# Mathematics

Apply NowThe Department of Mathematics offers programs leading to the degrees of Master of Science and Doctor of Philosophy in Mathematics and in Applied Mathematics. Students may opt for the Concentration in Computational Mathematics, which is attached to the program in applied mathematics. The Concentration in Interdisciplinary Mathematics (MAI) is available to Ph.D. students in either Mathematics or Applied Mathematics. It is not available to Masters Students. Joint research endeavors with industrial and governmental partners are facilitated and encouraged. The Department of Mathematics also offers a Certificate.

## Admissions Requirements

Applicants for admission should have an undergraduate or Master's degree in mathematics or applied mathematics. This should include courses in advanced calculus, analysis, modern algebra and linear algebra. Applicants with degrees in other subjects may be admitted but may be required to take certain undergraduate courses in mathematics without receiving graduate credit. GRE general scores are required. The GRE Subject Test in Mathematics is not required but a good score can be a positive factor in admission.

## Master of Science Requirements

The M.S. degree requires a minimum of 30 credit hours with courses chosen to satisfy certain requirements to cover material from three different areas in the department, and some level of depth of material.

## Ph.D. Requirements

The Ph.D. requires a minimum of 72 credit hours. A student will typically take 50-60 semester hours of course credits for the Ph.D. The written preliminary examination consists of examinations in three areas of mathematics chosen by the student from 12 possibilities. The research dissertation should represent a substantial contribution to an area of mathematics or its applications.

## Student Financial Support

Teaching assistantships and some research assistantships are available. Teaching assistants benefit from a structured program of training in university-level teaching.

## Faculty

### Full Professors

- Bojko Nentchev Bakalov
- Alina Emil Chertock
- Moody Ten-Chao Chu
- Jo-Ann D. Cohen
- Patrick Louis Combettes
- Pierre Alain Gremaud
- Mansoor Abbas Haider
- Hoon Hong
- Ilse Ipsen
- Kazufumi Ito
- Naihuan Jing
- Erich L. Kaltofen
- Carl Timothy Kelley
- Arkady Kheyfets
- Irina Aleksandrovna Kogan
- Zhilin Li
- Xiao-Biao Lin
- Alun L. Lloyd
- Sharon R. Lubkin
- Negash G. Medhin
- Kailash Chandra Misra
- Mette Olufsen
- Tao Pang
- Nathan P. Reading
- Jesus Rodriguez
- Michael Shearer
- Jack William Silverstein
- Ralph Conover Smith
- Ernest Lester Stitzinger
- Seth M. Sullivant
- Agnes Szanto
- Hien Trong Tran
- Semyon Victor Tsynkov
- Dmitry Valerievich Zenkov

### Associate Professors

- Lorena Viorica Bociu
- Min Jeong Kang
- Ricky Ini Liu
- Arvind Krishna Saibaba
- David Papp
- Cynthia Leslie Vinzant

### Assistant Professors

- Martin Helmer
- Alen Alexanderian
- Mohammad Mehdi Farazmand
- Kevin Bryant Flores
- P. Ivanisvili
- C. Jones
- Yerkin Kitapbayev
- Tye Lidman
- P. McGrath
- Ryan William Murray
- Tien Khai Nguyen
- A. Papanicolaou
- T. Saksala
- Radmila Sazdanovic

### Practice/Research/Teaching Professors

- Elisabeth M. M. Brown
- L. Castle
- Alina Nicoleta Duca
- Molly A. Fenn
- Bevin Laurel Maultsby
- S. Paul
- Brenda B. Williams

### Emeritus Faculty

- John William Bishir
- Stephen LaVern Campbell
- Richard E. Chandler
- H. Charlton
- Ethelbert N. Chukwu
- Lung-ock Chung
- Joseph C. Dunn
- Gary Doyle Faulkner
- John E. Franke
- Ronald O. Fulp
- Dennis E. Garoutte
- Robert E. Hartwig
- Aloysius G. Helminck
- Robert H. Martin Jr.
- Thomas J. Lada
- Joe A. Marlin
- Carl Meyer Jr.
- Larry Keith Norris
- Sandra Paur
- Lavon Barry Page
- E. Peterson
- Mohan Sastri Putcha
- N. Rose
- Stephen Schecter
- Jeffrey Scott Scroggs
- James Francis Selgrade
- C. Siewert
- Robert Silber
- Michael F. Singer
- R. White

### Adjunct Faculty

- Scott Christopher Batson
- Jonathan David Hauenstein
- Patricia L. Hersh
- John Lavery
- Jordan E. Massad
- Jessica Loock Matthews
- J. Ottesen

## Courses

**MA 501**

**Advanced Mathematics for Engineers and Scientists I**(3 credit hours)

Survey of mathematical methods for engineers and scientists. Ordinary differential equations and Green's functions; partial differential equations and separation of variables; special functions, Fourier series. Applications to engineering and science. May not be taken for graduate credit by Master's or Ph.D. students in Mathematics or Applied Mathematics. Credit for this course and MA 401 is not allowed.

