The Department offers the following wide range of graduate courses in most of the main areas of mathematics. Courses numbered 6000-6999 are taken by senior undergraduates as well as by beginning Masters degree students. These courses generally carry three hours of credit per semester. Courses numbered 8000-8999 are taken by Masters and Ph.D. students; they generally carry three hours of credit per semester.


Foundations

7900 Foundations for Graduate Mathematics  An intensive review of techniques and material essential for graduate study in mathematics, including background in calculus and linear algebra. Emphasis is on small group study and presentations. Topics include proofs, induction, the metric structure of the reals, the Bolzano-Weierstrass theorem, and the diagonalization theorem.
 


Algebra/Group Theory

6000    Modern Algebra and Geometry I  An introduction to the ideas and constructs of abstract aglebra, emphasizing geometric motivation and applications.  Beginning with a careful study of integers, modular arithmetic, the Euclidean algorithm, the course moves on to fields, isometries of the complex plain, polynomials, splitting fields, rings, homomorphisms, field extensions and compass and straightedge constructions.

6010    Modern Algebra and Geometry II  More advanced abstract algebraic structures and concepts, such as groups, symmetry, group actions, counting principles, symmetry groups of the regular polyhedra, Burnside's Theorem, isometries of R3 , Galois theory, affine and projective geometry.

6050    Advanced Linear Algebra  Orthogonal and unitary groups, spectral theorem; infinite-dimensional vector spaces; Jordan and rational conconical forms and applications.

6080    Advanced Algebra  A course in linear algebra, grouops, rings, and modules, intermediate in level between MATH 6010 and MATH 8000.  Topics include the finite-dimensional spectral theorem, group actions, classification of finitely generated modules over principal ideal domains, and canonical forms of linear operators.

8000    Algebra  A course in groups, fields and rings, designed to prepare the student for the algebra prelims.  Some topics covered include the Sylow theorems, solvable and simple groups, Galois theory, finite fields, Noetherian rings and modules.

8010    Representation Theory of Finite Groups Irreducible and indecomposable representations, Schur's Lemma, Maschke's theorem, the Wedderburn structure theorem, characters and orthogonality relations, induced representations and Frobenius reciprocity, central characters and central idempotents, Burnside's theorem, Frobenius normal p-complement theorem.

8020    Commutative Algebra  Localization and completion, Nakayama's lemma, Dedekind domains, Hilbert's basis theorem, Hilbert's Nullstellensatz, Krull dimension, depth and Cohen-Macaulay rings, regular local rings.

8030    Topics in Algebra  This course will present topics in abstract algebra at the level of current research.

8080    Lie Algebras  Nilpotent and solvable Lie algebras, structure and classification of semisimple Lie algebras, roots, weights, finite-dimensional representations

 


Analysis

6100    Real Analysis  Metric spaces and continuity; differentiable and integrable functions of one variable; sequences and series of functions.

6110    The Lebesgue Integral and Applications The Lebesgue integral, with applications to Fourier analysis and probability.

6120    Multivariable Analysis The continuation of MATH 4100 to the multivariable setting: the derivative as a linear map, inverse and implicit function theorems, change of variables in multiple integrals; manifolds, differential forms, and the generalized Stokes' Theorem.

6150    Complex Variables  Differential and integral calculus of functions of a complex variable, with applications.  Topics include the Cauchy integral formula, power series and Laurent series, and the residue theorem.

8100    Real Analysis I  Measureand integration theory with relevant examples from Lebesgue integration, Hilbert spaces (only with regard to L2 ), L2 spaces and the related Riesz representation theorem.  Hahn, Jordan and Lebesgue decomposition theorems, Radon-Nikodym Theorem and Fubini's Theorem.

8110    Real Analysis II  Topics including:  Haar Integral, change of variable formula, Hahn-Banach theorem for Hilbert spaces, Banach spaces and Fourier theory (series, transform, Gelfand-Fourier homomorphism).

8150    Complex Variables I  The Cauchy-Riemann Equations, linear fractional transformations and elementary conformal mappings, Cauchy's theorems and its consequences including: Morera's theorem, Taylor and Laurent expansions, maximum principle, residue theorem, argument principle, residue theorem, argument principle, Rouche's theorem and Liouville's theorem.

8160    Complex Variables II  Topics including Riemann Mapping Theorem, elliptic functions, Mittag-Leffler and Weierstrass Theorems, analytic continuation and Riemann surfaces.

8170    Functional Analysis I  Introduction to Hilbert spaces and Banach spaces, spectral theory, topological vector spaces, comvexity and its consequences including the Krein-Milman theorem.

8180    Functional Analysis II  Introduction to operator theory, spectral theorem for normal operators, distribution theory, the Schwartz spaces, topics from C*-algebras and von Neumann algebras.

8190    Lie Groups  Classical groups, exponential map, Poincare-Birkhoff-Witt Theorem, homogeneous spaces, adjoint representation, covering groups, compact groups, Peter-Weyl Theorem, Weyl character formula.

 


Applied Mathematics and Differential Equations

6700    Qualitative Ordinary Differential Equations  Transform methods, linear and nonlinear systems of ordinary differential equations, stability, and chaos.

