Graduate Courses for Mathematics (MATH)

Schedule of Classes: Fall | Winter | Spring | Summer
(Only current and next semester available)

MATH 400 Vectors and Matrices (3 credits)
Prerequisite: MATH221 or equivalent. Not open to students in the CMPS or Engineering Colleges. Credit will be granted for only one of the following: MATH240, MATH341, MATH400, or MATH461.
The essentials of matrix theory needed in the management, social and biological sciences. Main topics: systems of linear equations, linear independence, rank, orthogonal transformations, eigenvalues, the principal axes theorem. Typical applications: linear models in economics and in statistics, Markov chains, age-specific population growth.

MATH 401 Applications of Linear Algebra (3 credits)
Prerequisite: MATH240 or MATH461.
Various applications of linear algebra: theory of finite games, linear programming, matrix methods as applied to finite Markov chains, random walk, incidence matrices, graphs and directed graphs, networks and transportation problems.

MATH 402 Algebraic Structures (3 credits)
Prerequisite: MATH240 or equivalent. Not open to mathematics graduate students. Credit will be granted for only one of the following: MATH402 or MATH403.
For students having only limited experience with rigorous mathematical proofs. Parallels MATH403. Students planning graduate work in mathematics should take MATH403. Groups, rings, integral domains and fields, detailed study of several groups; properties of integers and polynomials. Emphasis is on the origin of the mathematical ideas studied and the logical structure of the subject.

MATH 403 Introduction to Abstract Algebra (3 credits)
Prerequisite: MATH240 and MATH241; or equivalent. Credit will be granted for only one of the following: MATH402 or MATH403.
Integers; groups, rings, integral domains, fields.

MATH 404 Field Theory (3 credits)
Prerequisite: MATH403.
Algebraic and transcendental elements, Galois theory, constructions with straight-edge and compass, solutions of equations of low degrees, insolubility of the quintic equation, Sylow theorems, fundamental theorem of finite Abelian groups.

MATH 405 Linear Algebra (3 credits)
Prerequisite: MATH240 or MATH461.
An abstract treatment of finite dimensional vector spaces. Linear transformations and their invariants.

MATH 406 Introduction to Number Theory (3 credits)
Prerequisite: MATH141 or permission of department.
Integers, divisibility, prime numbers, unique factorization, congruences, quadratic reciprocity, Diophantine equations and arithmetic functions.

MATH 410 Advanced Calculus I (3 credits)
Prerequisites: MATH240 and MATH241, with grade of C or better; and permission of department.
First semester of a year course. Subjects covered during the year are: sequences and series of numbers, continuity and differentiability of real valued functions of one variable, the Riemann integral, sequences of functions and power series. Functions of several variables including partial derivatives, multiple integrals, line and surface integrals. The implicit function theorem.

MATH 411 Advanced Calculus II (3 credits)
Prerequisite: MATH410 and permission of department. Credit will be granted for only one of the following: MATH411 or MATH412.
Continuation of MATH410.

MATH 412 Advanced Calculus with Applications (3 credits)
Prerequisite: MATH410 and permission of department. Not open to students who have completed MATH350 and MATH351. Credit will be granted for only one of the following: MATH411 or MATH412.
Analysis in several variables, and applications, from a computational perspective.

MATH 414 Differential Equations (3 credits)
Prerequisites: MATH410 and MATH240; or equivalent.
Existence and uniqueness theorems for initial value problems. Linear theory: fundamental matrix solutions, variation of constants formula, Floquet theory for periodic linear systems. Asymptotic orbital and Lyapunov stability with phase plane diagrams. Boundary value theory and series solutions.

MATH 416 Applied Harmonic Analysis: An Introduction to Signal Porcessing (3 credits)
Prerequisite: MATH141 and MATH240; or permission of department. Familiarity with MATLAB is also required.
Introduces students to the mathematical concepts arising in signal analysis from the applied harmonic anaylsis point of view. Topics include applied linear algebra, Fourier series, discrete Fourier transform, Fourier transform, Shannon Sampling Theorem, wavelet bases, multiresolution analysis, and discrete wavelet transfrom.

