and course 111A.

### Credits

5

## MATH114 Introduction to Financial Mathematics

Financial derivatives: contracts and options. Hedging and risk management. 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.

### Credits

5

## MATH115 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.

### Credits

5

## MATH116 Combinatorics

Based on induction and elementary counting techniques: counting subsets, partitions, and permutations; recurrence relations and generating functions; the principle of inclusion and exclusion; Polya enumeration; Ramsey theory or enumerative geometry.

### Credits

5

## MATH117 Advanced Linear Algebra

Review of abstract vector spaces. Dual spaces, bilinear forms, and the associated geometry. Normal forms of linear mappings. Introduction to tensor products and exterior algebras.

### Credits

5

## MATH118 Advanced Number Theory

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.

### Credits

5

## MATH120 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.

### Credits

5

## MATH121A Differential Geometry

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.

### Credits

5

## MATH121B Differential Geometry and Topology

Examples of surfaces of constant curvature, surfaces of revolutions, minimal surfaces. Abstract manifolds; integration theory; Riemannian manifolds. Total curvature and geodesics; the Euler characteristic, the Gauss-Bonnet theorem. Length-minimizing properties of geodesics, complete surfaces, curvature and conjugate points covering surfaces. Surfaces of constant curvature; the theorems of Bonnet and Hadamard.

### Credits

5

## MATH124 Introduction to Topology

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 fixed-point theorem.

### Credits

5

## MATH128A Classical Geometry: Euclidean and Non-Euclidean

Euclidean, projective, spherical, and hyperbolic (non-Euclidean) geometries. Begins with the thirteen books of Euclid. Surveys the other geometries. Attention paid to constructions and visual intuition as well as logical foundations. Rigid motions and projective transformations covered.

### Credits

5

## MATH128B Classical Geometry: Projective

Theorems of Desargue, Pascal, and Pappus; projectivities; homogeneous and affine coordinates; conics; relation to perspective drawing and some history.

### Credits

5

## MATH129 Algebraic Geometry

Algebraic geometry of affine and projective curves, including conics and elliptic curves; Bezout's theorem; coordinate rings and Hillbert's Nullstellensatz; affine and projective varieties; and regular and singular varieties. Other topics, such as blow-ups and algebraic surfaces as time permits.

### Credits

5

## MATH130 Celestial Mechanics

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.

### Credits

5

## MATH134 Cryptography

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.

### Credits

5

## MATH140 Industrial Mathematics

Introduction to mathematical modeling of industrial problems. Problems in air quality remediation, image capture and reproduction, and crystallization are modeled as ordinary and partial differential equations then analyzed using a combination of qualitative and quantitative methods.

### Credits

5

## MATH145 Introductory Chaos Theory

The Lorenz and Rossler attractors, measures of chaos, attractor reconstruction, and applications from the sciences. Students cannot receive credit for this course and AM 114.

### Credits

5

## MATH145L Introductory Chaos Laboratory

Laboratory sequence illustrating topics covered in course 145. One three-hour session per week in microcomputer laboratory.

### Credits

1

## MATH148 Numerical Analysis

A survey of the basic numerical methods which are used to solve scientific problems, including mathematical analysis and computing assignments. Some prior experience with Matlab (or similar) is helpful but not required. Some typical topics are: computer arithmetic; Newton's method for non-linear equations; linear algebra; interpolation and approximation; numerical differentiation and integration; numerical solutions of systems of ordinary differential equations and some partial differential equations; convergence and error bounds.

### Credits

5

## MATH148L Numerical Analysis Laboratory

Laboratory sequence illustrating topics covered in course 148. One three-hour session per week in the computer laboratory.

### Credits

1

## MATH152 Programming for Mathematics

Introduces programming in Python with applications to advanced mathematics. Students apply data structures and algorithms to topics such as numerical approximation, number theory, linear algebra, and combinatorics. No programming experience is necessary, but a strong mathematics background is required.

### Credits

5

## MATH160 Mathematical Logic I

Propositional and predicate calculus. Resolution, completeness, compactness, and Lowenheim-Skolem theorem. Recursive functions, Godel incompleteness theorem. Undecidable theories. Hilbert's 10th problem.

### Credits

5

## MATH161 Mathematical Logic II

Naive set theory and its limitations (Russell's paradox); construction of numbers as sets; cardinal and ordinal numbers; cardinal and ordinal arithmetic; transfinite induction; axiom systems for set theory, with particular emphasis on the axiom of choice and the regularity axiom and their consequences (such as, the Banach-Tarski paradox); continuum hypothesis.

### Credits

5

## MATH181 History of Mathematics

A survey from a historical point of view of various developments in mathematics. Specific topics and periods to vary yearly.

### Credits

5

## MATH188 Supervised Teaching

Supervised tutoring in self-paced courses. May not be repeated for credit. Students submit petition to sponsoring agency.

### Credits

5

## MATH192 Dir Stu Teach

### Credits

5

## MATH194 Senior Seminar

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.