6720    Introduction to Partial Differential Equations  The basic partial differential equations of mathematical physics: Laplace's equation, the wave equation, and the heat equation. Separation of variables and Fourier series techniques are featured.

6760    Mathematics and Music  Mathematical aspects of music.

6780    Mathematical Biology  Mathematical models in the biological sciences: compartmental flow models, dynamic system models, discrete and continuous models, deterministic and stochastic models.

8700    Applied Mathematics: Applications in Industry  Mathematical modeling of some real-world industrial problems.Topics will be selected from a list which includes air quality modeling, crystal precipitation, electron beam lithography, image processing, photographic film development, production planning in manufacturing, and optimal control of chemical reactions.

8710    Applied Mathematics: Variational Methods/Perturbation Theory  Calculus of variations, Euler-Lagrange equations, Hamilton's principle, approximate methods, eigenvalue problems, asymptotic expansions, method of steepest descent, method of stationary phase, perturbation of eigenvalues, nonlinear eigenvalue problems, oscillations and periodic solutions, Hopf bifurcation, singular perturbation theory, applications.

8740    Ordinary Differential Equations  Solutions of initial value problems: existence, uniqueness, and dependence on parameters, differential inequalities, maximal and minimal solutions, continuation of solutions, linear systems, self-adjoint eigenvalue problems, Floquet Theory.

8750    Introduction to Dynamical Systems  Continuous dynamical systems, trajectories, periodic orbits, invariant sets, structure of alpha and omega limit sets, applications to two-dimensional autonomous systems of ODE's, Poincare-Bendixson Theorem, discrete dynamical systems, infinite dimensional spaces, semi-dynamical systems, functional differential equations.

8770    Partial Differential Equations  Classification of second order linear partial differential equations, modern treatment of characteristics, function spaces, Sobolev spaces, Fourier transform of generalized functions, generalized and classical solutions, initial and boundary value problems, eigenvalue problems.

 


Algebraic Geometry

6300    Introduction to Algebraic Curves  Polynomials and resultants, projective spaces. The focus is on plane algebraic curves: intersection, Bezout's theorem, linear systems, rational curves, singularities, blowing up.

8300    Introduction to Algebraic Geometry  An invitation to algebraic through a study of examples. Affine and projective varieties, regular and rational maps, Nullstellensatz. Veronese and Segre varieties, Grassmannians, algebraic groups, quadrics.  Smoothness and tangent spaces, singularities and tangent cones.

8310    Geometry of Schemes  The language of Grothendieck's theory of schemes. Topics include the spectrum of a ring, "gluing" spectra to form schemes, products, quasi-coherent sheaves of ideals, and the functor of points.

8315    Sheaves and Cohomology Main results and techniques for sheaves on topological spaces and their cohomology.

8320    Algebraic Curves  The theory of curves, including linear series and the Riemann Roch theorem. Either the algebraic (variety), arithmetic (function field), or analytic (Riemann surface) aspect of the subject may be emphasized in different years.

8330    Topics in Algebraic Geometry  Advanced topics such as algebraic surfaces, or cohomology and sheaves.

 


Topology/Geometry

 

6200    Point Set Topology  Topological spaces, continuity; connectedness, compactness; separation axioms and Tietze extension theorem; function spaces.

 

6220    Differential Topology  Manifolds in Euclidean space:  fundamental ideas of transversality, homotopy, and intersection theory; differential forms, Stokes' Theorem, deRham cohomology, and degree theory.

6250    Differential Geometry  An introduction to the geometry of curves and surfaces in Euclidean space:  Frenet formulas for curves, notions of curvature for surfaces; Gauss-Bonnet Theorem; discussion of non-Euclidean geometries.

8200    Algebraic Topology  The fundamental group, van Kampen's theorem, and covering spaces.  Introduction to homology:  simplicial, singular, and cellular.  Applications.

8210    Topology of Manifolds  Poincar duality, deRham's theorem, topics from differential topology.

8220    Homotopy Theory  Topics in homotopy theory, including homotopy groups, CW complexes, and fibrations.

8230    Topics in Topology and Geometry  Advanced topics in topology and/or differential geometry leading to and including research level material.

8250    Differential Geometry I  Differentiable manifolds, vector bundles, tensors, flows, and Frobenius' theorem.  Introduction to Riemannian geometry.

8260    Differential Geometry II  Riemannian geometry:  connections, curvature, first and second variation; geometry of submanifolds.  Gauss-Bonnet theorem.  Additional topics, such as characteristic classes, complex manifolds, integral geometry.

 


Number Theory

6400    Number Theory  Euler's theorem, public key cryptology, pseudoprimes, multiplicative functions, primitive roots, quadratic reciprocity, continued fractions, sums of two squares and Gaussian integers.

6450    Cryptology and Computational Number Theory  Recognizing prime numbers, factoring composite numbers, finite fields, elliptic curves, discrete logarithms, private key cryptology, key exchange systems, signature authentication, public key cryptology.

8400    Algebraic/Analytic Number Theory I  The core material of algebraic number theory: number fields, rings of integers, discriminants, ideal class groups, Dirichlet's unit theorem, splitting of primes; p-adic fields, Hensel's lemma, adeles and ideles, the strong approximation theorem.

