Department of Mathematics

RICHARD McLAUGHLIN, Interim Chair

Professors

Idris Assani (45) Dynamical Systems, Ergodic Theory of Operators

Prakash Belkale (57) Algebraic Geometry

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

Sue E. Goodman (3) Topology, Dynamical Systems

Jane M. Hawkins (38) Ergodic Theory, Dynamical Systems

Jingfang Huang (51) Integral Equation Methods and Fast Algorithms

Christopher K. R. T. Jones (55) Applications of Dynamical Systems, Nonlinear Partial Differential Equations, Ocean Dynamics, Nonlinear Optics

Shrawan Kumar (46) Representation Theory, Geometry of Flag Varieties

Richard McLaughlin (50) Fluid Dynamics and Turbulent Transport

Peter J. Mucha (60) Network Analysis, Fluid Dynamics, Computer-Generated Animation

Karl E. Petersen (20) Ergodic Theory

Joseph F. Plante (23) Foliations, Dynamical Systems

Robert A. Proctor (43) Combinatorics, Representation Theory

Lev Rozansky (52) Three-Dimensional Topology

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

Associate Professors

David Adalsteinsson (1) Applied Mathematics and Scientific Computation

Jason Metcalfe (61) Partial Differential Equations

Laura Miller (22) Mathematical Biology and Fluid DynamicSorin Mitran (58) Computational Methods for Partial Differential Equations, Continuum-Kinetic Methods, Fluid Dynamics, Biological Fluid Dynamics and Mechanics

Richard Rimanyi (59), Topology, Geometry, Singularities

Assistant Professors

Hans Christianson (8) Semiclassical Analysis and Partial Differential Equations

Ellen Eischen (002) Algebraic Number Theory, Arithmetic Geometry

Robert Lipshitz (005) Topology

Jeremy Marzuola (9) Partial Differential Equations

Jason Metcalfe (61) Partial Differential Equations

Justin Sawon (64) Differential Geometry

Professors Emeriti

Ladnor Gessinger

William H. Graves

Robert G. Heyneman

Norberto Kerzman

Ancel C. Mewborn

John Pfaltzgraff

Michael Schlessinger

William W. Smith

Johann Sonner

James Stasheff

Warren R. Wogen

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 advisor in an allied discipline. The Department of Mathematics is housed in Phillips Hall and Chapman Hall. 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 filed by our posted deadline will receive full consideration.

Degree Requirements

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 Ph.D. 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 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 Ph.D. 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. A Ph.D. student can pass either the Pure Math option or the Applied Math option for the qualifying examination. To pass the Pure Math option a student must pass any three of the five qualifying exams. To pass the Applied Math option, a student is required to pass Methods of Applied Math and Scientific Computation.

During the second year a typical Ph.D. student will take the Ph.D. comprehensive exams and select courses from a list of 20 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 30 hours of graduate work in a program approved by the director of graduate studies. At least 15 of these hours must be in Mathematics Department courses numbered 600 or above.

4. Complete a master's project or thesis 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 coursework as well as the project or thesis.

6. A master's candidate must pass one of the written comprehensive exams given to doctoral students.

A candidate for a Ph.D. 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 one approved foreign language by passing an approved course or by passing a translation exam administered by the Mathematics Department.

4. Complete either the pure math option or the applied math option for qualifying examinations 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 that are not basic courses for any of the comprehensive exams passed by the student. Of these six courses at least three must be numbered over 700 and drawn from the second tier list.

6. Pass the TA Teaching Seminar and perform a minimum of two semesters of instructional service.

7. Pass a preliminary oral exam on the chosen Ph.D. specialty area.

8. Write a Ph.D. thesis and defend it successfully during a final oral exam chaired by the thesis advisor.

The student/faculty ratio of about 2/1 makes it possible for graduate students to take reading courses from individual faculty members that are tailored to meet the student's needs.

Minor in Mathematics

Graduate students in other departments who plan to offer mathematics as a (complete or partial) minor field for the Ph.D. should consult the director of graduate studies in mathematics for approval of their programs and for assignment of an advisor in the Department of Mathematics. This should be done at the earliest possible time, in order to prevent disappointment for the student.

Courses for Graduate and Advanced Undergraduate Students

MATH

401 Mathematical Concepts in Art (3). Mathematical theories of proportion, perspective (projective invariants and the mathematics of visual perception), symmetry, and aesthetics will be expounded and illustrated by examples from painting, architecture, and sculpture.

