PATRICK EBERLEIN, Chair
Idris Assani (45) Dynamical Systems, Ergodic Theory of Operators
Thomas H. Brylawski (2) Combinatorics
Roberto A. Camassa (16) Mathematical Modeling, Nonlinear Waves, Propagation, Dynamical Systems
Ivan V. Cherednik (48) Representation Theory, Mathematical Physics, Algebraic Combinatorics
Joseph A. Cima (4) Complex Analysis, Functional Analysis
James N. Damon (14) Singularity Theory, Differential Topology
Patrick B. Eberlein (6) Differential Geometry
M. Gregory Forest (7) Nonlinear Waves, Solitons, Fiber Flows of Complex Liquids
Ladnor D. Geissinger (9) Combinatorics, Group Characters
Sue E. Goodman (3) Topology, Dynamical Systems
Jane M. Hawkins (38) Ergodic Theory, Dynamical Systems
Christopher K. R. T. Jones (55) Applications of Dynamical Systems, Nonlinear Partial Differential Equations, Ocean Dynamics, Nonlinear Optics
Norberto Kerzman (32) Several Complex Variables, Partial Differential Equations
Shrawan Kumar (46) Representation Theory, Geometry of Flag Varieties
Richard McLaughlin (50) Fluid Dynamics and Turbulent Transport
Michael L. Minion (11) Scientific Computation, Computational Fluid Dynamics, Adaptive Mesh Refinement
Karl E. Petersen (20) Ergodic Theory
John A. Pfaltzgraff (22) Complex Analysis
Joseph F. Plante (23) Foliations, Dynamical Systems
Robert A. Proctor (43) Combinatorics, Representation Theory
William W. Smith (25) Commutative Algebra
Michael E. Taylor (40) Partial Differential Equations, Harmonic Analysis, Operator Theory
Alexandre N. Varchenko (47) Geometry, Mathematical Physics
Jonathan M. Wahl (28) Algebraic Geometry
Mark Williams (36) Partial Differential Equations
Warren R. Wogen (29) Operator Theory
David Adalsteinsson (1) Applied Mathematics and Scientific Computation
Jingfang Huang (51) Integral Equation Methods and Fast Algorithms
Lev Rozansky (52) Three-Dimensional Topology
Prakash Belkale (57) Algebraic Geometry
Sorin Mitran (58) Computational Methods for Partial Differential Equations, Continuum-Kinetic Methods, Fluid Dynamics, Biological Fluid Dynamics and Mechanics
Peter Mucha (60), Fluid Dynamics, Suspensions, Sedimentation, Granular Flows, Rheology, Computer-Generated Animation, Networks
Richard Rimanyi (59), Topology, Geometry, Singularities
Robert L. Davis
William H. Graves
Robert G. Heyneman
Ancel C. Mewborn
Michael Schlessinger
Johann Sonner
James Stasheff
Fred B. Wright
The Department of Mathematics offers graduate training leading to the degrees of master of arts, master of science, and doctor of philosophy. A master's degree may be included or bypassed in the doctoral program. All of a student's graduate work may be done within the department or, when appropriate, may be done under the direction of an approved adviser in an allied discipline. The MAT degree is also available with a major in mathematics in the School of Education.
The Department of Mathematics is housed in Phillips Hall, as are the Computation Center and the special library for the departments of Mathematics, Physics and Astronomy, Computer Science, and Statistics. This departmental library contains an unusually large and complete collection of mathematical books and journals.
The Department of Mathematics offers a number of teaching assistantships and teaching fellowships each year. Applicants for financial aid are also considered for several University fellowships awarded by The Graduate School in the university-wide competition. Applications for admission and financial assistance may be obtained from The Graduate School. Applications for financial aid should be filed by December 31.
The general regulations of The Graduate School govern the work for graduate degrees in mathematics. Specific requirements are explained below. In general, a graduate student in mathematics may receive credit only for mathematics courses numbered 600 and above.