8410    Algebraic/Analytic Number Theory II  A continuation of Algebraic and Analytic Number Theory I, introducing analytic methods: the Riemann Zeta function, its analytic continuation and functional equation, the Prime number theorem; sieves, the Bombieri-Vinogradov theorem, the Chebotarev density theorem.

8430    Topics in Arithmetic Geometry  Topics in Algebraic number theory and Arithmetic geometry, such as class field theory, Iwasawa theory, elliptic curves, complex multiplication, cohomology theories, Arakelov theory, diophantine geometry, automorphic forms, L-functions, representation theory.

8440    Topics in Combinatorial/Analytic Number Theory  Topics in combinatorial and analytic number theory, such as sieve methods, probabilistic models of prime numbers, the distribution of arithmetic functions, the circle method, additive number theory, transcendence methods.

8450    Topics in Algorithmic Number Theory  Topics in computational number theory and algebraic geometry, such as factoring and primality testing, cryptography and coding theory, algorithms in number theory and arithmetic geometry.

 


Numerical Analysis

6500    Numerical Analysis I  Methods for finding approximate numerical solutions to a variety of mathematical problems, featuring careful error analysis. A mathematical software package will be used to implement iterative techniques for nonlinear equations, polynomial interpolation, integration, and problems in linear algebra such as matrix inversion, eigenvalues and eigenvectors.

6510    Numerical Analysis II  Numerical solutions of ordinary and partial differential equations, higher-dimensional Newton's method, and splines.

8500    Advanced Numerical Analysis I  Numerical solution of nonlinear equations in one and several variables, numerical methods for constrained and unconstrained optimization, numerical solution of linear systems, numerical methods for computing eigenvalues and eigenvectors, numerical solution of linear least squares problems, computer applications for applied problems.

8510    Advanced Numerical Analysis II  Polynomial and spline interpolation and approximation theory, numerical integration methods, numerical solution of ordinary differential equations, computer applications for applied problems.

8520    Advanced Numerical Analysis III  Finite difference and finite element methods for elliptic, parabolic, and hyperbolic partial differential equations convergence and stability of those methods, numerical algorithms for the implementation of those methods.

8550    Special Topics in Numerical Analysis  Special topics in numerical analysis, including iterative methods for large linear systems, computer aided geometric design, multivariate splines, numerical solutions for pde's, numerical quadrature and cubature, numerical optimization, wavelet analysis for numerical imaging. In any semester, one of the above topics will be covered.
 


Probability, Stochastic Processes and Combinatorics

6600    Probability  Discrete and continuous random variables, expectation, independence and conditional probability; binomial, Bernoulli, normal, and Poisson distributions; law of large numbers and central limit theorem.

6630    Mathematical Analysis of Computer Algorithms  Discrete algorithms (number-theoretic, graph-theoretic, combinatorial, and algebraic) with an emphasis on techniques for their mathematical analysis.

6670    Combinatorics  Basic counting principles: permutations, combinations, probability, occupancy problems, and binomial coefficients.  More sophisticated methods include generating functions, recurrence relations, inclusion/exclusion principle, and the pigeonhole principle. Additional topics include asymptotic enumeration, Polya counting theory, combinatorial designs, coding theory, and combinatorial optimization.

6690    Graph Theory  Elementary theory of graphs and digraphs. Topics include connectivity, reconstruction, trees, Euler's problem, hamiltonicity, network flows, planarity, node and edge colorings, tournaments, matchings, and extremal graphs. A number of algorithms and applications are included.

8600    Probability  Probability spaces, random variables, distributions, expectation and higher moments, conditional probability and expectation, convergence of sequences and series of random variables, strong and weak laws of large numbers, characteristic functions, infinitely divisible distributions, weak convergence of measures, central limit theorems.

8620    Stochastic Processes  Conditional expectation, Markov processes, martingales and convergence theorems, stationary processes, introduction to stochastic integration.

8630    Stochastic Analysis  Conditional expectation, Brownian motion, semimartingales, stochastic calculus, stochastic differential equations, stochastic control, stochastic filtering.


Education

7040   Basic Ideas of Calculus I  Survey of one-variable calculus in preparation for teaching calculus at the secondary level: combines review of basic techniques with careful study of underlying concepts. This is MATH 2400H for graduate students in Mathematics Education.

7050   Basic Ideas of Calculus II  A continuation of Basic Ideas of Calculus I focusing on functions of several variables. This is MATH 2410H for graduate students in Mathematics Education.


Misc. Mathematics Courses

8800   Directed Reading  A directed reading and/or project at the doctoral level.

8850   Vertical Research Group and Graduate Student Seminar  Collaborative Research in a group setting on a topic of current research, under the direction of one or more faculty members. Emphasis will be given to problems that may be accessible to elementary methods. An introduction to basic research methods in mathematics as well as expository and literature-search skills.

7005/9005   Doctoral Graduate Student Seminar  Fall (Teaching seminar for first-time Math 1113 instructors) and Spring (Teaching seminar for first-time Math 2200 instructors).