# Courses

## MATH 1060Mathematics of Decision Making

Applications of modern mathematics to management and decision making including the solution of optimization problems using network theory, methods for optimal scheduling, voting methods, game theory, and related strategies. Applications include planning of postal…

## MATH 1101Introduction to Mathematical Modeling

Mathematical modeling using graphical, numerical, symbolic, and verbal techniques to describe and explore real-world data and phenomena. The investigation and analysis of applied problems and questions, and of effective communication of quantitative concepts and…

## MATH 1113Precalculus

Preparation for calculus, including an intensive study of algebraic, exponential, logarithmic, and trigonometric functions and their graphs. Applications include simple maximum/minimum problems, exponential growth and decay, and surveying problems.

Semester…

## MATH 1113Precalculus

Course ID: MATH 1113E. 3 hours. Course Title: Precalculus

Course Description: Preparation for calculus, including an intensive study of algebraic, exponential, logarithmic, and trigonometric functions and their graphs. Applications include simple maximum/…

## MATH 2001Geometry for Elementary School Teachers

Course ID: MATH 2001. 3 hours. Course Title: Geometry for Elementary School Teachers Course Description: A deep examination of mathematical topics designed for future elementary school teachers. Visualization. Properties of angles, circles, spheres, triangles, and…

## MATH 2002Numbers, Algebra, and Statistics for Elementary School Teachers

Course ID: MATH 2002. 3 hours. Course Title: Numbers, Algebra, and Statistics for Elementary School Teachers Course Description: A deep examination of mathematical topics designed for future elementary school teachers. Numbers, the decimal system, number lines.…

## MATH 2003Arithmetic for Elementary School Teachers

Course ID: MATH 2003. 3 hours. Course Title: Arithmetic for Elementary School Teachers Course Description: A deep examination of mathematical topics designed for future elementary school teachers. The meaning of addition, multiplication, and division. Adding,…

## MATH 2110Calculus for Economics

Topics specifically chosen to meet the needs of the student of economics: the definite integral, functions of several variables, partial derivatives, Lagrange multipliers, and matrices.

Duplicate Credit: Not open to students with credit in MATH 2210, MATH…

## MATH 2200Analytic Geometry and Calculus

Introductory differential calculus and its applications. Topics include limits, continuity, differentiability, derivatives of trigonometric, exponential and logarithmic functions, optimization, curve sketching, antiderivatives, differential equations, and…

## MATH 2250Calculus I for Science and Engineering

Limits, derivatives, differentiation of algebraic and transcendental functions; linear approximation, curve sketching, optimization, indeterminate forms. The integral, Fundamental Theorem of Calculus, areas. Emphasis on science and engineering applications.

## MATH 2260Calculus II for Science and Engineering

Volumes, arclength, work, separable differential equations. Techniques of integration. Sequences and series, convergence tests, power series and Taylor series. Vectors in three-dimensional space, dot product, cross product, lines and planes.

Duplicate Credit:…

## MATH 2270Calculus III for Science and Engineering

Calculus of functions of two and three variables: Parametric curves and applications to planetary motion. Derivatives, the gradient, Lagrange multipliers. Multiple integration, area, volume, and physical applications, polar, cylindrical, and spherical coordinates.…

## MATH 2270HCalculus III for Science and Mathematics (Honors)

Calculus of functions of two and three variables: Parametric curves and applications to planetary motion. Derivatives, the gradient, Lagrange multipliers. Multiple integration, area, volume, and physical applications; polar, cylindrical, and spherical coordinates.…

## MATH 2400Differential Calculus with Theory

Course Description: A rigorous and extensive treatment of differential calculus. Topics include the real numbers, the least upper bound property, limits, continuity, differentiability, and applications.

## MATH 2410Integral Calculus with Theory

Course Description: A rigorous and extensive treatment of integral calculus. Topics include the Fundamental Theorem of calculus, applications of integration, logarithms and exponentials, Taylor polynomials, `sequences, series, and uniform convergence.