These descriptions summarize the requirements for the master's and PhD degrees. More detailed statements may be obtained from the department. The director of graduate studies must approve all aspects of a student's program. The purpose of the graduate programs is to develop mathematical skills appropriate for competition in academia or industry.
The course schedule for all first-year students will depend upon each student's undergraduate training. The normal course load for a graduate student is three courses (nine credit hours) per semester. Graduate students must keep full time status in order to qualify for tuition and health insurance benefits. First-year students typically choose courses from five year-long sequences in algebra (676, 677), analysis (653, 656), geometry-topology (680, 681), scientific computation (661, 662), and methods of applied mathematics (668, 669). The PhD comprehensive exams are based on the content of the first-year sequences. These exams are offered in January and August of each year, just before the semester begins.
During the second year a typical PhD student will take the PhD comprehensive exams and select courses from a list of sixteen more advanced "second tier" courses. A typical master's student will complete that degree during the second year. The department considers two years to be the normal time needed to complete a master's degree.
A candidate for a master's degree must satisfy each of the following requirements:
1) Earn at least two semesters of residency credit and complete all requirements within five years.
2) Demonstrate computer programming ability by passing an approved undergraduate or graduate course in programming, or by passing an exam administered by the Mathematics Department.
3) Perform satisfactorily in thirty hours of graduate work in a program approved by the director of graduate studies. At least fifteen of these hours must be in Mathematics Department courses numbered 600 or above.
4) Complete a master's project for a master of science degree or a master's thesis for a master of arts degree.
5) Pass an oral examination upon completion of the master's project or master's thesis. The exam will cover course work as well as the project or thesis.
6) For graduate students entering UNC-Chapel Hill in the fall 2001 semester or later, a master's candidate must pass one of the written comprehensive exams given to doctoral students.
A candidate for a PhD degree must satisfy each of the following requirements:
1) Earn at least four semesters of residency credit and complete all requirements within eight years.
2) Satisfy the same computer programming requirement as a master's student.
3) Demonstrate reading competence in two approved foreign languages by passing an approved course or by passing a translation exam administered by the Mathematics Department.
4) Pass three PhD comprehensive exams by the beginning of the sixth semester.
5) Pass at least six courses from the following two lists: a) the second tier courses; or b) first-year comprehensive courses not required for the three comprehensive exams the student has passed. Of these six courses at least three must be numbered over 700 and drawn from the second tier list.
6) Pass a preliminary oral exam on the chosen PhD specialty area.
7) Write a PhD thesis and defend it successfully during a final oral exam chaired by the thesis adviser.
The student/faculty ratio of about 3/2 makes it possible for graduate students to take reading courses from individual faculty members that are tailored to meet the student's needs.
Graduate students in other departments who plan to offer mathematics as a (complete or partial) minor field for the PhD should consult the director of graduate studies in mathematics for approval of their programs and for assignment of an adviser in the Department of Mathematics. This should be done at the earliest possible time, in order to prevent disappointment for the student.
401 [101] MATHEMATICAL CONCEPTS IN ART (3). Mathematical theories of proportion, perspective (projective invariants and the mathematics of visual perception). Symmetry and aesthetics are expounded and illustrated by examples from painting, architecture, and sculpture. (Alternate years.) Brylawski.
406 [106] MATHEMATICAL METHODS IN BIOSTATISTICS (3). Prerequisite, MATH 232 or equivalent. Special mathematical techniques in the theory and methods of biostatistics as related to the life sciences and public health. Includes brief review of calculus, selected topics from intermediate calculus, and introductory matrix theory for applications in biostatistics. Fall.
411 [111] DEVELOPING MATHEMATICAL CONCEPTS (3). Prerequisite, consent of the instructor. An investigation of various ways elementary concepts in mathematics can be developed. Applications of the mathematics developed are considered. This course is ordinarily offered as an in-service course for teachers. Spring.
