# Mathematics (MA)

**MA 101**

**Intermediate Algebra**(4 credit hours)

Preparation for MA 103, MA 105, MA 107, MA 111, and MA 114. Reviews main topics from high school Algebra I and Algebra II emphasizing functions and problem solving. Other concepts and skills covered include algebraic operations, factoring, linear equations, graphs, exponents, radicals, complex numbers, quadratic equations, radical equations, inequalities, systems of equations, compound inequalities, absolute value in equations and inequalities. MA 101 may not be counted as credit toward meeting graduation. Credit for MA 101 is not allowed if student has prior credit in any other mathematical course.

*Typically offered in Summer only*

Primarily for students in Humanities and Social Sciences. Illustrations of contemporary uses of mathematics, varying from semester to semester, frequently including sets and logic, counting procedures, probability, modular arithmetic, and game theory.

Prerequisite: MA 101 or equivalent completed in high school

*GEP Mathematical Sciences*

*Typically offered in Fall, Spring, and Summer*

Primarily for students in Humanities and Social Sciences. Illustrations of contemporary uses of mathematics, varying from semester to semester, frequently including sets and logic, counting procedures, probability, modular arithmetic, and game theory.

Prerequisite: MA 101 or equivalent completed in high school

*GEP Mathematical Sciences*

*Typically offered in Fall and Spring*

**MA 105**

**Mathematics of Finance**(3 credit hours)

Simple and compound interest, annuities and their application to amortization and sinking fund problems, installment buying, calculation of premiums of life annuities and life insurance.

Prerequisite: MA 101 or equivalent completed in high school

*GEP Mathematical Sciences*

*Typically offered in Fall, Spring, and Summer*

**MA 107**

**Precalculus I**(3 credit hours)

Algebra and basic trigonometry; polynomial, rational, exponential, logarithmic and trigonometric functions and their graphs. Credit for MA 107 does not count toward graduation for students in Engineering, College of Sciences, Bio and Ag Engineering (Science Program), Bio Sci (all options), Math Edu, Sci Edu, Textiles, and B.S. degrees in CHASS. Credit is not allowed for both MA 107 and MA 111

Prerequisite: C- or better in MA 101, or a 450 or better on the SAT Subject Test in Mathematics Level 2 or the NCSU Math Skills Test.

*GEP Mathematical Sciences*

*Typically offered in Fall, Spring, and Summer*

**MA 108**

**Precalculus II**(3 credit hours)

Algebra, analytic geometry and trigonometry; inequalities, conic sections, complex numbers, sequences and series, solving triangles, polar coordinates, and applications.Credit for MA 108 does not count toward graduation for students in Engineering, College of Sciences, Design, Bio and Ag Engineering (Science Program), Bio Sci (all options), Math Edu, Sci Edu, Textiles, and B.S. degrees in CHASS. Credit is not allowed for both MA 108 and MA 111. Also, MA 108 should not be counted toward the GER mathematical sciences.

Prerequisite: C- or better in MA 107

*Typically offered in Spring only*

**MA 111**

**Precalculus Algebra and Trigonometry**(3 credit hours)

Real numbers, functions and their graphs (special attention to polynomial, rational, exponential, logarithmic, and trigonometric functions), analytic trigonometry. Credit in MA 111 does not count toward graduation for students in Engr., College of Sciences., Design, Biological & Ag. Engr. (Science Program), Biological Sci.(all options),Math. Edu., Forestry, & Textiles. Credit is not allowed for both MA 111 and either MA 107 or MA 108.

Prerequisite: C- or better in MA 101, or 480 or better on the SAT Subject Test in Mathematics Level 2 or the NCSU Math Skills Test.

*GEP Mathematical Sciences*

*Typically offered in Fall, Spring, and Summer*

**MA 114**

**Introduction to Finite Mathematics with Applications**(3 credit hours)

Elementary matrix algebra including arithmetic operations, inverses, and systems of equations; introduction to linear programming including simplex method; sets and counting techniques, elementary probability including conditional probability; Markov chains; applications in the behavioral, managerial and biological sciences. Computer use for completion of assignments.