MATH 420 Mathematical Modeling (3 credits)
Prerequisite: MATH241, MATH246, STAT400, MATH240 or MATH461; and permission of department. Also offered as AMSC420. Credit will be granted for only one of the following: AMSC420, MAPL420, or MATH420.
The course will develop skills in mathematical modeling through practical experience. Students will work in groups on specific projects involving real-life problems that are accessible to their existing mathematical backgrounds. In addition to the development of mathematical models, emphasis will be placed on the use of computational methods to investigate these models, and effective oral and written presentation of the results.

MATH 424 Introduction to the Mathematics of Finance (3 credits)
Prerequisites: MATH141; and either STAT400 or BMGT231 and permission of department. Recommended: MATH240, MATH241, or MATH246. Credit will be granted for only one of the following: BMGT444, MATH424 or MATH498F. Formerly MATH 498F.
Introduction to the mathematical models used in finance and economics with emphasis on pricing derivative instruments. Designed for students in mathematics, computer science, engineering, finance and physics. Financial markets and instruments; elements from basic probability theory; interest rates and present value analysis; normal distribution of stock returns; option pricing; arbitrage pricing theory; the multiperiod binomial model; the Black-Scholes option pricing formula; proof of the Black-Scholes option pricing formula and applications; trading and hedging of options; Delta hedging; utility functions and portfolio theory; elementary stochastic calculus; Ito's Lemma; the Black-Scholes equation and its conversion to the heat equation.

MATH 430 Euclidean and Non-Euclidean Geometries (3 credits)
Prerequisite: MATH141.
Hilbert's axioms for Euclidean geometry. Neutral geometry: the consistency of the hyperbolic parallel postulate and the inconsistency of the elliptic parallel postulate with neutral geometry. Models of hyerbolic geometry. Existence and properties of isometries.

MATH 431 Geometry for Computer Graphics (3 credits)
Prerequisite: MATH240 or MATH461.
Topics from projective geometry and transformation geometry, emphasizing the two-dimensional representation of three-dimensional objects and objects moving about in the plane and space. The emphasis will be on formulas and algorithms of immediate use in computer graphics.

MATH 432 Introduction to Topology (3 credits)
Prerequisite: MATH410 or equivalent.
Metric spaces, topological spaces, connectedness, compactness (including Heine-Borel and Bolzano-Weierstrass theorems), Cantor sets, continuous maps and homeomorphisms, fundamental group (homotopy, covering spaces, the fundamental theorem of algebra, Brouwer fixed point theorem), surfaces (e.g., Euler characteristic, the index of a vector field, hairy sphere theorem), elements of combinatorial topology (graphs and trees, planarity, coloring problems).

MATH 436 Differential Geometry of Curves and Surfaces I (3 credits)
Prerequisites: MATH241; and either MATH240 or MATH461; and two 400-level MATH courses (not including MATH400, 461 and 478).
Curves in the plane and Euclidean space, moving frames, surfaces in Euclidean space, orientability of surfaces; Gaussian and mean curvatures; surfaces of revolution, ruled surfaces, minimal surfaces, special curves on surfaces, "Theorema Egregium"; the intrinsic geometry of surfaces.

MATH 437 Differential Forms (3 credits)
Prerequisite: MATH241; and either MATH240 or MATH461. Recommended: One of the following - MATH403, MATH405, MATH410, MATH432, or MATH436.
Introduction to differential forms and their applications, and unites the fundamental theorems of multivariable calculus in a general Stokes Theorem that is valid in great generality. It develops this theory and technique to perform calculations in analysis and geometry. Topics include an introduction to topological spaces, the Gauss-Bonnet Theorem, Gauss's formula for the linking number, and the Cauchy Integral Theorem. Applications include Maxwell's equations of electromagnetism, connections and guage theory, and symplectic geometry and Hamiltonian dynamics.

MATH 445 Elementary Mathematical Logic (3 credits)
Prerequisite: MATH141. Credit will be granted for only one of the following: MATH445 or MATH450/CMSC450.
Elementary development of propositional and predicate logic, including semantics and deductive systems and with a discussion of completeness, incompleteness and the decision problem.

MATH 446 Axiomatic Set Theory (3 credits)
Prerequisite: MATH403 or MATH410.
Development of a system of axiomatic set theory, choice principles, induction principles, ordinal arithmetic including discussion of cancellation laws, divisibility, canonical expansions, cardinal arithmetic including connections with the axiom of choice, Hartog's theorem, Konig's theorem, properties of regular, singular and inaccessible cardinals.