### Credits

5

## MATH195 Senior Thesis

Students research a mathematical topic under the guidance of a faculty sponsor and write a senior thesis demonstrating knowledge of the material. Prerequisite(s): satisfaction of the Entry Level Writing and Composition requirements. Students submit petition to sponsoring agency.

### Credits

5

## MATH199 Tutorial

Students submit petition to sponsoring agency.

### Credits

5

## MATH199F Tutorial

Tutorial

### Credits

2

## MATH200 Algebra I

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): courses 111A and 117 are recommended as preparation.

### Credits

5

## MATH201 Algebra II

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.

### Credits

5

## MATH202 Algebra III

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.

### Credits

5

## MATH203 Algebra IV

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.

### Credits

5

## MATH204 Analysis I

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.

### Credits

5

## MATH205 Analysis II

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.### Credits

5

## MATH206 Analysis III

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.

### Credits

5

## MATH207 Complex Analysis

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

### Credits

5

## MATH208 Manifolds I

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. Course 204 recommended for preparation.

### Credits

5

## MATH209 Manifolds II

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.

### Credits

5

## MATH210 Manifolds III

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.

### Credits

5

## MATH211 Algebraic Topology

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

### Credits

5

## MATH212 Differential Geometry

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.

### Credits

5

## MATH213A Partial Differential Equations I

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.### Credits

5

## MATH213B Partial Differential Equations II

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.

### Credits

5

## MATH214 Theory of Finite Groups

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.

### Credits

5

## MATH215 Operator Theory

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

### Credits

5

## MATH216 Advanced Analysis

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.

### Credits

5

## MATH217 Advanced Elliptic Partial Differential Equations

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.

### Credits

5

## MATH218 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.

### Credits

5

## MATH219 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.

### Credits

5

## MATH220A Representation Theory I

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

### Credits

5

## MATH220B Representation Theory II

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

### Credits

5

## MATH222A Algebraic Number Theory

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.

### Credits

5

## MATH222B 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.

### Credits

5

## MATH223A 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.

### Credits

5

## MATH223B Algebraic Geometry II

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. Course 223A is recommended as preparation.

### Credits

5

## MATH225A Lie Algebras

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.

### Credits

5

## MATH225B Infinite Dimensional Lie Algebras

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.

### Credits

5

## MATH226A 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, the Kac determinant formula, and unitary and discrete series representations.

### Credits

5

## MATH226B Infinite Dimensional Lie Algebras and Quantum Field Theory II

Continuation of course 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.

### Credits

5

## MATH227 Lie Groups

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.

### Credits

5

## MATH228 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.

### Credits

5

## MATH229 Kac-Moody Algebras

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.

### Credits

5

## MATH232 Morse Theory

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.

### Credits

5

## MATH233 Random Matrix Theory

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.

### Credits

5

## MATH234 Riemann Surfaces

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

### Credits

5

## MATH235 Dynamical Systems Theory

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

### Credits

5

## MATH238 Elliptic Functions and Modular Forms

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.

### Credits

5

## MATH239 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.

### Credits

5

## MATH240A 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 Brauer's induction theorems. Recommended: successful completion of courses 200-202.

### Credits

5

## MATH240B Representations of Finite Groups II

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 200-203 and 240A.

### Credits

5

## MATH246 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.

### Credits

5

## MATH248 Symplectic Geometry

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.

### Credits

5

## MATH249A Mechanics I

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. Courses 208 and 209 are recommended as preparation.

### Credits

5

## MATH249B 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.

### Credits

5

## MATH249C 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.

### Credits

5

## MATH252 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.

### Credits

5

## MATH254 Geometric Analysis

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.

### Credits

5

## MATH256 Algebraic Curves

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

### Credits

5

## MATH260 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.

### Credits

5

## MATH264 Inverse Problems and Integral Geometry

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. Courses 148, 204, 205, 206, and 208, are recommended for preparation.

### Credits

5

## MATH280 Topics in Analysis

### Credits

5

## MATH281 Topics in Algebra

### Credits

5

## MATH282 Topics in Geometry

### Credits

5

## MATH283 Topics in Combinatorial Theory

### Credits

5

## MATH284 Topics in Dynamics

### Credits

5

## MATH285 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.

### Credits

5

## MATH286 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.

### Credits

5

## MATH287 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.

### Credits

5

## MATH288A Pedagogy of Mathematics

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.

### Credits

2

## MATH288B Pedagogy of Mathematics

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.

### Credits

2

## MATH292 Seminar

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

### Credits

0

## MATH296 Special Student Seminar

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

### Credits

5

## MATH297A Independent Study

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

### Credits

5

## MATH297B Independent Study

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

### Credits

10

## MATH297C Independent Study

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

### Credits

15

## MATH298 Master's Thesis Research

Enrollment restricted to graduate students.

### Credits

5

## MATH299A Thesis Research

Enrollment restricted to graduate students.

### Credits

5

## MATH299B Thesis Research

Enrollment restricted to graduate students.

### Credits

10

## MATH299C Thesis Research

Enrollment restricted to graduate students.

### Credits

15