MATH 501
PROBABILITY
Basic probability notions, classical combinatorial methods, conditional probabilities. Random variables and properties of distributions. Moments, moment generating functions, covariances, correlations. Transformations, order statistics. Convergence in probability and large sample properties. Prerequisites: MATH 323 or equivalent.

MATH 502
STATISTICAL INFERENCE
Likelihood functions and sufficient statistics. Theory of estimation; completeness and UMVU estimators. Blackwell-Rao theorem; information inequality. MLEs and their asymptotic properties. Confidence intervals. Testing hypotheses and Neyman-Pearson theory. Introduction to linear models and non-parametric methods. Prerequisites: MATH 501 or equivalent.

MATH 503-504
ALGEBRA
Higher algebra, especially groups, rings, fields and modules. Prerequisites: MATH 401 and 402, or consent of department.

MATH 505-506
ANALYSIS
Real analysis including theory of Lebesgue measure, integration and elementary theory of Banach and Hilbert spaces. Complex analysis. Prerequisites: MATH 478 and 479, or consent of department.

MATH 507
LINEAR ALGEBRA AND MATRIX THEORY
Linear algebra over the complex numbers and finite fields, eigenvectors and eigenvalues, quadratic forms, normal forms of matrices, selected topics in matrix theory. Prerequisites: consent of department.

MATH 508
COMPLEX ANALYSIS
A rigorous introduction to complex analysis. Rational functions; conformal maps; Cauchy's Integral Theorem with applications; representations of analytic functions as series, products, integrals; topics selected by instructor. Prerequisites: MATH 479 or consent of department.

MATH 509
GRADUATE COMPUTER SCIENCE FOR MATHEMATICIANS
Graduate-level introduction to computer science from mathematician's point of view, models of computation, automata theory, programming languages, program semantics, proof theory for programs.

MATH 513-514
GENERAL TOPOLOGY
Topological spaces, metric spaces, separation axioms, compactness, connectedness, quotient spaces. Topics from geometric topology, including fundamental group, complexes and homotopy.

MATH 517-518
ALGEBRAIC TOPOLOGY
Concept of homotopy, fundamental group, covering spaces, categories and functors, simplicial complexes, simplicial homology and cohomology, singular homology and cohomology, cup product structure, CW-complexes, higher homotopy groups. Prerequisites: MATH 461, 513-514 or equivalent.

MATH 519
THEORY OF FIBER SPACES
Various types of fibrations (Serre, Hurewicz, Dold fibrations, fiber bundles, covering spaces), applications of homotopy theory, topics from classical theory of bundles, classification theorems, spectral sequences. Prerequisites: MATH 513, 514, consent of department.

MATH 520
HOMOLOGICAL ALGEBRA
Modules, chain complexes, tensor products, derived functors, homology of groups, other topics selected by the instructor. Prerequisites: MATH 504 or consent of department.

MATH 521-522
DIFFERENTIAL TOPOLOGY
Differentiable manifolds, imbeddings and immersions, Whitney's imbedding theorem, tangent and cotangent bundles, Morse theory. Prerequisites: MATH 513-514.

MATH 523-524
GROUP THEORY
Properties of groups, extensions, transfer, generators, defining relations. Prerequisites: MATH 503-504 or equivalent.

MATH 525-526
RINGS AND ALGEBRAS
Advanced study of rings and algebras; special topics selected from current literature. Prerequisites: MATH 503-504 or equivalent.

MATH 527
REPRESENTATION THEORY
Representations of groups and rings by linear transformations, characters, applications in structure theory of groups and rings. Prerequisites: consent of department.

MATH 532
ADVANCED NUMERICAL ANALYSIS
Solution to non-linear equations, differential equations, eigenvalue problems, finite element method, discretization error, iterative methods, computer implementation. Prerequisites: undergraduate differential equations, linear algebra, advanced calculus, some programming experience.

MATH 537
ANALYSIS OF ALGORITHMS
Time and space analysis of algorithms for applications such as sorting, searching, graphics manipulation, pattern matching and algebraic calculation. Statistical analysis. Empirical analysis of complex algorithms arising in computer systems. Prerequisites: MATH 509.

MATH 538
FORMAL LANGUAGES
Formal description of syntax and semantics of computer languages. Transition from formal description to implementation as compiler or interpreter. Various languages compared as to their data structures, procedures and input-output. Prerequisites: MATH 509.

MATH 545
TOPOLOGICAL GROUPS
Locally compact topological groups, open homomorphism and closed graph theorems, measure and integration on locally compact topological groups. Prerequisites: MATH 505-506, 513-514 or consent of department.

MATH 547-548
DECOMPOSITION SPACES
Upper and lower semi-continuous decompositions, properties inherited by decomposition spaces, applications (in particular to manifolds). Prerequisites: MATH 513-514 and consent of department.

