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# Graduate Courses

Topics in mathematics designed for future elementary school teachers. Problem solving. Number systems: whole numbers, integers, rational numbers (fractions) and real numbers (decimals), and the relationships between these systems. Understanding multiplication and…

A deep examination of mathematical topics designed for future elementary school teachers. Length, area, and volume. Geometric shapes and their properties. Probability. Elementary number theory. Applications of elementary mathematics.

A deep examination of topics in mathematics that are relevant for elementary school teaching. Probability, number theory, algebra and functions, including ratio and proportion. Posing and modifying problems.

Operations of arithmetic for middle school teachers; number systems; set theory to study mappings, functions, and equivalence relations.

Principles of geometry and measurement for middle school teachers.

Advanced elementary geometry for prospective teachers of secondary school mathematics: axiom systems and models; the parallel postulate; neutral, Euclidean, and non-Euclidean geometries.

Further development of the axioms and models for Euclidean and non-Euclidean geometry; transformation geometry. Often includes advanced topics in geometry.

Abstract algebra, emphasizing geometric motivation and applications. Beginning with a careful study of integers, modular arithmetic, and the Euclidean algorithm, the course moves on to fields, isometries of the complex plane, polynomials, splitting fields, rings,…

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 R^3, Galois Theory, and affine and projective geometry.

Orthogonal and unitary groups, spectral theorem; infinite dimensional vector spaces; Jordan and rational canonical forms and applications.

Linear algebra, groups, rings, and modules, intermediate in level between Modern Algebra and Geometry II and Algebra. Topics include the finite-dimensional spectral theorem, group actions, classification of finitely generated modules over principal ideal domains, and…

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

The Lebesgue integral with applications to Fourier analysis and probability.

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.

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.

Topological spaces, continuity; connectedness, compactness; separation axioms and Tietze extension theorem; function spaces.

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

**Not offered on a regular basis.**

Recognizing prime numbers, factoring composite numbers, finite fields, elliptic curves, discrete logarithms, private key cryptology, key exchange systems, signature authentication, public key cryptology.

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…

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

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

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…

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…

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…

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…

Transform methods, linear and nonlinear systems of ordinary differential equations, stability, and chaos.

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.

Basic mathematical models describing the physical, chemical, and biological interactions that affect climate. Mathematical and computational tools for analyzing these models.

Basic mathematical models describing the physical, chemical, and biological interactions that affect climate. Mathematical and computational tools for analyzing these models.

Foundations in the most commonly used transforms in mathematics, science, and engineering. Eigenvector decompositions, Fourier transforms, singular value decompositions, and the Radon transform, with emphasis on mathematical structure and applications.

This course is intended for undergraduates (math majors, music majors, and others) interested in the mathematical aspects of music. At least some familiarity with musical notation is a prerequisite. Topics to be discussed include the structure of sound, the…

Mathematical models in the biological sciences, systems, phase-plane analysis, diffusion, convective transport, bifurcation analysis. Possible applications will include population models, infectious disease and epidemic models, acquired immunity and drug distribution…

Mathematical models in the biological sciences, systems, phase-plane analysis, diffusion, convective transport, bifurcation analysis. Possible applications will include population models, infectious disease and epidemic models, acquired immunity and drug distribution…

Bonds, stock markets, derivatives, arbitrage, and binomial tree models for stocks and options, Black-Scholes formula for options pricing, hedging. Computational methods will be incorporated.

Topics in mathematics designed for future elementary school teachers. Problem solving. Number systems: whole numbers, integers, rational numbers (fractions) and real numbers (decimals), and the relationships between these systems. Understanding multiplication and…

A deep examination of mathematical topics designed for future elementary school teachers. Length, area, and volume. Geometric shapes and their properties. Probability. Elementary number theory. Applications of elementary mathematics.

A deep examination of topics in mathematics that are relevant for elementary school teaching. Probability, number theory, algebra and functions, including ratio and proportion. Posing and modifying problems.

Operations of arithmetic for middle school teachers; number systems; set theory to study mappings, functions, and equivalence relations.

Principles of geometry and measurement for middle school teachers.

Advanced elementary geometry for prospective teachers of secondary school mathematics: axiom systems and models; the parallel postulate; neutral, Euclidean, and non-Euclidean geometries.

Further development of the axioms and models for Euclidean and non-Euclidean geometry; transformation geometry. Often includes advanced topics in geometry.

Groups and rings, including Sylow theorems, classifying small groups, finitely generated abelian groups, Jordan-Holder theorem, solvable groups, simplicity of the alternating group, euclidean domains, principal ideal domains, unique factorization domains, noetherian…

Modules and fields, including noetherian modules, finitely generated modules over principal ideal domains, canonical forms of matrices, spectral theorems, tensor products, algebraic and transcendental field extensions, galois theory, solvability of polynomials,…

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 p^a q…

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.

Topics in abstract algebra at the level of current research.

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

Measure and integration theory with relevant examples from Lebesgue integration, Hilbert spaces (only with regard to L^2), L^p spaces and the related Riesz representation theorem. Hahn, Jordan and Lebesgue decomposition theorems, Radon-Nikodym Theorem and Fubini's…

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

Topics in analysis at the level of current research.

Description: Topics including Riemann Mapping Theorem, elliptic functions, Mittag-Leffler and Weierstrass Theorems, analytic continuation and Riemann surfaces.

Hilbert spaces and Banach spaces, spectral theory, topological vector spaces, convexity and its consequences, including the Krein-Milman theorem.

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

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

MATH 8200 carries 3 credit hours; students are expected to sign up for MATH 8205 for 1 credit hour.

Poincare duality, deRham's theorem, topics from differential topology.

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

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

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

An invitation to algebraic geometry 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…

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.

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

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.

Advanced topics such as algebraic surfaces, or cohomology and sheaves.

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.

Description: 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…

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.

…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.

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…

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

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…

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…

The selection of topics varies from year to year. Students will make presentations based on journal articles or original research.

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

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

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…

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…

Description: 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.

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,…

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,…