*Typically offered in Fall, Spring, and Summer*

**MA 502**

**Advanced Mathematics for Engineers and Scientists II**(3 credit hours)

Determinants and matrices; line and surface integrals, integral theorems; complex integrals and residues; distribution functions of probability. Not for credit by mathematics majors. Any student receiving credit for MA 502 may receive credit for, atmost, one of the following: MA 405, MA 512, MA 513

Prerequisite: MA 341.

*Typically offered in Fall, Spring, and Summer*

Basic concepts of linear, nonlinear and dynamic programming theory. Not for majors in OR at Ph.D. level.

*Typically offered in Fall only*

Introduction including: applications to economics and engineering; the simplex and interior-point methods; parametric programming and post-optimality analysis; duality matrix games, linear systems solvability theory and linear systems duality theory; polyhedral sets and cones, including their convexity and separation properties and dual representations; equilibrium prices, Lagrange multipliers, subgradients and sensitivity analysis.

Prerequisite: MA 405

*Typically offered in Fall only*

**MA 507**

**Survey of Real Analysis**(3 credit hours)

A broad overview of topics in analysis. Historical development, logical refinement and applications of concepts such as limits, continuity, differentiation and integration. May not be taken for graduate credit by Master's or Ph.D. students in Mathematics or Applied Mathematics.

Prerequisite: Graduate standing

*Typically offered in Fall, Spring, and Summer*

**MA 508**

**Survey of Geometry**(3 credit hours)

A broad overview of topics in geometry. Various approaches to study of geometry, including vector geometry, transformational geometry and axiomatics. May not be taken for graduate credit by Master's or Ph.D. students in Mathematics or Applied Mathematics.

Prerequisite: Graduate standing

*Typically offered in Fall, Spring, and Summer*

**MA 509**

**Survey of Abstract Algebra**(3 credit hours)

A broad overview of topics in abstract algebra. Theory of equations, polynomial rings, rational functions and elementary number theory. May not be taken for graduate credit by Master's or Ph.D. students in Mathematics or Applied Mathematics.

Prerequisite: Graduate standing

*Typically offered in Fall, Spring, and Summer*

**MA 510**

**Selected Topics In Mathematics For Secondary Teachers**(1-6 credit hours)

Coverage of various topics in mathematics of concern to secondary teachers. Topics selected from areas such as mathematics of finance, probability, statistics, linear programming and theory of games, intuitive topology, recreational math, computers and applications of mathematics. Course may be taken for graduate credit for certification renewal by secondary school teachers. Credit towards a graduate degree may be allowed only for students in mathematics education.

Prerequisite: Graduate standing

*Typically offered in Spring and Summer*

**MA 511**

**Advanced Calculus I**(3 credit hours)

Fundamental theorems on continuous functions; convergence theory of sequences, series and integrals; the Riemann integral. Credit for both MA 425 and MA 511 is not allowed

Prerequisite: MA 341

*Typically offered in Fall and Spring*

**MA 513**

**Introduction To Complex Variables**(3 credit hours)

Operations with complex numbers, derivatives, analytic functions, integrals, definitions and properties of elementary functions, multivalued functions, power series, residue theory and applications, conformal mapping.

Prerequisite: MA 242

*Typically offered in Fall and Spring*

Cryptography is the study of mathematical techniques for securing digital information, systems and distributed computation against adversarial attacks. In this class you will learn the concepts and the algorithms behind the most used cryptographic protocols: you will learn how to formally define security properties and how to formally prove/disprove that a cryptographic protocol achieves a certain security property. You will also discover that cryptography has a much broader range of applications. It solves absolutely paradoxical problems such as proving knowledge of a secret without ever revealing the secret (zero-knowledge proof), or computing the output of a function without ever knowing the input of the function (secure computation). Finally, we will look closely at one of the recent popular application of cryptography: the blockchain technology. Additionally, graduate students will study some of the topics in greater depth.