## MATH 2500Multivariable Calculus

Calculus of functions of two and three variables: Parametric curves, derivatives, gradient, Lagrange multipliers. Multiple integration, area, volume, polar, cylindrical, and spherical coordinates. Line integrals and Green's Theorem. Introduction to surface integrals…

## MATH 2610Discrete Mathematics for Computer Science

Course ID: CSCI(MATH) 2610. 4 hours. Course Title: Discrete Mathematics for Computer Science

Course Description: The fundamental mathematical tools used in computer science: sets, relations, and functions; propositional logic, predicate logic, and inductive…

## MATH 2700Elementary Differential Equations

Course ID: MATH 2700. 3 hours. Course Title: Elementary Differential Equations

Course Description: First and second order ordinary differential equations, including physical and biological applications, numerical solutions, and mathematical modeling.

## MATH 3000Introduction to Linear Algebra

Course ID: MATH 3000.

3 hours.

Course Title: Introduction to Linear Algebra

Course Description: Theory and applications of systems of linear equations, vector spaces, and linear transformations. Fundamental concepts include: linear independence,…

## MATH 3100Sequences and Series

Course ID: MATH 3100. 3 hours. Course Title: Sequences and Series Course Description: Precise definitions of limit and convergence concepts; practical tests for convergence of infinite series; power series representations and numerical error estimates;…

## MATH 3200Introduction to Higher Mathematics

Course ID: MATH 3200. 3 hours. Course Title: Introduction to Higher Mathematics Course Description: Preparation in mathematical reasoning and proof-writing necessary for upper division course work in mathematics. Topics include logic, integers and induction, sets and…

Course Description: Strategies and tactics for solving advanced problems in undergraduate mathematics. Methods will be demonstrated through examples and exercises. (Will not count towards the mathematics degree.)

## MATH 3300Applied Linear Algebra

Course Description: Linear algebra from an applied and computational viewpoint. Linear equations, vector spaces, linear transformations; linear independence, basis, dimension; orthogonality, projections, and least squares solutions; eigenvalues, eigenvectors,…

## MATH 3500Multivariable Mathematics I (Honors)

Vector algebra and geometry, fundamental concepts of linear algebra, linear transformations, differential calculus of functions of several variables, solutions of linear systems and linear independence, extremum problems and projections. This course and its sequel…

## MATH 3500Multivariable Mathematics I

Course Description: Vector algebra and geometry, fundamental concepts of linear algebra, linear transformations, differential calculus of functions of several variables, solutions of linear systems and linear independence, extremum problems and projections. This…

## MATH 3510Multivariable Mathematics II (Honors)

Inverse function theorem and manifolds, integration in several variables, the change of variables theorem. Differential forms, line integrals, surface integrals, and Stokes's Theorem; applications to physics. Eigenvalues, eigenvectors, spectral theorem, and…

## MATH 3510Multivariable Mathematics II

Inverse function theorem and manifolds, integration in several variables, the change of variables theorem. Differential forms, line integrals, surface integrals, and Stokes's Theorem; applications to physics. Eigenvalues, eigenvectors, spectral theorem, and…

## MATH 4000Modern Algebra and Geometry I

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,…

## MATH 4010Modern 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 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…

## MATH 4100Real Analysis

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

## MATH 4110The Lebesgue Integral and Applications

The Lebesgue integral with applications to Fourier analysis and probability.

## MATH 4120Multivariable Analysis

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.

## MATH 4150Complex 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.

## MATH 4200Point Set Topology

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

## MATH 4220Differential Topology

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

## MATH 4250Differential Geometry

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.

## MATH 4300Introduction 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.

Not offered on a regular basis.

## MATH 4400Number Theory

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.

## MATH 4450Cryptology 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.

## MATH 4500Numerical 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…

## MATH 4510Numerical Analysis II

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

## MATH 4600Probability

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

## CSCI 4670Combinatorics

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…

## MATH 4670Combinatorics

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…

## CSCI 4690Graph 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…

## MATH 4690Graph 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…

## MATH 4700Qualitative Ordinary Differential Equations

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

## MATH 4720Introduction 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.

## MARS 4730Mathematics of Climate

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

## MATH 4730Mathematics of Climate

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

## MATH 4750Matrix and Integral Transforms with Applications

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.

## MATH 4760Mathematics and Music

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…

## BINF 4780Mathematical Biology

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…

## MATH 4780Mathematical Biology

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…

## MATH 4790The Mathematics of Option Pricing

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.

## MATH 4850History of Mathematics

The development of mathematical thought from ancient times to the present, paying particular attention to the context of today's mathematics curriculum.

Not offered on a regular basis.