406 Mathematical Methods in Biostatistics (1). Prerequisite, MATH 232. 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.

410 Teaching and Learning Mathematics (4). Study of how people learn and understand mathematics, based on research in mathematics, mathematics education, psychology, and cognitive science. This course is designed to prepare undergraduate mathematics majors to become excellent high school mathematics teachers. It involves fieldwork in both the high school and college environments.

411 Developing Mathematical Concepts (3). Permission of the instructor. An investigation of various ways elementary concepts in mathematics can be developed. Applications of the mathematics developed will be considered.

418 Basic Concepts of Analysis for High School Teachers (3). Prerequisites, MATH 233 and 381. An examination of high school mathematics from an advanced perspective, including number systems and the behavior of functions and equations. Designed primarily for prospective or practicing high school teachers.

452 Mathematical and Computational Models in Biology (BIOL 452) (4). See BIOL 452 for description.

515 History of Mathematics (3). Prerequisite, MATH 381. A general survey of the history of mathematics with emphasis on elementary mathematics. Some special problems will be treated in depth.

521 Advanced Calculus I (3). Prerequisites, MATH 233 and 381. The real numbers, continuity and differentiability of functions of one variable, infinite series, integration.

522 Advanced Calculus II (3). Prerequisites, MATH 383 and 521. Functions of several variables, the derivative as a linear transformation, inverse and implicit function theorems, multiple integration.

523 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.

524 Elementary Differential Equations (3). Prerequisite, MATH 383. Linear differential equations, power series solutions, Laplace transforms, numerical methods.

528 Mathematical Methods for the Physical Sciences I (3). Prerequisites, MATH 383; and PHYS 104 and 105, or PHYS 116 and 117. Theory and applications of Laplace transform, Fourier series and transform, Sturm-Liouville problems. Students will be expected to do some numerical calculations on either a programmable calculator or a computer.

529 Mathematical Methods for the Physical Sciences II (3). Prerequisites, PHYS 104 and 105, and one of MATH 521, 524, or 528. 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.

533 Elementary Theory of Numbers (3). Prerequisite, MATH 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.

534 Elements of Modern Algebra (3). Prerequisite, MATH 381. Binary operations, groups, subgroups, cosets, quotient groups, rings, polynomials.

535 Introduction to Probability (STOR 435) (3). See STOR 435 for description.

547 Linear Algebra for Applications (3). Prerequisite, MATH 233 or 283. Algebra of matrices with applications: determinants, solution 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.

548 Combinatorial Mathematics (3). Prerequisite, MATH 381. Topics chosen from generating functions, Polya's theory of counting, partial orderings and incidence algebras, principle of inclusion-exclusion, Moebius inversion, combinatorial problems in physics and other branches of science.

550 Topology (3). Prerequisite, MATH 233; corequisite, MATH 383. Introduction to topics in topology, particularly surface topology, including classification of compact surfaces, Euler characteristic, orientability, vector fields on surfaces, tessellations, and fundamental group.

551 Euclidean and Non-Euclidean Geometries (3). Prerequisite, MATH 381. Critical study of basic notions and models of Euclidean and non-Euclidean geometries: order, congruence, and distance.

555 Introduction to Dynamics (3). Prerequisite, MATH 383. Topics will vary and may include iteration of maps, orbits, periodic points, attractors, symbolic dynamics, bifurcations, fractal sets, chaotic systems, systems arising from differential equations, iterated function systems, and applications.

564 Mathematical Modeling (3). Prerequisite, MATH 283 or 383. Requires some knowledge of computer programming. 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 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 to differential equations. Introduction to appropriate programming language; emphasis on graphics.

566 Introduction to Numerical Analysis (3). Prerequisite, MATH 383. Requires some knowledge of computer programming. Iterative methods, interpolation, polynomial and spline approximations, numerical differentiation and integration, numerical solution of ordinary and partial differential equations.

577 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. A much more abstract course than MATH 416 or 547.

578 Algebraic Structures (3). Prerequisite, MATH 547 or 577. 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.

590 Topics in Mathematics (3). Permission of the instructor. Topics may focus on matrix theory, analysis, algebra, geometry, or applied and computational mathematics.

594 Nonlinear Dynamics (PHYS 594) (3). See PHYS 594 for description.

635 Probability (STOR 635) (3). See STOR 635 for description.

641 Enumerative Combinatorics (3). Prerequisite, MATH 578. Basic counting; partitions; recursions and generating functions; signed enumeration; counting with respect to symmetry, plane partitions, and tableaux.