Prerequisite: MA 101 or equivalent completed in high school.

*GEP Mathematical Sciences*

*Typically offered in Fall, Spring, and Summer*

**MA 116**

**Introduction to Scientific Programming (Math)**(3 credit hours)

Computer-based mathematical problem solving and simulation techniques using MATLAB. Emphasizes scientific programming constructs that utilize good practices in code development, including documentation and style. Covers user-defined functions, data abstractions, data visualization and appropriate use of pre-defined functions. Applications are from science and engineering.

*Typically offered in Fall and Spring*

**MA 121**

**Elements of Calculus**(3 credit hours)

For students who require only a single semester of calculus. Emphasis on concepts and applications of calculus, along with basic skills. Algebra review, functions, graphs, limits, derivatives, integrals, logarithmic and exponential functions, functions of several variables, applications in management, applications in biological and social sciences. Credit is not allowed in more than one of MA 121, 131, 141. MA 121 may not be substituted for MA 131 or MA 141 as a curricular requirement

Prerequisite: MA 107 or 111 with a C- or better, or 480 on the SAT Subject Test in Mathematics Level 2 or the NCSU Math Skills Test, or 2 or better on an AP Calculus exam. Credit is not allowed for both MA 121 and MA 131 or MA 141.

*GEP Mathematical Sciences*

*Typically offered in Fall, Spring, and Summer*

**MA 131**

**Calculus for Life and Management Sciences A**(3 credit hours)

First order finite difference models; derivatives - limits, power rule, graphing, and optimization; exponential and logarithmic functions - growth and decay models; integrals - computation, area, total change; applications in life, management, and social sciences. Credit not allowed for more than one of MA 121, 131, and 141

Prerequisite: C- or better in MA 107 or MA 111, or 520 or better on the SAT Subject Test in Mathematcs Level 2 or the NCSU Math Skills Test, or 2 or better on an AP Calculus exam. Credit is not allowed for both MA 131 and MA 121 or MA 141.

*GEP Mathematical Sciences*

*Typically offered in Fall, Spring, and Summer*

**MA 132**

**Computational Mathematics for Life and Management Sciences**(1 credit hours)

Computational aspects of calculus for the life and management sciences; use of spreadsheets and a computer algebra system; applications to data models, differential equation models, and optimization.

*Typically offered in Fall and Spring*

**MA 141**

**Calculus I**(4 credit hours)

First of three semesters in a calculus sequence for science and engineering majors. Functions, graphs, limits, derivatives, rules of differentiation, definite integrals, fundamental theorem of calculus, applications of derivatives and integrals.

Prerequisite: MA 111 or MA 108 with a grade C- or better or qualifying score on math placement exam. Credit is not allowed for both MA 141 and MA 121 or MA 131.

*GEP Mathematical Sciences*

*Typically offered in Fall, Spring, and Summer*

**MA 151**

**Calculus for Elementary Education I**(3 credit hours)

Calculus for Elementary Education I is the first semester of a two semester sequence of courses designed for the Elementary Education Program. Topics will include sequences, limit, and derivative. Also, topics related to teaching elementary mathematics will be discussed. Students cannot receive credit for more than one of MA 151, MA 121, MA 131, or MA 141. MA 151 is not an accepted prerequisite for MA 231 and MA 241. This course is restricted to Elementary Education majors only.

Prerequisite: C- or better in MA 107 or MA 111, or 520 or better on the SAT Subject Test in Mathematcs Level 2 or the NCSU Math Skills Test, or 2 or better on an AP Calculus exam. Credit is not allowed for both MA 131 and MA 121 or MA 141.