MATH 549
KNOT THEORY
Knots and knot types, presentation of a knot group, combinatorial covering spaces, absolute calculus, cubes with holes. Prerequisites: MATH 513-514 and consent of department.

MATH 551-552
POLYHEDRAL TOPOLOGY
Regular neighborhood theory, general position, unknotting balls and spheres, engulfing techniques, handlebody theory and s-cobordism. Prerequisites: MATH 513-514 and consent of department.

MATH 553
NON-PARAMETRIC INFERENCE
Order statistics and quantiles, non-parametric confidence intervals, non-parametric measures of association, tests based on ranks, tests of independence, symmetry, location differences, chi-square and Kolmogorov-Smirnov goodness of fit tests, non-parametric regression, robustness, asymptotic relative efficiency of tests, concepts of non-parametric density estimation. Prerequisites: MATH 448 or 502.

MATH 554
SAMPLING FROM FINITE POPULATIONS
The classical model and sampling strategies. Sampling distributions of estimators of population quantities. Simple random sampling, stratified sampling, two-stage and multistage cluster sampling, optimal allocation of resources, and other design aspects. Sampling inspection techniques for quality control. Other topics as time permits. Prerequisites: MATH 447 or 501.

MATH 555
LINEAR MODELS
Inference in linear models based on the least squares approach: Point estimation, confidence regions, hypothesis testing, model building and verification, residual analysis, selection of best regression, influential observations. Prerequisites: MATH 448 or 502 and MATH 404 or 507.

MATH 556
DESIGN OF EXPERIMENTS
The role and principles of DE in scientific research. Reference distributions, ANOVA, multiple comparisons. Randomized complete block designs, Latin squares, Pn factorial design and the calculus of factorial experiments. Balanced incomplete block designs, the recovery of intrablock information. Exploration of response surfaces. Prerequisites: MATH 555.

MATH 558
MULTIVARIATE STATISTICAL ANALYSIS
Multivariate normal distributions, Wishart distributions, inferences on means and covariances, Hotelling's T2, multivariate linear models, regression, ANOVA, tests of independence, discriminant analysis, principal components, canonical correlations and variables, factor analysis. Prerequisites: MATH 555.

MATH 559
TIME-SERIES ANALYSIS
Trend analysis and smoothing. Estimation, testing, modeling and forecasting for ARMA and ARIMA models. Prerequisites: MATH 555.

MATH 561
ALGEBRA SEMINAR
Prerequisites: consent of department.
1-4 cr.

MATH 564
PROBABILITY SEMINAR
Prerequisites: consent of department.
1-4 cr.

MATH 565
TOPOLOGY SEMINAR I
Prerequisites: consent of department.
1-4 cr.

MATH 567
TOPOLOGY SEMINAR II
Prerequisites: consent of department.
1-4 cr.

MATH 570
APPLIED MULTIVARIATE ANALYSIS
Multivariate normal distributions, Wishart distributions, Hotelling's T, tests of independence, large sample distribution theory, multivariate linear models, discriminant analysis, factor analysis, principal components and other selected topics. Prerequisites: MATH 558.

MATH 571
ADVANCED PROBABILITY THEORY
Measure theoretic probability. Axiomatic foundations, random variables, conditional probability and expectation, characteristic functions, infinite divisibility and stable laws, types of convergence, law of large numbers, central limit theorem, other topics as time permits. Prerequisites: MATH 447 or 501, and MATH 506 or consent of instructor.
5 cr.

MATH 572
STOCHASTIC PROCESSES
A continuation of the subject matter presented in MATH 571. Martingales and Markov processes (if not covered in MATH 571), orthogonality, stationary processes, other topics as time permits. Prerequisites: MATH 571.
5 cr.

MATH 573
APPLIED PROBABILITY AND STOCHASTIC PROCESSES
Introduction to Markov chains, Markov processes with emphasis on applications. Classification of states, stationarity. Continuity, integration, and differentiation of second order processes. Stochastic differential equations. Prerequisites: MATH 501

MATH 574
NUMBER THEORY (MAT/MSED)
Elementary number theory, divisibility, fundamental theorem of arithmetic, prime numbers, quadratic reciprocity, Diophantine equations. Prerequisites: consent of instructor.

MATH 575
SPECIAL TOPICS FOR TEACHERS (MAT/MSED)
Special topics of interest to teachers. Prerequisites: consent of instructor.
1-4 cr.

MATH 576
COMPUTER APPLICATIONS IN MATHEMATICS EDUCATION (MAT/MSED)
Computer usage in education from historical point of view, evaluation of various levels of computer usage in learning situation (low-key approach, interactive CAI approach, artificial intelligence approach). Prerequisites: consent of instructor.