416 [116] LINEAR ALGEBRA (3). An introduction to the theory of vector spaces, linear transformations, systems of linear equations, matrices, determinants, eigenvectors, diagonalizations.
418 [118] BASIC CONCEPTS OF ANALYSIS (3). Prerequisites, MATH 232 and consent of the instructor. Limits, continuity, differentiability, uniform continuity. Riemann integration. Infinite sequences and series; uniform convergence; power series. A student cannot receive credit for this course after receiving credit for MATH 193. Summer.
435 [126] INTRO TO PROBABIL (STAT 435) (3). Prerequisite, MATH 233. An introduction to the mathematical theory of probability, covering random variables; moments; binomial, Poisson, normal, and related distribution; generating functions; sums and sequences of random variables; combinatorial and statistical applications. Fall and spring.
452 [107] MATHEMATICAL BIOLOGY (BIOL 452) (4).
515 [115] HISTORY OF MATHEMATICS (3). Prerequisites, calculus and abstract algebra; graduate students by permission only. A brief general survey of the history of mathematics. Some special problems in depth. Problems in the history of mathematics. Spring.
521 [121] ADVANCED CALCULUS I (3). Prerequisites, MATH 233 and 381. The real numbers; continuity and differentiability of functions of one variable; infinite series; integration. Fall and spring.
522 [122] ADVANCED CALCULUS II (3). Prerequisite, MATH 521. Functions of several variables; derivative as linear transformation; inverse and implicit function theorems; multiple integration. Spring.
523 [123] FUNCTIONS OF A COMPLEX VARIABLE WITH APPLICATIONS (3). Prerequisite, MATH 383. The algebra of complex numbers, elementary functions and their mapping properties, complex limits, power series, analytic functions; contour integrals, Cauchy's theorem and formulae, Laurent series and residue calculus, elementary conformal mapping and boundary value problems. Poisson integral formula for the disk and the half plane. Spring.
524 [124] ELEMENTARY DIFFERENTIAL EQUATIONS (3). Prerequisite, MATH 383. Linear differential systems, power series solutions, Laplace transforms, numerical methods. Fall and spring.
528 [128] MATHEMATICAL METHODS FOR THE PHYSICAL SCIENCES I (3). Prerequisites, MATH 383 and PHYS 104-105, or equivalent. Theory and applications of Fourier series and transform. Laplace transform; Sturm-Liouville problems. Students are expected to do some numerical calculations with a programmable calculator or a computer. Fall.
529 [129] MATHEMATICAL METHODS FOR THE PHYSICAL SCIENCES II (3). Prerequisites, PHYS 104-105, and one of MATH 521, 524, or 528 or equivalent. Introduction to boundary value problems for the diffusion, Laplace, and wave partial differential equations. Bessel functions and Legendre functions. Introduction to complex variables including the calculus of residues. Spring.
533 [133] ELEMENTARY THEORY OF NUMBERS (3). Prerequisites, MATH 232 and 381. Divisibility, Euclidean algorithm, congruences, residue classes, Euler's function, primitive roots, Chinese remainder theorem, quadratic residues, number theoretic functions. Farey and continued fractions, Gaussian integers. Fall and spring.
534 [134] ELEMENTS OF MODERN ALGEBRA (3). Prerequisite, MATH 381. Sets and functions, rings, ordered integral domains, integers, fields and rational numbers, real and complex numbers, polynomials, groups. Fall and spring.
547 [147] LINEAR ALGEBRA FOR APPLICATIONS (3). Prerequisite, MATH 233. Algebra of matrices with applications; determinants; solutions of linear systems by Gaussian elimination; Gram-Schmidt procedure; eigenvalues. MATH 416 may not be taken for credit after credit has been granted for Math 547. Fall, spring, and summer.