*Typically offered in Fall only*

**MA 515**

**Analysis I**(3 credit hours)

Metric spaces: contraction mapping principle, Tietze extension theorem, Ascoli-Arzela lemma, Baire category theorem, Stone-Weierstrass theorem, LP spaces. Banach spaces: linear operators, Hahn-Banach theorem, open mapping and closed graph theorems. Hilbert spaces: projection theorem, Riesz representation theorem, Lax-Milgram theorem, complete orthonormal sets.

Prerequisite: MA 426

*Typically offered in Fall and Spring*

**MA 518**

**Geometry of Curves and Surfaces**(3 credit hours)

Geometry of curves and surfaces in space; Arclength, torsion, and curvature of curves; Tangent spaces, shape operators, and curvatures of surfaces; metrics, covariant derivatives, geodesics, and holonomy. Applications in the physical sciences and/or projects using computer algebra.

*Typically offered in Spring only*

**MA 520**

**Linear Algebra**(3 credit hours)

Vector spaces. Bases and dimension. Changes of basis. Linear transformations and their matrices. Linear functionals. Simultaneous triangularization and diagonalization. Rational and Jordan canonical forms. Bilinear forms.

Prerequisite: MA 405

*Typically offered in Fall and Spring*

**MA 521**

**Abstract Algebra I**(3 credit hours)

Groups, quotient groups, group actions, Sylow's Theorems. Rings, ideals and quotient rings, factorization, principal ideal domains. Fields, field extensions, Galois theory.

*Typically offered in Fall only*

**MA 522**

**Computer Algebra**(3 credit hours)

Basic techniques and algorithms of computer algebra. Integer arithmetic, primality tests and factorization of integers, polynomial arithmetic, polynomial factorization, Groebner bases, integration in finite terms.

*Typically offered in Fall only*

**MA 523**

**Linear Transformations and Matrix Theory**(3 credit hours)

Vector spaces, linear transformations and matrices, orthogonality, orthogonal transformations with emphasis on rotations and reflections, matrix norms, projectors, least squares, generalized inverses, definite matrices, singular values.

Prerequisite: MA 405

*Typically offered in Fall and Spring*

**MA 524**

**Combinatorics I**(3 credit hours)

Enumerative combinatorics, including placements of balls in bins, the twelvefold way, inclusion/exclusion, sign-reversing involutions and lattice path enumeration. Partically ordered sets, lattices, distributive lattices, Moebius functions, and rational generating functions.

*Typically offered in Fall only*

The course covers (i) structure and operation of derivative markets, (ii) valuation of derivatives, (iii) hedging of derivatives, and (iv) applications of derivatives in areas of risk management and financial engineering. Models and pricing techniques include Black-Scholes model, binomial trees, Monte-Carlo simulation. Specific topics include simple no-arbitrage pricing relations for futures/forward contracts; put-call parity relationship; delta, gamma, and vega hedging; implied volatility and statistical properties; dynamic hedging strategies; interest-rate risk, pricing of fixed-income product; credit risk, pricing of defaultable securities.

*Typically offered in Fall only*

Introduction to modeling, analysis and control of linear discrete-time and continuous-time dynamical systems. State space representations and transfer methods. Controllability and observability. Realization. Applications to biological, chemical, economic, electrical, mechanical and sociological systems.

*Typically offered in Fall only*

**MA 532**

**Ordinary Differential Equations I**(3 credit hours)

Existence and uniqueness theorems, systems of linear equations, fundamental matrices, matrix exponential, nonlinear systems, plane autonomous systems, stability theory.

*Typically offered in Fall only*

**MA 534**

**Introduction To Partial Differential Equations**(3 credit hours)

Linear first order equations, method of characteristics. Classification of second order equations. Solution techniques for the heat equation, wave equation and Laplace's equation. Maximum principles. Green's functions and fundamental solutions.

*Typically offered in Fall only*

**MA 537**

**Nonlinear Dynamics and Chaos**(3 credit hours)

Usage of computer experiments for demonstration of nonlinear dynamics and chaos and motivation of mathematical definitions and concepts. Examples from finance and ecology as well as traditional science and engineering. Difference equations and iteration of functions as nonlinear dynamical systems. Fixed points, periodic points and general orbits. Bifurcations and transition to chaos. Symbolic dynamics, chaos, Sarkovskii's Theorem, Schwarzian derivative, Newton's method and fractals.