MATH 450 Logic for Computer Science (3 credits)
Prerequisites: CMSC251 and MATH141, with grade of C or better and permission of department. Also offered as CMSC450. Credit will be granted for only one of the following: MATH445 or MATH450/CMSC450.
Elementary development of propositional and first-order logic accessible to the advanced undergraduate computer science student, including the resolution method in propositional logic and Herbrand's Unsatisfiability Theorem in first-order logic. Included are the concepts of truth, interpretation, validity, provability, soundness, completeness, incompleteness, decidability and semi-decidability.

MATH 452 Introduction to Dynamics and Chaos (3 credits)
Prerequisite: MATH240 and MATH246. Also offered as AMSC452. Credit will be granted for only one of the following: AMSC452, MAPL452 or MATH452.
An introduction to mathematical dynamics and chaos. Orbits, bifurcations, Cantor sets and horseshoes, symbolic dynamics, fractal dimension, notions of stability, flows and chaos. Includes motivation and historical perspectives, as well as examples of fundamental maps studied in dynamics and applications of dynamics.

MATH 456 Cryptology (3 credits)
Prerequisites: Any two 400-level MATH courses; or CMSC330 and CMSC351 and permission of department. Also offered as CMSC456. Credit will be granted for only one of the following: MATH456 or CMSC456.
Importance in protecting data in communications between computers. The subject lies on the border between mathematics and computer science. Mathematical topics include number theory and probability. Computer science topics include complexity theory.

MATH 461 Linear Algebra for Scientists and Engineers (3 credits)
Prerequisites: MATH141 and one MATH/STAT course for which MATH141 is a prerequisite. This course cannot be used toward the upper level math requirements for MATH/STAT majors. Credit will be granted for only one of the following: MATH240, MATH341, MATH400 or MATH461.
Basic concepts of linear algebra. This course is similar to MATH 240, but with more extensive coverage of the topics needed in applied linear algebra: change of basis, complex eigenvalues, diagonalization, the Jordan canonical form.

MATH 462 Partial Differential Equations for Scientists and Engineers (3 credits)
Prerequisites: MATH241 and MATH246.
Linear spaces and operators, orthogonality, Sturm-Liouville problems and eigenfunction expansions for ordinary differential equations. Introduction to partial differential equations, including the heat equation, wave equation and Laplace's equation. Boundary value problems, initial value problems and initial-boundary value problems.

MATH 463 Complex Variables for Scientists and Engineers (3 credits)
Prerequisite: MATH241 or equivalent.
The algebra of complex numbers, analytic functions, mapping properties of the elementary functions. Cauchy integral formula. Theory of residues and application to evaluation of integrals. Conformal mapping.

MATH 464 Transform Methods for Scientists and Engineers (3 credits)
Prerequisite: MATH246.
Fourier series, Fourier and Laplace transforms. Evaluation of the complex inversion integral by the theory of residues. Applications to ordinary and partial differential equations of mathematical physics: solutions using transforms and separation of variables. Additional topics such as Bessel functions and calculus of variations.

MATH 470 Mathematics for Secondary Education (3 credits)
Prerequisite: MATH140, MATH141, and one 400-level Math course. Not open to students who have completed MATH498E. Credit will be granted for only one of the following: MATH470 or MATH498E. Formerly MATH 498E.
An advanced perspective on some of the core mathematics underlying high school mathematics courses. Topics include number sytems, functions of one variable, equations, inequalities, trigonometric functions, curve fitting, and polynomials. The course includes an analysis of alternate approaches to mathematical ideas and problems, and makes connections between ideas that may have been studied separately in different high school and college courses.

MATH 475 Combinatorics and Graph Theory (3 credits)
Prerequisites: MATH240 and MATH241; and permission of department. Also offered as CMSC475. Credit will be granted for only one of the following: MATH475 or CMSC475.
General enumeration methods, difference equations, generating functions. Elements of graph theory, matrix representations of graphs, applications of graph theory to transport networks, matching theory and graphical algorithms.

MATH 478 Selected Topics For Teachers of Mathematics (1-3 credits)
Prerequisite: one year of college mathematics or permission of department. This course cannot be used toward the upper level math requirements for MATH/STAT majors.