548 [148] COMBINATORIAL MATHEMATICS (STAT 156) (3). Prerequisite, MATH 381 or permission of the instructor. Recurrence relations and generating functions; graph and graph algorithms, principle of inclusion-exclusion. Fall.
550 [130] TOPOLOGY (3). Prerequisite, MATH 233; corequisite, MATH 383 or permission of the instructor. Introduction to topics in topology, particularly surface topology, including classification of compact surfaces, Euler characteristic, orientability, vector fields on surfaces, tesselations and fundamental group. Research problems discussed at elementary level. Spring.
551 [131] EUCLIDEAN AND NON-EUCLIDEAN GEOMETRIES (3). Prerequisite, MATH 381 or permission of the instructor. Critical study of basic notions and models of Euclidean and non-Euclidean geometries, order, congruence, and distance. Summer and spring.
555 [155] INTRODUCTION TO DYNAMICS (3). Topics will vary and may include iteration of maps, orbits, periodic points, attractors, symbolic dynamics, bifurcations, fractal sets, chaotic systems, systems arising from different equations, iterated function systems and applications.
564 [145] MATHEMATICAL MODELING (3). Prerequisites, MATH 383, some knowledge of computer programming or permission of instructor. Model validation and numerical simulations using differential equations, probability, and iterated maps. Applications may include conservation laws, dynamics, mixing, geophysical flows and climate change, fluid motion, epidemics, ecological models, population biology, cell biology, and neuron dynamics.
565 [125] COMPUTER-ASSISTED MATHEMATICAL PROBLEM SOLVING (3). Prerequisite, MATH 383. Personal computer as tool in solving a variety of mathematical problems, e.g., finding roots of equations and approximate solutions of differential equations. Introduction to appropriate programming language. Emphasis on graphics. Fall.
566 [166] INTRODUCTION TO NUMERICAL ANALYSIS (3). Prerequisites, MATH 383 and some knowledge of computer programming. Iterative methods, interpolation, polynomial and spline approximations, numerical differentiation and integration, numerical solution of ordinary and partial differential equations. The subject matter of this course includes the material covered in the Society of Actuaries examination on numerical methods. Fall.
577 [137] LINEAR ALGEBRA (3). Prerequisites, MATH 381 and 383. Vector spaces, linear transformations, duality, diagonalization, primary and cyclic decomposition, Jordan canonical form, inner product spaces, orthogonal reduction of symmetric matrices, spectral theorem, bilinear forms, multilinear functions. Fall and spring.
578 [138] ALGEBRAIC STRUCTURES (3). Prerequisite, MATH 577 or 547. Permutation groups, matrix groups, groups of linear transformations, symmetry groups, finite abelian groups; residue class rings, algebra of matrices, linear maps, and polynomials; real and complex numbers, rational functions, quadratic fields, finite fields. Fall.
579 [157] TOPICS IN MATRIX THEORY (3). Prerequisites, MATH 577 or 547 or equivalent, and some computer programming language. Quadratic and hermitian series; applications to systems of differential equations; nonnegative matrices. Perron-Frobenius Theorem; integer matrices, some applications in combinatorics. Spring.
590 [175] TOPICS IN ANALYSIS (3). Prerequisite, MATH 522 or consent of the instructor. Topics may include linear spaces, convexity, mathematical programming, duality, algorithms, and other subjects related to the mathematical theory of optimization. Fall.
591 [176] TOPICS IN ALGEBRA (3). Prerequisite, consent of the instructor. Topics may include number theory, algebraic number theory, field theory, and algebraic geometry.
592 [177] TOPICS IN GEOMETRY (3). Prerequisite, consent of the instructor. Topics may include non-Euclidean geometries, linear geometry, finite geometries, topology, and algebraic geometry.
635 [195] PROBABILITY (STAT 635) (3). Prerequisite, STAT 634 or permission of the instructor. Foundations of probability theory. Basic classical theorems. Modes of probabilistic convergence. Central limit problem. Generating functions, characteristic functions. Introduction to stochastic processes. Spring.