*Typically offered in Spring only*

**MA 540**

**Uncertainty Quantification for Physical and Biological Models**(3 credit hours)

Introduction to uncertainty quantification for physical and biological models. Parameter selection techniques, Bayesian model calibration, propagation of uncertainties, surrogate model construction, local and global sensitivity analysis.

Prerequisite: MA 341 and basic knowledge of probability, linear algebra, and scientific computation

*Typically offered in Fall and Spring*

**MA 542**

**Convex Optimization Methods in Data Science**(3 credit hours)

Convex optimization methods and their applications in various areas of data science including, but not limited to, signal and image processing, inverse problems, statistical data analysis, machine learning and classification. Basic theory, algorithm design and concrete applications.

Prerequisite: MA 141, 241, 242, or equivalent and MA 405 or equivalent; Some notions of elementary convex analysis are an asset but are neither required nor assumed known.

*Typically offered in Fall only*

**MA 544**

**Computer Experiments In Mathematical Probability**(3 credit hours)

Exposure of student to practice of performing mathematical experiments on computer, with emphasis on probability. Programming in an interactive language such as APL, MATLAB or Mathematica. Mathematical treatment of random number generation and application of these tools to mathematical topics in Monte Carlo method, limit theorems and stochastic processes for purpose of gaining mathematical insight.

Prerequisite: MA 421

*Typically offered in Spring only*

Modern introduction to Probability Theory and Stochastic Processes. The choice of material is motivated by applications to problems such as queueing networks, filtering and financial mathematics. Topics include: review of discrete probability and continuous random variables, random walks, markov chains, martingales, stopping times, erodicity, conditional expectations, continuous-time Markov chains, laws of large numbers, central limit theorem and large deviations.

*Typically offered in Fall only*

This course explores stochastics calculus with its applications in pricing and hedging problems for financial derivatives such as options. Topics to be covered in the course include 1) discrete and continuous martingales, 2) Brownian motions and Ito's stochastic calculus, and 3) Black-Scholas framework for financial derivatives pricing and hedging.

Prerequisite: FIM 528 and MA(ST) 546

*Typically offered in Spring only*

Monte Carlo (MC) methods for accurate option pricing, hedging and risk management. Modeling using stochastic asset models (e.g. geometric Brownian motion) and parameter estimation. Stochastic models, including use of random number generators, random paths and discretization methods (e.g. Euler-Maruyama method), and variance reduction. Implementation using Matlab. Incorporation of the latest developments regarding MC methods and their uses in Finance.

*Typically offered in Spring only*

This course focuses on mathematical methods to analyze and manage risks associated with financial derivatives. Topics covered include aggregate loss distributions, extreme value theory, default probabilities, Value-at-Risk and expected shortfall, coherent risk measures, correlation and copula, applications of principle component analysis and Monte Carlo simulations in financial risk management, how to use stochastic differential equations to price financial risk derivatives, and how to back-test and stress-test models.

*Typically offered in Spring only*

**MA 551**

**Introduction to Topology**(3 credit hours)

Set theory, topological spaces, metric spaces, continuous functions, separation, cardinality properties, product and quotient topologies, compactness, connectedness.

Prerequisite: MA 426

*Typically offered in Fall only*

**MA 555**

**Introduction to Manifold Theory**(3 credit hours)

An introduction to smooth manifolds. Topics include: topological and smooth manifolds, smooth maps and differentials, vector fields and flows, Lie derivatives, vector bundles, tensors, differential forms, exterior calculus, and integration on manifolds.

*Typically offered in Fall only*

**MA 561**

**Set Theory and Foundations Of Mathematics**(3 credit hours)

Logic and axiomatic approach, the Zermelo-Fraenkel axioms and other systems, algebra of sets and order relations, equivalents of the Axiom of Choice, one-to-one correspondences, cardinal and ordinal numbers, the Continuum Hypothesis.

Prerequisite: MA 407

*Typically offered in Spring only*

Basic concepts of graph theory. Trees and forests. Vector spaces associated with a graph. Representation of graphs by binary matrices and list structures. Traversability. Connectivity. Matchings and assignment problems. Planar graphs. Colorability. Directed graphs. Applications of graph theory with emphasis on organizing problems in a form suitable for computer solution.