*GEP Mathematical Sciences*

*Typically offered in Spring only*

**MA 152**

**Calculus for Elementary Education II**(3 credit hours)

Calculus for Elementary Education II is the second semester of a two semester sequence of courses designed for the Elementary Education Program. Topics will include derivative, integrals, difference equations, and differential equations. Also, topics related to teaching elementary mathematics will be discussed. This course is restricted to Elementary Education majors only. Students cannot receive credit for both MA 152 and MA 121, MA 131, or MA 141. MA 152 is not an accepted prerequisite for MA 241.

Prerequisite: MA 151

*GEP Mathematical Sciences*

*Typically offered in Fall only*

**MA 225**

**Foundations of Advanced Mathematics**(3 credit hours)

Introduction to mathematical proof with focus on properties of the real number system. Elementary symbolic logic, mathematical induction, algebra of sets, relations, functions, countability. Algebraic and completeness properties of the reals.

Prerequisite: MA 241

*Typically offered in Fall, Spring, and Summer*

**MA 231**

**Calculus for Life and Management Sciences B**(3 credit hours)

Functions of several variables - partial derivatives, optimization, least squares, Lagrange multiplier method; differential equations - population growth, finance and investment models, systems, numerical methods; MA 121 is not an accepted prerequisite for MA 231.

*GEP Mathematical Sciences*

*Typically offered in Fall, Spring, and Summer*

**MA 241**

**Calculus II**(4 credit hours)

Second of three semesters in a calculus sequence for science and engineering majors. Techniques and applications of integration, elementary differential equations, sequences, series, power series, and Taylor's Theorem. Use of computational tools.

Prerequisite: MA 141 with grade of C- or better or AP Calculus credit. Credit is not allowed for both MA 241 and MA 231.

*GEP Mathematical Sciences*

*Typically offered in Fall, Spring, and Summer*

**MA 242**

**Calculus III**(4 credit hours)

Third of three semesters in a calculus sequence for science and engineering majors. Vectors, vector algebra, and vector functions. Functions of several variables, partial derivatives, gradients, directional derivatives, maxima and mimima. Multiple integration. Line and surface integrals, Green's Theorem, Divergence Theorems, Stokes' Theorem, and applications. Use of computational tools.

Prerequisite: MA 241 with grade of C- or better or AP Calculus credit, or Higher Level IB credit.

*Typically offered in Fall, Spring, and Summer*

**MA 302**

**Numerical Applications to Differential Equations**(1 credit hours)

Numerical methods for approximating solutions for differential equations, with an emphasis on Runge-Kutta-Fehlberg methods with stepsize control. Applications to population, economic, orbital and mechanical models.

Prerequisite: MA 241

*Typically offered in Fall and Spring*

**MA 303**

**Linear Analysis**(3 credit hours)

Linear difference equations of first and second order, compound interest and amortization. Matrices and systems of linear equations, eigenvalues, diagonalization, systems of difference and differential equations, transform methods, population problems. Credit not allowed if credit has been obtained for MA 341 or MA 405

Prerequisite: MA 241

*Typically offered in Fall and Spring*

**MA 305**

**Introductory Linear Algebra and Matrices**(3 credit hours)

The course is an elementary introduction to matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, Euclidean vector spaces, determinants, eigenvalues and eigenvectors, linear transformations, similarity, and applications such as numerical solutions of equations and computer graphics. Compares with MA 405 Introductory Linear Algebra, more emphasis is placed on methods and calculations,. Credit is not allowed for both MA 305 and MA 405.

*Typically offered in Fall, Spring, and Summer*

For sophomore meteorology and marine science students. A complement to MA 242 designed to prepare students for quantitative atmospheric applications. Topics include an introduction to vectors and vector calculus, atmospheric waves, phase and group velocity, perturbation analysis, fourier decomposition, matrix operations, chaos and predictability. For MY, MMY, and MRM majors only.