MATH 480 Algebra for Middle School Teachers (3 credits)
Restricted to middle school teachers. This course cannot be used toward the upper level math requirement for MATH and STAT majors. Prerequisite: MATH214 or equivalent. Not open to students who have completed MATH498C. Credit will be granted for only one of the following: MATH480, MATH483, or MATH498C. Formerly MATH 498C.
Prepares teachers with elementary certification to teach Algebra 1 in middle school. Focuses on basic algebra concepts and related theoretical ideas.

MATH 481 Statistics and Data Analysis for Middle School Teachers (3 credits)
Prerequisite: MATH214 or equivalent. Course for middle school teachers. This course cannot be used toward the upper level math requirements for MATH/STAT majors. Not open to students who have completed MATH498B. Credit will be granted for only one of the following: MATH481, MATH498B, or MATH485. Formerly MATH 498B.
Prepares teachers with elementary certification to teach simple data analysis and probability in middle school. Focuses on understanding basic statistics, data analysis, and related theoretical ideas.

MATH 482 Geometry for Middle School Teachers (3 credits)
Prerequisite: MATH214 or equivalent. Course for middle school teachers. This course cannot be used toward the upper level math requirements for MATH/STAT majors. Senior standing. Not open to students who have completed MATH498E. Credit will be granted for only one of the following: MATH482, MATH484, or MATH498E.
Prepares teachers with elementary certification to teach geometry in middle school. Focuses on understanding basic geometry concepts and related theoretical ideas.

MATH 483 Algebra for School Teachers (3 credits)
Prerequisite: MATH141. Cannot be used toward the upper-level math requirements for MATH/STAT majors. Senior standing. Not open to students who have completed MATH498C. Credit will be granted for only one of the following: MATH498C, MATH483, or MATH480. Formerly MATH 498C.
Focuses on concepts related to algebra and trigonometry, including functions, equations, inequalities, and data analysis. Assumes a good understanding of calculus.

MATH 484 Geometry for High School Teachers (3 credits)
Prerequisite: MATH141. Cannot be used toward the upper-level math requirement for MATH/STAT majors. Senior standing. Not open to students who have completed MATH498E. Credit will be granted for only one of the following: MATH482, MATH484, or MATH498E. Formerly MATH 498E.
Focuses on concepts related to geometry, including several geometry axiom schemes, transformations, and similarity. Includes constructions with Geometer's Sketchpad.

MATH 485 Statistics for High School Teachers (3 credits)
Prerequisite: MATH141. Cannot be used toward the upper level math requirements for MATH/STAT majors. Not open to students who have completed MATH498S. Credit will be granted for only one of the following: MATH481, MATH485, or MATH98S. Formerly MATH 498S.
Focuses on concepts related to statitics and data analysis, including probability, sampling, distribution of data, and inference.

MATH 486 Calculus for High School Teachers (3 credits)
Prerequisite: MATH141. Cannot be used toward the upper level math requirements for MATH/STAT majors.
Focuses on concepts related to one-variable calculus including limits, continuity, derivative, integrals, series, and applications of these topics.

MATH 489 Research Interactions in Mathematics (1-3 credits)
Prerequisite: permission of department. Repeatable to 10 credits if content differs.
Students participate in a vertically integrated (undergraduate, graduate and/or postdoctoral, faculty) mathematics research group. Format varies. Students and supervising faculty will agree to a contract which must be approved by the department. Up to three credits of MATH489 may be applied to the mathematics degree requirements. See the department's MATH489 online syllabus for further information.

MATH 498 Selected Topics in Mathematics (1-9 credits)
Honors students register for reading courses under this number. Repeatable to 9 credits if content differs.
Topics of special interest to advanced undergraduate students will be offered occasionally under the general guidance of the departmental committee on undergraduate studies.

MATH 499 Honors Seminar (2 credits)
Prerequisite: permission of department. Not open to graduate students. Formerly MATH 398.
Faculty supervised reports by students on mathematical literature. Both oral and written presentation on special topics of current interest.

MATH 598 Topics for Teachers Workshops (1-3 credits)
Prerequisite: current status as school teacher or permission of instructor.
Workshops offered to school teachers for enrichment in various topics in modern mathematics.

MATH 600 Abstract Algebra I (3 credits)
Prerequisite: MATH 403 and MATH405; or equivalent.
Groups with operators, homomorphism and isomorphism theorems, normal series, Sylow theorems, free groups, Abelian groups, rings, integral domains, fields, modules. Topics may include HOM (A,B), Tensor products, exterior algebra.