*Typically offered in Spring only*

Introduction to model development for physical and biological applications. Mathematical and statistical aspects of parameter estimation. Compartmental analysis and conservation laws, heat transfer, and population and disease models. Analytic and numerical solution techniques and experimental validation of models. Knowledge of high-level programming languages required.

Prerequisite: MA 341 and knowledge of high-level programming language.

*Typically offered in Fall only*

Model development, using Newtonian and Hamiltonian principles, for acoustic and fluid applications, and structural systems including membranes, rods, beams, and shells. Fundamental aspects of electromagnetic theory. Analytic and numerical solution techniques and experimental validation of models.

Prerequisite: MA/BMA 573

*Typically offered in Spring only*

Algorithm behavior and applicability. Effect of roundoff errors, systems of linear equations and direct methods, least squares via Givens and Householder transformations, stationary and Krylov iterative methods, the conjugate gradient and GMRES methods, convergence of method.

*Typically offered in Fall and Spring*

Introduction to basic parallel architectures, algorithms and programming paradigms; message passing collectives and communicators; parallel matrix products, domain decomposition with direct and iterative methods for linear systems; analysis of efficiency, complexity and errors; applications such as 2D heat and mass transfer.

*Typically offered in Spring only*

**MA 584**

**Numerical Solution of Partial Differential Equations--Finite Difference Methods**(3 credit hours)

Survey of finite difference methods for partial differential equations including elliptic, parabolic and hyperbolic PDE's. Consideration of both linear and nonlinear problems. Theoretical foundations described; however, emphasis on algorithm design and implementation.

Prerequisite: MA 501; knowledge of a high level programming language

*Typically offered in Fall only*

**MA 587**

**Numerical Solution of Partial Differential Equations--Finite Element Method**(3 credit hours)

Introduction to finite element method. Applications to both linear and nonlinear elliptic and parabolic partial differential equations. Theoretical foundations described; however, emphasis on algorithm design and implementation.

Prerequisite: MA 501; knowledge of a high level programming language

*Typically offered in Spring only*

**MA 591**

**Special Topics**(1-6 credit hours)

*Typically offered in Fall and Spring*

**MA 601**

**Seminar**(1 credit hours)

Review and discussion of scientific articles, progress reports on research and special problems of interest to mathematicians.

P: Graduate Standing

*Typically offered in Fall and Spring*

**MA 676**

**Master's Project**(3 credit hours)

Investigation of some topic in mathematics to a deeper and broader extent than typically done in a classroom situation. For the applied mathematics student the topic usually consists of a realistic application of mathematics to student's minor area.A written and oral report on the project required.

*Typically offered in Fall, Spring, and Summer*

**MA 685**

**Master's Supervised Teaching**(1-3 credit hours)

Teaching experience under the mentorship of faculty who assist the student in planning for the teaching assignment, observe and provide feedback to the student during the teaching assignment, and evaluate the student upon completion of the assignment.

Prerequisite: Master's student

*Typically offered in Fall and Spring*

**MA 689**

**Non-Thesis Master Continuous Registration - Full Time Registration**(3 credit hours)

For students in non-thesis master's programs who have completed all credit hour requirements for their degree but need to maintain full-time continuous registration to complete incomplete grades, projects, final master's exam, etc. Students may register for this course a maximum of one semester.

Prerequisite: Master's student

*Typically offered in Fall, Spring, and Summer*

**MA 690**

**Master's Examination**(1-9 credit hours)

For students in non thesis master's programs who have completed all other requirements of the degree except preparing for and taking the final master's exam.

Prerequisite: Master's student

*Typically offered in Fall and Spring*

**MA 693**

**Master's Supervised Research**(1-9 credit hours)

Instruction in research and research under the mentorship of a member of the Graduate Faculty.

Prerequisite: Master's student

*Typically offered in Fall, Spring, and Summer*

**MA 695**

**Master's Thesis Research**(1-9 credit hours)

Thesis Research

Prerequisite: Master's student

*Typically offered in Fall and Spring*

**MA 696**

**Summer Thesis Research**(1 credit hours)

For graduate students whose programs of work specify no formal course work during a summer session and who will be devoting full time to thesis research.

Prerequisite: Master's student

*Typically offered in Summer only*

**MA 699**

**Master's Thesis Preparation**(1-9 credit hours)

For students who have completed all credit hour requirements and full-time enrollment for the master's degree and are writing and defending their thesis. Credits Arranged

Prerequisite: Master's student

*Typically offered in Fall, Spring, and Summer*

An advanced mathematical treatment of analytical and algorithmic aspects of finite dimensional nonlinear programming. Including an examination of structure and effectiveness of computational methods for unconstrained and constrained minimization. Special attention directed toward current research and recent developments in the field.