## MATH 4900Topics in Mathematics

A special topic not otherwise offered in the mathematics curriculum.

Not offered on a regular basis.

## MATH 4950Research in Mathematics

Research in mathematics directed by a faculty member in the department of mathematics. A final report is required.

1-3 credit hours. Repeatable for maximum 6 hours credit. 2-6 hours lab per week.

Nontraditional Format: Students will meet with faculty…

## MATH 5001Arithmetic and Problem Solving

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…

## MATH 5002Geometry and Problem Solving

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.

## MATH 5003Algebra and Problem Solving

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.

## MATH 5020Arithmetic for Middle School Teachers

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

## MATH 5030Geometry and Measurement for Middle School Teachers

Principles of geometry and measurement for middle school teachers.

## MATH 5200Foundations of Geometry I

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

## MATH 5210Foundations of Geometry II

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

## MATH 6000Modern Algebra and Geometry I

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,…

## MATH 6010Modern 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 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…

## MATH 6100Real Analysis

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

## MATH 6110The Lebesgue Integral and Applications

The Lebesgue integral with applications to Fourier analysis and probability.

## MATH 6120Multivariable Analysis

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.

## MATH 6150Complex 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.

## MATH 6200Point Set Topology

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

## MATH 6400Number Theory

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.

## MATH 6450Cryptology 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.

## MATH 6500Numerical 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…

## MATH 6510Numerical Analysis II

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

## MATH 6600Probability

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

## CSCI 6670Combinatorics

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…

## MATH 6670Combinatorics

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…

## CSCI 6690Graph 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…

## MATH 6690Graph 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…

## MATH 6700Qualitative Ordinary Differential Equations

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

## MATH 6720Introduction 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.

## MARS 6730Mathematics of Climate

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

## MATH 6730Mathematics of Climate

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

## MATH 6750Matrix and Integral Transforms with Applications

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.

## MATH 6760Mathematics and Music

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…

## BINF 6780Mathematical Biology

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…

## MATH 6780Mathematical Biology

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…

## MATH 6790The Mathematics of Option Pricing

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.

## MATH 7001Arithmetic and Problem Solving

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…

## MATH 7002Geometry and Problem Solving

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.

## MATH 7003Algebra and Problem Solving

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.

## MATH 7020Arithmetic for Middle School Teachers

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

## MATH 7030Geometry and Measurement for Middle School Teachers

Principles of geometry and measurement for middle school teachers.

## MATH 7200Foundations of Geometry I

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

## MATH 7210Foundations of Geometry II

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

## MATH 8000Algebra

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…

## MATH 8010Algebra II

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,…

## MATH 8010Representation 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 p^a q…

## MATH 8020Commutative 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.

## MATH 8030Topics in Algebra

Topics in abstract algebra at the level of current research.

## MATH 8080Lie Algebras

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

## MATH 8100Real Analysis I

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…

## MATH 8110Real 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).

## MATH 8130Topics in Analysis

Topics in analysis at the level of current research.

## MATH 8150Complex Analysis 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, Rouche's…

## MATH 8160Complex Variables II

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

## MATH 8170Functional Analysis I

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

## MATH 8190Lie Groups

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

## MATH 8200Algebraic Topology

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.

## MATH 8210Topology of Manifolds

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

## MATH 8230Topics in Topology and Geometry

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

## MATH 8250Differential Geometry I

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

## MATH 8260Differential 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.

## MATH 8300Introduction to Algebraic 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…

## MATH 8310Geometry 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.

## MATH 8315Sheaves and Cohomology

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

## MATH 8320Algebraic 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.

## MATH 8330Topics in Algebraic Geometry

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

## MATH 8400Algebraic/Analytic Number Theory II

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…

## MATH 8400Algebraic/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.

## MATH 8430Topics 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.

## MATH 8440Topics 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.

## MATH 8500Advanced 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…

## MATH 8510Advanced Numerical Analysis II

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

## MATH 8550Special 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…

## MATH 8600Probability

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…

## MATH 8612Topics in Quantum Computation

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

## MATH 8620Stochastic Processes

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

## MATH 8630Stochastic Analysis

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

## MATH 8700Applied 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…

## MATH 8710Applied 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…

## MATH 8740Ordinary Differential Equations

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.

## MATH 8750Introduction 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,…

## MATH 8770Partial 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,…