MATH 601 Abstract Algebra II (3 credits)
Prerequisite: MATH 600.
Field theory, Galois theory, multilinear algebra. Further topics from: Dedekind domains, Noetherian domains, rings with minimum condition, homological algebra.

MATH 602 Homological Algebra (3 credits)
Prerequisite: MATH 600.
Projective and injective modules, homological dimensions, derived functors, spectral sequence of a composite functor. Applications.

MATH 603 Commutative Algebra (3 credits)
Prerequisite: MATH 600.
Ideal theory of Noetherian rings, valuations, localizations, complete local rings, Dedekind domains.

MATH 606 Algebraic Geometry I (3 credits)
Prerequisite: MATH 600 and MATH 601.
Prime and primary ideals in Noetherian rings, Hilbert Nullstellensatz, places and valuations, prevarieties (in the sense of Serre), dimension, morphisms, singularities, varieties, schemes, rationality.

MATH 607 Algebraic Geometry II (3 credits)
Prerequisite: MATH 606.
Topics in contemporary algebraic geometry chosen from among: theory of algebraic curves and surfaces, elliptic curves, Abelian varieties, theory of schemes, theory of zeta functions, formal cohomology, algebraic groups, reduction theory.

MATH 608 Selected Topics in Algebra (1-3 credits)
Prerequisite: permission of instructor.

MATH 620 Algebraic Number Theory I (3 credits)
Prerequisite: MATH 601.
Algebraic numbers and algebraic integers, algebraic number fields of finite degree, ideals and units, fundamental theorem of algebraic number theory, theory of residue classes, Minkowski's theorem on linear forms, class numbers, Dirichlet's theorem on units, relative algebraic number fields, decomposition group, inertia group and ramification group of prime ideals with respect to a relatively Galois extension.

MATH 621 Algebraic Number Theory II (3 credits)
Prerequisites: MATH 600; and MATH 620 or equivalent.
Valuation of a field, algebraic function fields, completion of a valuation field, ramification exponent and residue class degree, ramification theory, elements, differents, discriminants, product formula and characterization of fields by the formula, Gauss sum, class number formula of cyclotomic fields.

MATH 630 Real Analysis I (3 credits)
Prerequisite: MATH 411 or equivalent.
Lebesgue measure and the Lebesgue integral on R, differentiation of functions of bounded variation, absolute continuity and fundamental theorem of calculus, Lp spaces on R, Riesz-Fischer theorem, bounded linear functionals on Lp, measure and outer measure, Fubini's theorem.

MATH 631 Real Analysis II (3 credits)
Prerequisite: MATH 630.
Abstract measure and integration theory, metric spaces, Baire category theorem and uniform boundedness principle, Radon-Nikodym theorem, Riesz Representation theorem, Lebesgue decomposition, Banach and Hilbert Spaces, Banach-Steinhaus theorem, topological spaces, Arzela-Ascoli and Stone-Weierstrass theorems, compact sets and Tychonoff's theorem.

MATH 632 Functional Analysis (3 credits)
Prerequisite: MATH 631.
Introduction to functional analysis and operator theory: normed linear spaces, basic principles of functional analysis, bounded linear operators on Hilbert spaces, spectral theory of selfadjoint operators, applications to differential and integral equations, additional topics as time permits.

MATH 634 Harmonic Analysis (3 credits)
Prerequisite: MATH 631.
L1 theory: Fejer theorem, inversion theorem, ideal structure, Tauberian theorem. L2 theory: Plancherel-Parseval theorems, Paley-Wiener theorem. Lp theory: Hausdorff-Young theorem. Distribution theory: Bochner's theorem, Wiener continuous measures theorem, Malliavin theorem, Schwartz theory, almost periodic functions.

MATH 636 Representation Theory (3 credits)
Prerequisite: MATH 631.
Introduction to representation theory of Lie groups and Lie algebras; initiation into non-abelian harmonic analysis through a detailed study of the most basic examples, such as unitary and orthogonal groups, the Heisenberg group, Euclidean motion groups, the special linear group. Additional topics from the theory of nilpotent Lie groups, semisimple Lie groups, p-adic groups or C*-algebras.