MATH 577
RECREATIONAL MATHEMATICS (MAT/MSED)
Sources of recreational mathematics, magic squares, dissection problems, map coloring problems, traversing of mazes, chessboard recreations, instant insanity, arithmetical and geometrical fallacies. Prerequisites: consent of instructor.

MATH 578
COMBINATORICS (MAT/MSED)
Combinations and permutations, enumeration techniques, recursion, sum and difference sequences, partitions, applications to pre-college mathematics. Prerequisites: consent of instructor.

MATH 579
ADVANCED STATISTICAL INFERENCE
Weak convergence of probability measures on Euclidean spaces. Interval estimation, point estimation, and hypothesis testing. General decision theory including the minimax theorem, the complete class theorem, the abstract Rao-Blackwell theorem, the theorem of Hunt and Stein, and Bayes methods. Asymptotic decision theory. Prerequisites: MATH 571 and 502.

MATH 580
TOPICS IN COMBINATORIAL ANALYSIS
Variable subject matter chosen from field of combinatorial analysis. Prerequisites: MATH 401. May be repeated for credit with consent of department.

MATH 581
TOPICS IN GRAPH THEORY
Theoretical and applied graph theory. Applications including personnel assignment problem, construction of reliable communications networks, chromatic polynomials. Prerequisites: MATH 401 or consent of instructor. May be repeated for credit with consent of department.

MATH 582
ALGEBRA (MAT/MSED)
Classical theory of equations, algebraic systems (including groups, rings, fields, modules) and their properties. Prerequisites: consent of instructor.

MATH 583
METRIC AND AFFINE GEOMETRY (MAT/MSED)
Affine and metric geometry from transformational point of view. Finite and infinite geometries, Euclidean geometry, applications to pre-college mathematics. Prerequisites: consent of instructor.

MATH 584
EUCLIDEAN AND NON-EUCLIDEAN GEOMETRY (MAT/MSED)
Algebraic (analytic) approach to classical geometries (Euclidean, hyperbolic, projective). Prerequisites: consent of instructor.

MATH 588
PROBABILITY AND STATISTICS (MAT/MSED)
Finite probability and probability-related statistical problems. Mixture of formal development and problem solving with applications to pre-college mathematics. Prerequisites: consent of instructor.

MATH 589
HISTORY AND CONCEPTUAL DEVELOPMENT OF THE CALCULUS (MAT/MSED)
Historical and conceptual development of mathematical ideas underlying modern calculus, including problems of infinity and of continuity as treated in ancient and modern times. Applications to pre-college mathematics wherever appropriate.

MATH 590
TOPICS IN MODERN MATHEMATICS
Study (at graduate level) of some topic in mathematics not a part of regular graduate curriculum. Content changes from term to term. With consent of department, students may repeat course for credit. Prerequisites: consent of department.
1-4 cr.

MATH 591
THE TEACHING OF COLLEGE MATHEMATICS
Required for teaching assistants, suggested for graduate assistants interested in college teaching. Does not count toward required number of courses for MA or PhD.
1-4 cr.

MATH 597
INDEPENDENT WORK
Reading and research on special topic, under direction of adviser. May be repeated for credit with consent of department. Commonly taught topics under Independent Work include but are not limited to the following:
1-4 cr.

MATH 599
THESIS
1-4 cr.

MATH 601
TOPICS IN TOPOLOGY
Variable subject matter chosen from field of topology. May be repeated for credit with consent of department.

MATH 603
TOPICS IN ALGEBRA
Variable subject matter chosen from field of algebra. May be repeated for credit with consent of department.
1-4 cr.

MATH 604
ADVANCED TOPICS IN THE THEORY OF GROUPS
Topics selected from current research. May be repeated for credit with consent of department.

MATH 605
SEMINAR IN STATISTICS
Variable subject matter chosen from field of statistics. Topics selected from current research. May be repeated for credit with consent of department.
1-4 cr.

MATH 698
PRE-DISSERTATION RESEARCH
Independent reading and/or research in preparation for comprehensive examinations for admission to PhD candidacy and/or preparation of dissertation prospectus. Graded on Satisfactory/Unsatisfactory basis only.
1-9 cr./sem.

MATH 699
DISSERTATION
Research for and preparation of the dissertation.
1-12 cr./sem.

MATH 700
CONTINUOUS REGISTRATION
Required for maintenance of matriculated status in graduate program. No credit toward graduate degree requirements.
1 cr./sem.

MATH 707
RESEARCH SKILLS
Development of research skills required within graduate programs. May not be applied toward course credits for any graduate degree. Prerequisites: approval of relevant graduate program directors or department chairs.
1-4 cr.