641 [189] ENUMERATIVE COMBINATORICS (3). Prerequisite, MATH 578 or permission of the instructor. Basic counting, partitions, recursions and generating functions, signed enumeration, counting with respect to symmetry, plane partitions and tableaux.
643 [190] COMBINATORIAL STRUCTURES (3). Prerequisite, MATH 578 or permission of the instructor. Graph theory, matchings, Ramsey theory, extremal set theory, network flows, lattices, Mobius inversion, q-analogs, combinatorial and projective geometries, codes and designs.
653 [193] INTRODUCTORY ANALYSIS (3). Prerequisite, advanced calculus. Elementary metric space topology; continuous functions; differentiation of vector-valued functions; implicit, inverse function theorems; series of functions. Measure theory, convergence theorems, L1 spaces. Fall.
656 [196] COMPLEX ANALYSIS (3). Prerequisite, MATH 653. A rigorous treatment of complex integration, including the Cauchy theory. Elementary special functions, power series, local behavior of analytic functions. Maximum modulus theorem. Normal families. Spring.
657 [197] QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS (3). Prerequisites, linear algebra and MATH 653, or consent of the instructor. Existence and uniqueness theorems, linear and nonlinear systems, differential equations in the plane and on surfaces, Poincare-Bendixson Theory, Liapounov stability and structural stability, critical point analysis. Spring.
661 [191] SCIENTIFIC COMPUTATION (ENVR 661) (3). Error in computation. Solution of nonlinear equations. Interpolation. Approximation of functions. Fourier methods. Numerical integration and differentiation. Introduction to numerical solution of ODEs. Introduction to numerical linear algebra. Fall.
662 [192] SCIENTIFIC COMPUTATION II (COMP 760) (ENVR 662) (3). Direct methods for linear systems. Least squares problems. Iterative methods for linear systems. Direct and iterative methods for eigenvalue problems. The singular value decomposition. Methods for (stiff) systems of ODEs. Spring.
668 [198] METHODS OF APPLIED MATHEMATICS I (ENVR 668) (3). Topics: Contour integration in the complex plane, asymptotic expansions and steepest descent/stationary phase methods, special functions often arising in physical applications, elliptic functions and theta functions, Sturm-Liouville spectral theory. Fall.
669 [199] METHODS OF APPLIED MATHEMATICS II (ENVR 669) (3). Topics: Perturbation methods for ODE and PDE; WKBJ method, averaging, modulation theory for linear dispersive PDEs and nonlinear wave equations; long-time asymptotics of Fourier integral representations of PDEs; Green's functions; physical applications. Spring.
676 [186] MODULES, LINEAR ALGEBRA, AND GROUPS (3). Prerequisite, MATH 578 or permission of the instructor. Modules over rings, canonical forms for linear operators and bilinear forms, multilinear algebra, group and group actions. Fall.
677 [187] GROUPS, REPRESENTATIONS, AND FIELDS (3). Prerequisite, MATH 676. Internal structure of groups, Sylow theorems, generators and relations, group representations, fields, Galois theory, category theory. Spring.
680 [180] GEOMETRY OF CURVES AND SURFACES (3). Prerequisite, advanced calculus. Topics include: (curves) Frenet formulas, isoperimetric inequality, theorems of Crofton, Fenchel, Fary-Milnor; (surfaces) fundamental forms, Gaussian and mean curvature, special surfaces, geodesics, Gauss-Bonnet theorem. Fall.
681 [181] INTRODUCTORY TOPOLOGY (3). Prerequisites, MATH 653 and 676 or permission of the instructor. Topological spaces, connectedness, separation axioms, product spaces, extension theorems. Classification of surfaces, fundamental group and covering spaces. Spring.