643 Combinatorial Structures (3). Prerequisite, MATH 578. Graph theory, matchings, Ramsey theory, extremal set theory, network flows, lattices, Moebius inversion, q-analogs, combinatorial and projective geometries, codes, and designs.

653 Introductory Analysis (3). Requires knowledge of advanced calculus. Elementary metric space topology, continuous functions, differentiation of vector-valued functions, implicit and inverse function theorems. Topics from Weierstrass theorem, existence and uniqueness theorems for differential equations, series of functions.

656 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.

657 Qualitative Theory of Differential Equations (3). Prerequisite, MATH 653. Requires knowledge of linear algebra. Existence and uniqueness theorems, linear and nonlinear systems, differential equations in the plane and on surfaces, Poincare-Bendixson theory, Lyapunov stability and structural stability, critical point analysis.

661 Scientific Computation I (ENVR 661) (3). Requires some programming experience and basic numerical analysis. Error in computation, solutions of nonlinear equations, interpolation, approximation of functions, Fourier methods, numerical integration and differentiation, introduction to numerical solution of ODEs, Gaussian elimination.

662 Scientific Computation II (COMP 662, ENVR 662) (3). Prerequisite, MATH 661. Theory and practical issues arising in linear algebra problems derived from physical applications, e.g., discretization of ODEs and PDEs. Linear systems, linear least squares, eigenvalue problems, singular value decomposition.

668 Methods of Applied Mathematics I (ENVR 668) (3). Requires an undergraduate course in differential equations. Contour integration, asymptotic expansions, steepest descent/stationary phase methods, special functions arising in physical applications, elliptic and theta functions, elementary bifurcation theory.

669 Methods of Applied Mathematics II (ENVR 669) (3). Prerequisite, MATH 668. Perturbation methods for ODEs and PDEs, WKBJ method, averaging and modulation theory for linear and nonlinear wave equations, long-time asymptotics of Fourier integral representations of PDEs, Green's functions, dynamical systems tools.

676 Modules, Linear Algebra, and Groups (3). Modules over rings, canonical forms for linear operators and bilinear forms, multilinear algebra, groups and group actions.

677 Groups, Representations, and Fields (3). Internal structure of groups, Sylow theorems, generators and relations, group representations, fields, Galois theory, category theory.

680 Geometry of Curves and Surfaces (3). Requires 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.

681 Introductory Topology (3). Prerequisites, MATH 653 and 680. Topological spaces, connectedness, separation axioms, product spaces, extension theorems. Classification of surfaces, fundamental group, covering spaces.

690 Topics in Mathematics (3). Permission of the department. Directed study of an advanced topic in mathematics. Topics will vary.

Courses for Graduate Students

MATH

751 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 Measure and Integration (3). Prerequisite, MATH 653. Permission of the instructor for students lacking the prerequisite. 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.

754 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.

755 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.

756 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.

761 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.

762 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.

768 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.

769 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).

771 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.

773 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.

774 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.

775 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.

776 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.

781 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.

782 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.

853 Harmonic Analysis (3). Permission 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.

854 Advanced Functional Analysis (3). Permission of the instructor. Subjects may include operator theory on Hilbert space, operators on Banach spaces, locally convex spaces, vector measures, Banach algebras.

857 Theory of Dynamical Systems (3). Permission of the instructor. Topics may include: ergodic theory, topological dynamics, stability theory of differential equations, classical dynamical systems, differentiable dynamics.

891 Special Topics (1–3). Advance topics in current research in statistics and operations research.

892 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.

893 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 Topics in Combinatorial Mathematics (3). Prerequisite, MATH 642. Permission of the instructor for students lacking the prerequisite. 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.

895 Special Topics in Geometry (3). 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.

896 Topics in Algebraic Topology (3). Prerequisite, MATH 776. Permission of the instructor for students lacking the prerequisite. 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.

920 Seminar and Directed Readings (1–3).

921 Seminar (3).

925 Practical Training Course in Mathematics (3–5). Required preparation, passed Ph.D. written comprehensive exam. An opportunity for the practical training of a graduate student interested in mathematics is identified. Typically this opportunity is expected to take the form of a summer internship.

992 Master's Project (3–21).

993 Master's Thesis (3–6). This should not be taken by students electing non-thesis master's projects.

994 Doctoral Dissertation (3–9).