MATH 642 Dynamical Systems I (3 credits)
Prerequisites: MATH 432; and MATH 630 or equivalent.
Foundations of topological dynamics, homeomorphisms, flows, periodic and recurrent points, transitivity and minimality, symbolic dynamics. Elements of ergodic theory, invariant measures and sets, ergodicity, ergodic theorems, mixing, spectral theory, flows and sections. Applications of dynamical systems to number theory, the Weyl theorem, the distribution of values of polynomials, Vander Waerden's theorem on arithmetic progressions.

MATH 643 Dynamical Systems II (3 credits)
Prerequisite: MATH 642 or equivalent.
Entropy theory, variational principle for the entropy, expansiveness, measures with maximal entropy. Smooth systems on manifolds, diffeomorphisms and flows, periodic points, stable and unstable manifolds, homoclinic points, transversality, the Krupka-Smale theorem, Morse-Smale systems. Hyperbolicity, Anosov systems, distributions and foliations, strange attractors, Bowen's measure.

MATH 648 Selected Topics in Analysis (1-3 credits)
Prerequisite: permission of instructor.

MATH 660 Complex Analysis I (3 credits)
Prerequisite: MATH 410 or equivalent.
Linear transformations, analytic functions, conformal mappings, Cauchy's theorem and applications, power series, partial fractions and factorization, elementary Riemann surfaces, Riemann's mapping theorem.

MATH 661 Complex Analysis II (3 credits)
Prerequisites: MATH 630; and MATH 660.
Topics in conformal mappings, normal families, Picard's theorem, classes of univalent functions, extremal properties, variational methods, elliptic functions, Riemann surfaces.

MATH 668 Selected Topics in Complex Analysis (1-3 credits)
Repeatable if content differs.. Prerequisite: permission of instructor.
Material selected to suit interests and background of the students. Typical topics: Kaehler geometry, automorphic functions, several complex variables, symmetric spaces.

MATH 669 Selected Topics in Riemann Surfaces (1-3 credits)
Prerequisite: permission of instructor. Repeatable if content differs.
Construction of Riemann surfaces, hyperbolic geometry, Fuchsian and Kleinian groups, potential theory, uniformisation spaces of meromorphic functions, line bundles, Picard variety, Riemann-Roch, Teichmueller theory.

MATH 670 Ordinary Differential Equations I (3 credits)
Prerequisites: MATH 405; and MATH 410 or the equivalent. Also offered as AMSC 670. Credit will be granted for only one of the following: AMSC 670, MAPL 670 OR MATH 670.
Existence and uniqueness, linear systems usually with Floquet theory for periodic systems, linearization and stability, planar systems usually with Poincare-Bendixson theorem.

MATH 671 Ordinary Differential Equations II (3 credits)
Prerequisites: MATH 630; and AMSC/MATH/MAPL 670 or the equivalent. Also offered as AMSC 671. Credit will be granted for only one of the following: AMSC 671, MAPL 671 or MATH 671.
The content of this course varies with the interests of the instructor and the class. Stability theory, control, time delay systems, Hamiltonian systems, bifurcation theory, and boundary value problems, and the like.

MATH 673 Partial Differential Equations I (3 credits)
Prerequisite: MATH 411 or equivalent. Also offered as AMSC 673. Credit will be granted for only one of the following: AMSC 673, MAPL 673 or MATH 673.
Analysis of boundary value problems for Laplace's equation, initial value problems for the heat and wave equations. Fundamental solutions, maximum principles, energy methods. First order nonlinear PDE, conservation laws. Characteristics, shock formation, weak solutions. Distributions, Fourier transform.

MATH 674 Partial Differential Equations II (3 credits)
Prerequisite: AMSC/MATH/MAPL 673 or permission of instructor. Also offered as AMSC 674. Credit will be granted for only one of the following: AMSC 674, MAPL 674 or MATH 674.
Boundary value problems for elliptic partial differential equations via operator-theoretic methods. Hilbert spaces of functions. Duality, weak convergence. Sobolev spaces. Spectral theory of compact operators. Eigenfunction expansions.

MATH 680 Eigenvalue and Boundary Value Problems I (3 credits)
Prerequisites: MATH 405; and MATH 410 or equivalent. Also offered as AMSC 680. Credit will be granted for only one of the following: AMSC 680, MAPL 680 or MATH 680.
Operational methods applied to ordinary differential equations. Introduction to linear spaces, compact operators in Hilbert space, study of eigenvalues.