751 [201] INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS (3). Prerequisite, MATH 653. Basic methods in partial differential equations. Topics may include: Cauchy-Kowalewski Theorem, Holmgren's Uniqueness Theorem, Laplace's equation, Maximum Principle, Dirichlet problem, harmonic functions, wave equation, heat equation.
753 [203] MEASURE AND INTEGRATION (3). Prerequisite, MATH 653 or consent of the instructor. Lebesgue and abstract measure and integration, convergence theorems, differentiation, Radon-Nikodym theorem, product measures, Fubini theorem, Lebesgue spaces, invariance under transformations, Haar measure and convolution. Fall.
754 [204] INTRODUCTORY FUNCTIONAL ANALYSIS (3). Prerequisite, MATH 753. Hahn-Banach and separation theorems. Normed and locally convex spaces, duals of spaces and maps, weak topologies; closed graph and open mapping theorems, uniform boundedness theorem, linear operators. Spring.
755 [205] ADVANCED COMPLEX ANALYSIS (3). Prerequisite, MATH 656. Laurent series; Mittag-Leffler and Weierstrass Theorems; Riemann mapping theorem; Runge's theorem; additional topics chosen from: harmonic, elliptic, univalent, entire, meromorphic functions; Dirichlet problem; Riemann surfaces. Fall.
756 [206] SEVERAL COMPLEX VARIABLES (3). Prerequisite, MATH 656. Elementary theory, the Cousin problems, domains of holomorphy, Runge domains and polynomial approximation, local theory, complex analytic structures, coherent analytic sheaves and Stein manifolds, Cartan's theorems. Spring. (Alternate years.)
761 [221] NUMERICAL ODE/PDE, I (ENVR 761) (MASC 781) (3). Prerequisites, MATH 661 and 662. Single, multistep methods for ODEs: stability regions, the root condition; stiff systems, backward difference formulas; two-point BVPs; stability theory; finite difference methods for linear advection diffusion equations. Fall.
762 [222] NUMERICAL ODE/PDE, II (ENVR 762) (MASC 782) (3). Prerequisite, MATH 761. Elliptic equation methods (finite differences, elements, integral equations); hyperbolic conservation law methods (Lax-Fiedrich, characteristics, entropy condition, shock tracking/capturing); spectral, pseudo-spectral methods; particle methods, fast summation, fast multipole/vortex methods. Spring.
768 [228] MATHEMATICAL MODELING I (ENVR 763) (MASC 783) (3). Prerequisites, MATH 668, 669, 661, and 662. Nondimensionalization and identification of leading order physical effects with respect to relevant scales and phenomena; derivation of classical models of fluid mechanics (lubrication, slender filament, thin films, Stokes flow); derivation of weakly nonlinear envelope equations. Fall.
769 [229] MATHEMATICAL MODELING II (ENVR 764) (MASC 784) (3). Prerequisites, MATH 668, 669, 661, and 662. Current models in science and technology: topics ranging from material science applications (e.g., flow of polymers and LCPs); geophysical applications (e.g., ocean circulation, quasi-geostrophic models, atmospheric vortices). Spring.
771 [231] COMMUTATIVE ALGEBRA (3). Prerequisite, MATH 677. Field extensions, integral ring extensions, Nullstellensatz and normalization theorem, derivations and separability, local rings, valuations, completions, filtrations and graded rings, dimension theory. Spring.
773 [273] LIE GROUPS (3). Prerequisites, MATH 676 and 781. Lie groups, closed subgroups, Lie algebra of a Lie group, exponential map, compact groups, Haar measure, orthogonality relations, Peter-Weyl theorem, maximal torus, representations, Weyl character formula, homogeneous spaces. Spring.
774 [274] LIE ALGEBRAS (3). Prerequisite, MATH 676. Nilpotent, solvable, and semisimple Lie algebras, structure theorems, root systems, Weyl groups, weights, classification of semisimple Lie algebras and their finite dimensional representations, character formulas. Fall.