Prerequisite: OR(IE,MA) 505 and MA 425

*Typically offered in Spring only*

General integer programming problems and principal methods of solving them. Emphasis on intuitive presentation of ideas underlying various algorithms rather than detailed description of computer codes. Students have some "hands on" computing experience that should enable them to adapt ideas presented in course to integer programming problems they may encounter.

*Typically offered in Spring only*

**MA 715**

**Analysis II**(3 credit hours)

Integration: Legesgue measure and integration, Lebesgue-dominated convergence and monotone convergence theorems, Fubini's theorem, extension of the fundamental theorem of calculus. Banach spaces: Lp spaces, weak convergence, adjoint operators, compact linear operators, Fredholm-Fiesz Schauder theory and spectral theorem.

Prerequisite: MA 515

*Typically offered in Spring only*

**MA 716**

**Advanced Functional Analysis**(3 credit hours)

Advanced topics in functional analysis such as linear topological spaces; Banach algebra, spectral theory and abstract measure theory and integration.

Prerequisite: MA 715

*Typically offered in Fall only*

Introduction to algebraic and function-analytic concepts used in system modeling and optimization: vector space, linear mappings, spectral decomposition, adjoints, orthogonal projection, quality, fixed points and differentials. Emphasis on geometricinsight. Topics include least square optimization of linear systems, minimum norm problems in Banach space, linearization in Hilbert space, iterative solution of system equations and optimization problems. Broad range of applications in operations research and system engineering including control theory, mathematical programming, econometrics, statistical estimation, circuit theory and numerical analysis.

*Typically offered in Fall only*

**MA 720**

**Lie Algebras**(3 credit hours)

Definition of Lie algebras and examples. Nilpotent, solvable and semisimple Lie algebras. Engel's theorem, Lie's Theorem, Killing form and Cartan's criterion. Weyl's theorem on complete reducibility. Representations of s1(2,C). Root space decomposition of semisimple Lie algebras. Root system and Weyl group.

Prerequisite: MA 520

*Typically offered in Spring only*

**MA 721**

**Abstract Algebra II**(3 credit hours)

This course covers: Module theory including the structure theory of modules over a PID and primary decomposition; Tensor, exterior, and symmetric algebras; introductory homological algebra including: complexes, derived functors, Ext and Tor; and the representation theory of groups. Further topics will be covered as time permits.

Prerequisite: MA 521

*Typically offered in Spring only*

**MA 722**

**Computer Algebra II**(3 credit hours)

Effective algorithms for symbolic matrices, commutative algebra, real and complex algebraic geometry, and differential and difference equations. The emphasis is on the algorithmic aspects.

Prerequisite: MA 522

*Typically offered in Spring only*

**MA 723**

**Theory of Matrices and Applications**(3 credit hours)

Canonical forms, functions of matrices, variational methods, perturbation theory, numerical methods, nonnegative matrices, applications to differential equations, Markov chains.

*Typically offered in Spring only*

**MA 724**

**Combinatorics II**(3 credit hours)

Polytopes(V-polytopes and H-polytopes). Fourier-Motzkin elimination, Farkas Lemma, face numbers of polytopes, graphs of polytopes, linear programming for geometers, Balinski's Theorem, Steinitz' Theorem, Schlegel diagrams, polyhedral complexes, shellability, and face rings.

Prerequisite: MA 524

*Typically offered in Spring only*

**MA 725**

**Lie Algebra Representation Theory**(3 credit hours)

Semisimple Lie algebras, root systems, Weyl groups, Cartan matrices and Dynkin diagrams, universal enveloping algebras, Serre's Theorem, Kac-Moody algebras, highest weight representations of finite dimensional semisimple algebras and affine Lie algebras, Kac-Weyl character formula.

Prerequisite: MA 720

*Typically offered in Fall only*

**MA 726**

**Algebraic Geometry**(3 credit hours)

Abstract theory of solutions of systems of polynomial equations. Topics covered: ideals and affine varieties, the Nullstellensatz, irreducible varieties and primary decomposition, morphisms and rational maps, computational aspects including Groebner bases and elimination theory, projective varieties and homogeneous ideals, Grassmannians, graded modules, the Hilbert function, Bezout's theorem.

