# Courses

### MATH 635 – Probability II

Credits: 3
Description: Foundations of probability. Basic classical theorems. Modes of probabilistic convergence. Central limit problem. Generating functions, characteristic functions. Conditional probability and expectation.
Prerequisite: STOR 634
Permission: Permission of the instructor for students lacking the prerequisite
Same As: STOR 635

### MATH 641 – Enumerative Combinatorics

Credits: 3
Description: Basic counting; partitions; recursions and generating functions; signed enumeration; counting with respect to symmetry, plane partitions, and tableaux.
Prerequisite: MATH 578

### MATH 643 – Combinatorial Structures

Credits: 3
Description: Graph theory, matchings, Ramsey theory, extremal set theory, network flows, lattices, Moebius inversion, q-analogs, combinatorial and projective geometries, codes, and designs.
Prerequisite: MATH 578

### MATH 653 – Introductory Analysis

Credits: 3
Description: 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.

### MATH 656 – Complex Analysis

Credits: 3
Description: A rigorous treatment of complex integration, including the Cauchy theory. Elementary special functions, power series, local behavior of analytic functions.
Prerequisite: MATH 653

### MATH 657 – Qualitative Theory of Differential Equations

Credits: 3
Description: 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.
Prerequisite: MATH 653

### MATH 661 – Scientific Computation I

Credits: 3
Description: 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.
Same As: ENVR 661

### MATH 662 – Scientific Computation II

Credits: 3
Description: Theory and practical issues arising in linear algebra problems derived from physical applications, for example, discretization of ODEs and PDEs. Linear systems, linear least squares, eigenvalue problems, singular value decomposition.
Prerequisite: MATH 661
Same As: COMP 662; ENVR 662

### MATH 668 – Methods of Applied Mathematics I

Credits: 3
Description: 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.
Same As: ENVR 668

### MATH 669 – Methods of Applied Mathematics II

Credits: 3
Description: 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.
Prerequisite: MATH 668
Same As: ENVR 669

### MATH 676 – Modules, Linear Algebra, and Groups

Credits: 3
Description: Requires knowledge of linear algebra and algebraic structures. Modules over rings, canonical forms for linear operators and bilinear forms, multilinear algebra, groups and group actions.
Repeat Rules: May be repeated for credit; 6 total credits; 2 total completions.

### MATH 677 – Groups, Representations, and Fields

Credits: 3
Description: Internal structure of groups, Sylow theorems, generators and relations, group representations, fields, Galois theory, category theory.
Prerequisite: MATH 676

### MATH 680 – Differentiable Manifolds.

Credits: 3
Description: Calculus on manifolds, vector bundles, vector fields and differential equations, de Rham cohomology.
Prerequisite: MATH 681

### MATH 681 – Introductory Topology

Credits: 3
Description: Topological spaces, connectedness, separation axioms, product spaces, extension theorems. Classification of surfaces, fundamental group, covering spaces.

### MATH 690 – Topics In Mathematics

Credits: 3
Description: Permission of the department. Directed study of an advanced topic in mathematics. Topics will vary.
Repeat Rules: May be repeated for credit; may be repeated in the same term for different topics; 12 total credits. 4 total completions.

### MATH 691H – Honors Research in Mathematics

Credits: 3
Description: Honors Research in Mathematics
Prerequisite: Permission of the director of undergraduate studies. Readings in mathematics and the beginning of directed research on an honors thesis.
Gen Ed: Making Connections: EE- Mentored Research; IDEAs in Action: RESEARCH.

### MATH 692H – Honors Thesis in Mathematics

Credits: 3
Description: Honors Thesis in Mathematics
Prerequisite: Permission of the director of undergraduate studies. Completion of an honors thesis under the direction of a member of the faculty. Required of all candidates for graduation with honors in mathematics.
Gen Ed: Making Connections: EE- Mentored Research; IDEAs in Action: RESEARCH.