MATH 687 Minicourse Series in the Mathematical Sciences (1 credits)
Also offered as AMSC 687 and STAT 687. Credit will be granted for only one of the following: AMSC 687, MATH 687 or STAT 687.
This series will consist of up to sixteen 3-lecture presentations covering a broad range of topics in the mathematical sciences. Each minicourse is intended to be self-contained and accessible to first year graduate students and advanced undergraduates. The goal of each minicourse is to present an active research area or significant result and the necessary vocabulary and perspective for students to appreciate it. The goal of the Minicourse Series is to broaden a student's awareness of the mathematical sciences and to inform them of research directions.

MATH 689 Research Interactions in Mathematics (1-3 credits)
Prerequisite: consent of the instructor. Repeatable to 06 credits if content differs.
The students participate in a vertically integrated (undergraduate, graduate and/or postdoctoral, faculty) research group. Format varies, but includes regular meetings, readings and presentations of material. See graduate program's online syllabus or contact the graduate program director for more information.

MATH 695 Teaching Seminar (1 credits)
For MATH majors only.
A course intended for first year teaching assistants. Topics include: everyday mechanics of teaching; teaching methods and styles; technology; course enrichment, diversity in the classroom; sexual harassment; teacher-student interactions; presentations by students.

MATH 710 Consistency Proofs in Set Theory (3 credits)
Prerequisites: MATH 446; and MATH 447.
Consistency and independence of such fundamental principles of set theory as the laws of choice, of cardinal arithmetic of constructability and regularity. Godel's model of constructible sets, inner models, Cohen's generic models.

MATH 712 Mathematical Logic I (3 credits)
Sentential logic, first-order languages, models and formal deductions. Basic model theory including completeness and compactness theorems, other methods of constructing models, and applications such as non-standard analysis.

MATH 713 Mathematical Logic II (3 credits)
Prerequisite: MATH 712 or MATH 447.
Incompleteness and undecidability results of Godel, Church, Tarski and others. Recursive function. Basic proof theory and axiomatic set theory.

MATH 718 Selected Topics in Mathematical Logic (1-3 credits)
Prerequisite: permission of instructor.

MATH 730 Fundamental Concepts of Topology (3 credits)
Prerequisites: MATH 410, and MATH 411, and MATH 403; or equivalent.
Survey of basic point set topology, fundamental group, covering spaces, Van Kampen's theorem, simplicial complexes, simplicial homology, Euler characteristics and classification of surfaces.

MATH 734 Algebraic Topology (3 credits)
Prerequisite: MATH 403 or equivalent. Recommended: MATH 730.
Singular homology and cohomology, cup products, Poincare duality, Eilenberg-Steenrod axioms, Whitehead and Hurewicz theorems, universal coefficient theorem, cellular homology.

MATH 740 Riemannian Geometry (3 credits)
Prerequisites: {MATH 405; and MATH 411} or equivalent.
Manifolds, tangent vectors and differential forms, Riemannian metrics, connections, curvature, structure equations, geodesics, completeness, immersions, tensor algebra, Lie derivative.

MATH 742 Differential Topology (3 credits)
Prerequisites: {MATH 410; and MATH 411} or equivalent.
Inverse and implicit function theorems, Sard's theorem, orientability, degrees, smooth vector bundles, imbeddings and immersions, transversality approximation theorems and applications, isotopy extension theorem, tubular neighborhoods.

MATH 744 Lie Groups I (3 credits)
Prerequisite: MATH 403, MATH 405, MATH 411 and MATH 432; or equivalent.
An introduction to the fundamentals of Lie groups, including some material on groups of matrices and Lie algebras.

MATH 745 Lie Groups II (3 credits)
Prerequisite: MATH 744.
A continuation of Lie groups I in which some of the following topics will be emphasized: solvable Lie groups, compact Lie groups, classifications of semi-simple Lie groups, representation theory, homogeneous spaces.

MATH 748 Selected Topics in Geometry and Topology (1-3 credits)
Prerequisite: permission of instructor.

MATH 799 Master's Thesis Research (1-6 credits)

MATH 898 Pre-Candidacy Research (1-8 credits)

MATH 899 Doctoral Dissertation Research (1-8 credits)