Prerequisite: MA 521

*Typically offered in Spring only*

Stability of equilibrium points for nonlinear systems. Liapunov functions. Unconstrained and constrained optimal control problems. Pontryagin's maximum principle and dynamic programming. Computation with gradient methods and Newton methods. Multidisciplinary applications.

Prerequisite: OR(E,MA) 531

*Typically offered in Spring only*

**MA 732**

**Ordinary Differential Equations II**(3 credit hours)

Existence-uniqueness theory, periodic solutions, invariant manifolds, bifurcations, Fredholm's alternative.

*Typically offered in Spring only*

**MA 734**

**Partial Differential Equations**(3 credit hours)

Linear second order parabolic, elliptic and hyperbolic equations. Initial value problems and boundary value problems. Iterative and variational methods. Existence, uniqueness and regularity. Nonlinear equations and systems.

*Typically offered in Spring only*

Markov chains and Markov processes, Poisson process, birth and death processes, queuing theory, renewal theory, stationary processes, Brownian motion.

Prerequisite: MA 405 and MA(ST) 546 or ST 521

*Typically offered in Spring only*

Fundamental mathematical results of probabilistic measure theory needed for advanced applications in stochastic processes. Probability measures, sigma-algebras, random variables, Lebesgue integration, expectation and conditional expectations w.r.t.sigma algebras, characteristic functions, notions of convergence of sequences of random variables, weak convergence of measures, Gaussian systems, Poisson processes, mixing properties, discrete-time martingales, continuous-time markov chains.

Prerequisite: MA(ST) 546

*Typically offered in Spring only*

Theory of stochastic differential equations driven by Brownian motions. Current techniques in filtering and financial mathematics. Construction and properties of Brownian motion, wiener measure, Ito's integrals, martingale representation theorem, stochastic differential equations and diffusion processes, Girsanov's theorem, relation to partial differential equations, the Feynman-Kac formula.

Prerequisite: MA(ST) 747

*Typically offered in Fall only*

**MA 753**

**Algebraic Topology**(3 credit hours)

Homotopy, fundamental group, covering spaces, classification of surfaces, homology and cohomology.

*Typically offered in Spring only*

**MA 755**

**Introduction to Riemannian Geometry**(3 credit hours)

An introduction to smooth manifolds with metric. Topics include: Riemannian metric and generalizations, connections, covariant derivatives, parallel translation, Riemannian (or Levi-Civita) connection, geodesics and distance, curvature tensor, Bianchi identities, Ricci and scalar curvatures, isometric embeddings, Riemannian submanifolds, hypersurfaces, Gauss Bonnet Theorem; applications and connections to other fields.

Prerequisite: MA 555

*Typically offered in Spring only*

Study of problems of flows in networks. These problems include the determination of shortest chain, maximal flow and minimal cost flow in networks. Relationship between network flows and linear programming developed as well as problems with nonlinear cost functions, multi-commodity flows and problem of network synthesis.

Prerequisite: OR(IE,MA) 505

*Typically offered in Spring only*

Role of theory construction and model building in development of experimental science. Historical development of mathematical theories and models for growth of one-species populations (logistic and off-shoots), including considerations of age distributions (matrix models, Leslie and Lopez; continuous theory, renewal equation). Some of the more elementary theories on the growth of organisms (von Bertalanffy and others; allometric theories; cultures grown in a chemostat). Mathematical theories oftwo and more species systems (predator-prey, competition, symbosis; leading up to present-day research) and discussion of some similar models for chemical kinetics. Much emphasis on scrutiny of biological concepts as well as of mathematical structureof models in order to uncover both weak and strong points of models discussed. Mathematical treatment of differential equations in models stressing qualitative and graphical aspects, as well as certain aspects of discretization. Difference equation models.

Prerequisite: Advanced calculus, reasonable background in biology

*Typically offered in Fall only*

Continuation of topics of BMA 771. Some more advanced mathematical techniques concerning nonlinear differential equations of types encountered in BMA 771: several concepts of stability, asymptotic directions, Liapunov functions; different time-scales. Comparison of deterministic and stochastic models for several biological problems including birth and death processes. Discussion of various other applications of mathematics to biology, some recent research.

Prerequisite: BMA 771, elementary probability theory

*Typically offered in Spring only*

Survey of modeling approaches and analysis methods for data from continuous state random processes. Emphasis on differential and difference equations with noisy input. Doob-Meyer decomposition of process into its signal and noise components. Examples from biological and physical sciences, and engineering. Student project.