### MATH 751 – Introduction to Partial Differential Equations

Credits: 3
Description: 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.
Prerequisite: MATH 653

### MATH 753 – Measure and Integration

Credits: 3
Description: 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.
Prerequisite: MATH 653, Permission of the instructor for students lacking the prerequisite

### MATH 754 – Introductory Functional Analysis

Credits: 3
Description: 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.
Prerequisite: MATH 753

### MATH 755 – Advanced Complex Analysis

Credits: 3
Description: 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.
Prerequisite: MATH 656

### MATH 756 – Several Complex Variables

Credits: 3
Description: 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.
Prerequisite: MATH 656

### MATH 761 – Numerical ODE/PDE I

Credits: 3
Description: 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.
Prerequisite: MATH 661, MATH 662
Same As: ENVR 761, and MASC 781

### MATH 762 – Numerical ODE/PDE II

Credits: 3
Description: 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.
Prerequisite: MATH 761
Same As: ENVR 762, and MASC 782

### MATH 768 – Mathematical Modeling I

Credits: 3
Description: 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.
Prerequisite: MATH 661, MATH 662, MATH 668, and MATH 669
Same As: ENVR 763, and MASC 783

### MATH 769 – Mathematical Modeling II

Credits: 3
Description: Current models in science and technology: topics ranging from material science applications, for example, flow of polymers and LCPs; geophysical applications, for example, ocean circulation, quasi-geostrophic models, atmospheric vortices.
Prerequisite: MATH 661, MATH 662, MATH 668, and MATH 669
Same As: ENVR 764, and MASC 784

### MATH 771 – Commutative Algebra

Credits: 3
Description: Field extensions, integral ring extensions, Nullstellensatz and normalization theorem, derivations and separability, local rings, valuations, completions, filtrations and graded rings, dimension theory.
Prerequisite: MATH 677

### MATH 773 – Lie Groups

Credits: 3
Description: 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.
Prerequisite: MATH 676, MATH 781

### MATH 774 – Lie Algebras

Credits: 3
Description: 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.
Prerequisite: MATH 676

### MATH 775 – Algebraic Geometry

Credits: 3
Description: Topics may include: algebraic varieties, algebraic functions, abelian varieties, projective and complete varieties, algebraic groups, schemes and the Grothendieck theory, Riemann-Roch theorem.
Prerequisite: MATH 771

### MATH 776 – Algebraic Topology

Credits: 3
Description: Homotopy and homology; simplicial complexes and singular homology; other topics may include cohomology, universal coefficient theorems, higher homotopy groups, fibre spaces.
Prerequisite: MATH 676, MATH 681

### MATH 782 – Differential Geometry

Credits: 3
Description: 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.
Prerequisite: MATH 781

### MATH 853 – Harmonic Analysis

Credits: 3
Description: 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.

### MATH 854 – Advanced Functional Analysis

Credits: 3
Description: Permission of the instructor. Subjects may include operator theory on Hilbert space, operators on Banach spaces, locally convex spaces, vector measures, Banach algebras.

### MATH 857 – Theory of Dynamical Systems

Credits: 3
Description: Permission of the instructor. Topics may include: ergodic theory, topological dynamics, stability theory of differential equations, classical dynamical systems, differentiable dynamics.

### MATH 891 – Special Topics

Credits: 1-3
Description: Advance topics in current research in statistics and operations research.
Same As: GNET 891, BCB 891
Repeat Rules: May be repeated for credit; may be repeated in the same term for different topics.

### MATH 892 – Topics in Computational Mathematics

Credits: 3
Description: Topics may include: finite element method; numerical methods for hyperbolic conservation laws, infinite dimensional optimization problems, variational inequalities, inverse problems.
Prerequisite: MATH 661, MATH 662

### MATH 893 – Topics in Algebra

Credits: 3
Description: Topics from the theory of rings, theory of bialgebras, homological algebra, algebraic number theory, categories and functions.
Prerequisite: MATH 677

### MATH 894 – Topics in Combinatorial Mathematics

Credits: 3
Description: 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.
Prerequisite: MATH 641, Permission of the instructor for students lacking the prerequisite.

### MATH 895 – Special Topics in Geometry

Credits: 3
Description: Topics may include elliptic operators, complex manifolds, exterior differential systems, homogeneous spaces, integral geometry, submanifolds of Euclidean space, geometrical aspects of mathematical physics.
Prerequisite: MATH 781

### MATH 896 – Topics in Algebraic Topology

Credits: 3
Description: 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.
Prerequisite: MATH 776, Permission of the instructor for students lacking the prerequisite.

Credits: 1-3

Credits: 3

### MATH 925 – Practical Training Course in Mathematics

Credits: 1-3
Description: Required preparation, passed Ph.D. or M.S. 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.
Prerequisite: Prerequisite, Successful completion of the written comprehensive examination degree requirement
Repeat Rules: May be repeated for credit

Credits: 3

### MATH 993 – Master's Research and Thesis

Credits: 3
Description: This should not be taken by students electing non-thesis master's projects.
Repeat Rules: May be repeated for credit.

### MATH 994 – Doctoral Research and Dissertation

Credits: 3 