775 [257] ALGEBRAIC GEOMETRY (3). Prerequisite, MATH 771. Topics may include: algebraic varieties, algebraic functions, abelian varieties, projective and complete varieties, algebraic groups, schemes and the Grothendieck theory, Riemann-Roch theorem. Spring. (Alternate years.)
776 [286] ALGEBRAIC TOPOLOGY (3). Prerequisites, MATH 681 and 676. Homotopy and homology; simplicial complexes and singular homology; other topics may include cohomology, universal coefficient theorems, higher homotopy groups, fibre spaces. Spring.
781 [271] DIFFERENTIABLE MANIFOLDS (3). Prerequisites, MATH 681, 676, and 653. Calculus on manifolds, vector bundles, vector fields and differential equations, Lie Groups, connections, de Rham cohomology. Fall.
782 [272] DIFFERENTIAL GEOMETRY (3). Prerequisite, MATH 781. Riemannian geometry, first and second variation of area and applications, effect of curvature on homology and homotopy, Chern-Weil theory of characteristic classes, Chern-Gauss-Bonnet theorem. Spring.
853 [224] HARMONIC ANALYSIS (3). Prerequisite, consent of the instructor. Subjects may include topological groups, abstract harmonic analysis, Fourier analysis, noncommutative harmonic analysis and group representation, automorphic forms, and analytic number theory. Fall. (Alternate years.)
854 [213] ADVANCED FUNCTIONAL ANALYSIS (3). Prerequisite, consent of the instructor. Subjects may include operator theory on Hilbert space, operators on Banach spaces, locally convex spaces, vector measures, Banach algebras. Spring. (Alternate years.)
857 [261] THEORY OF DYNAMICAL SYSTEMS (3). Prerequisite, consent of the instructor. Topics may include: ergodic theory, topological dynamics, stability theory of differential equations, classical dynamical systems, differentiable dynamics.
891 [210] TOPICS IN ANALYSIS (3). Prerequisite, consent of the instructor. Subjects may include geometric function theory, Riemann surfaces, calculus of variations, distribution theory, partial differential equations, or Fourier analysis. Spring.
892 [215] TOPICS IN COMPUTATIONAL MATHEMATICS (3). Prerequisites, MATH 661 and 662. Topics may include: finite element method; numerical methods for hyperbolic conservation laws, infinite dimensional optimization problems, variational inequalities, inverse problems. Spring.
893 [234] TOPICS IN ALGEBRA (3). Prerequisite, MATH 677. Topics from the theory of rings, theory of bialgebras, homological algebra, algebraic number theory, categories and functions.
894 [253] TOPICS IN COMBINATORIAL MATHEMATICS (3). Prerequisite, MATH 642 or consent of the instructor. Topics may include: combinatorial geometries, coloring and the critical problem, the bracket algebra, reduced incidence algebras and generating functions, binomial enumeration, designs, valuation module of a lattice, lattice theory. Spring. (Alternate years.)
895 [277] SPECIAL TOPICS IN GEOMETRY (3 each). Prerequisite, MATH 781. Topics may include elliptic operators, complex manifolds, exterior differential systems, homogeneous spaces, integral geometry, submanifolds of Euclidean space, geometrical aspects of mathematical physics. Fall. (Alternate years.)
896 [287] TOPICS IN ALGEBRAIC TOPOLOGY (3). Prerequisite, MATH 776 or permission of the instructor. Topics primarily from algebraic or differential topology, such as cohomology operations, homotopy groups, fibre bundles, spectral sequences, K-theory, cobordism, Morse Theory, surgery, topology of singularities. Fall and spring. (Alternate years.)
920 [390] SEMINAR AND DIRECTED READINGS (1-3).
921 [391] SEMINAR (3).
922 [392] MASTER'S PROJECT (3 or more).
923 [393] MASTER'S THESIS (3 or more). (This should not be taken by students electing nonthesis master's projects.)
924 [394] DOCTORAL DISSERTATION (3 or more).