Prerequisite: BMA 772 or ST (MA) 746

*Typically offered in Spring only*

Modeling with and analysis of partial differential equations as applied to real problems in biology. Review of diffusion and conservation laws. Waves and pattern formation. Chemotaxis and other forms of cell and organism movement. Introduction to solid and fluid mechanics/dynamics. Introductory numerical methods. Scaling. Perturbations, Asymptotics, Cartesian, polar and spherical geometries. Case studies.

*Typically offered in Spring only*

Approximation and interpolation, Fast Fourier Transform, numerical differentiation and integration, numerical solution of initial value problems for ordinary differential equations.

Prerequisite: MA 580

*Typically offered in Spring only*

**MA 784**

**Nonlinear Equations and Unconstrained Optimization**(3 credit hours)

Newton's method and Quasi-Newton methods for nonlinear equations and optimization problems, globally convergent extensions, methods for sparse problems, applications to differential equations, integral equations and general minimization problems. Methods appropriate for boundary value problems.

Prerequisite: MA 580

*Typically offered in Spring only*

**MA 788**

**Numerical Nonlinear Partial Differential Equations**(3 credit hours)

Nonlinear discrete equations; Newton and monotone methods for nonlinear equations; computational algorithms and applications; finite difference method-convergence, stability and error estimates; multiplicity of solutions and bifurcation; asymptotic behavior of solutions; and coupled systems of equations.

*Typically offered in Spring only*

**MA 791**

**Special Topics In Real Analysis**(1-6 credit hours)

*Typically offered in Fall and Spring*

**MA 792**

**Special Topics In Algebra**(1-6 credit hours)

*Typically offered in Fall and Spring*

**MA 793**

**Special Topics In Differential Equations**(1-6 credit hours)

**MA 796**

**Special Topics In Combinatorial Analysis**(1-6 credit hours)

**MA 797**

**Special Topics In Applied Mathematics**(1-6 credit hours)

**MA 798**

**Special Topics In Numerical Analysis**(1-6 credit hours)

**MA 810**

**Special Topics**(1-6 credit hours)

*Typically offered in Fall and Spring*

Study of special advanced topics in area of mathematical programming. Discussion of new techniques and current research in this area. The faculty responsible for this course select areas to be covered during semester according to their preference and interest. This course not necessarily taught by an individual faculty member but can, on occasion, be joint effort of several faculty members from this university as well as visiting faculty from other institutions. To date, a course of Theory of Networks and another on Integer Programming offered under the umbrella of this course. Anticipation that these two topics will be repeated in future together with other topics.

Prerequisite: IE(MA,OR) 505

*Typically offered in Spring only*

Advanced topics in some phase of system optimization. Identification of various specific topics and prerequisite for each section from term to term.

*Typically offered in Fall and Spring*

**MA 885**

**Doctoral Supervised Teaching**(1-3 credit hours)

Teaching experience under the mentorship of faculty who assist the student in planning for the teaching assignment, observe and provide feedback to the student during the teaching assignment, and evaluate the student upon completion of the assignment.

Prerequisite: Doctoral student

*Typically offered in Fall and Spring*

**MA 890**

**Doctoral Preliminary Examination**(1-9 credit hours)

For students who are preparing for and taking written and/or oral preliminary exams.

Prerequisite: Doctoral student

*Typically offered in Fall and Spring*

**MA 893**

**Doctoral Supervised Research**(1-9 credit hours)

Instruction in research and research under the mentorship of a member of the Graduate Faculty.

Prerequisite: Doctoral student

*Typically offered in Fall, Spring, and Summer*

**MA 895**

**Doctoral Dissertation Research**(1-9 credit hours)

Dissertation Research

Prerequisite: Doctoral student

*Typically offered in Fall and Spring*

**MA 896**

**Summer Dissertation Research**(1 credit hours)

For graduate students whose programs of work specify no formal course work during a summer session and who will be devoting full time to thesis research.

Prerequisite: Doctoral student

*Typically offered in Summer only*

**MA 899**

**Doctoral Dissertation Preparation**(1-9 credit hours)

For students who have completed all credit hour requirements, full-time enrollment, preliminary examination, and residency requirements for the doctoral degree, and are writing and defending their dissertations.

Prerequisite: Doctoral student

*Typically offered in Fall, Spring, and Summer*