*Typically offered in Spring only*

**MA 325**

**Introduction to Applied Mathematics**(3 credit hours)

Introduces students with multivariable calculus to five different areas of applied mathematics. These areas will be five three-week modules, which lead to higher level courses in the application areas. Topics will vary, and examples of modules areheat and mass transfer, biology and population, probability and finance, acoustic models, cryptography as well as others.

*Typically offered in Spring only*

**MA 326**

**Mathematical Foundations of Data Science**(3 credit hours)

The course covers foundational mathematical concepts fundamental to data science and data-driven mathematical modeling. The course includes the following topics: basics of machine learning, unconstrained optimization, neural networks and overfitting, parameter estimation and sensitivity analysis for mathematical models, and an introduction to topological data analysis. Programming familiarity (MATLAB) is recommended.

*Typically offered in Fall and Spring*

**MA 331**

**Differential Equations for the Life Sciences**(3 credit hours)

This course provides students with an understanding of how mathematics and life sciences can stimulate and enrich each other. The course topics include first order differential equations, separable equations, second order systems, vector and matrix systems, eigenvectors/eigenvalues, graphical and qualitative methods. The methods are motivated with examples from the biological sciences (growth models, kinetics and compartmental models, epidemic models, predator-prey, etc). Computational modeling will be carried out using SimBiology, a MATLAB toolbox based graphical user interface, which which automates and simplifies the process of modeling biological systems. Credit cannot be given for both MA 341 and MA 331.

*Typically offered in Fall only*

Intermediate level introduction to modern symbolic logic focusing on standard first-order logic; topics include proofs, interpretations, applications and basic metalogical results.

*GEP Mathematical Sciences*

*Typically offered in Fall only*

**MA 341**

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

Differential equations and systems of differential equations. Methods for solving ordinary differential equations including Laplace transforms, phase plane analysis, and numerical methods. Matrix techniques for systems of linear ordinary differential equations. Credit is not allowed for both MA 301 and MA 341

*Typically offered in Fall, Spring, and Summer*

**MA 351**

**Introduction to Discrete Mathematical Models**(3 credit hours)

Basic concepts of discrete mathematics, including graph theory, Markov chains, game theory, with emphasis on applications; problems and models from areas such as traffic flow, genetics, population growth, economics, and ecosystem analysis.

Prerequisite: MA 224, 225, 231 or 241

*Typically offered in Fall only*

**MA 401**

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

Wave, heat and Laplace equations. Solutions by separation of variables and expansion in Fourier Series or other appropriate orthogonal sets. Sturm-Liouville problems. Introduction to methods for solving some classical partial differential equations.Use of power series as a tool in solving ordinary differential equations. Credit for both MA 401 and MA 501 will not be given

*Typically offered in Fall, Spring, and Summer*

**MA 402**

**Mathematics of Scientific Computing**(3 credit hours)

This course will provide an overview of methods to solve quantitative problems and analyze data. The tools to be introduced are mathematical in nature and have links to Algebra, Analysis, Geometry, Graph Theory, Probability and Topology. Students will acquire an appreciation of (I) the fundamental role played by mathematics in countless applications and (II) the exciting challenges in mathematical research that lie ahead in the analysis of large data and uncertainties. Students will work on a project for each unit. While this is not a programming class, the students will do some programming through their projects.

*Typically offered in Fall and Spring*

**MA 403**

**Introduction to Modern Algebra**(3 credit hours)

Sets and mappings, equivalence relations, rings, integral domains, ordered integral domains, ring of integers. Other topics selected from fields, polynomial rings, real and complex numbers, groups, permutation groups, ideals, and quotient rings. Credit is not allowed for both MA 403 and MA 407

Prerequisite: MA 225

*Typically offered in Fall only*

**MA 405**

**Introduction to Linear Algebra**(3 credit hours)

This course offers a rigorous treatment of linear algebra, including systems of linear equations, matrices, determinants, abstract vector spaces, bases, linear independence, spanning sets, linear transformations, eigenvalues and eigenvectors, similarity, inner product spaces, orthogonality and orthogonal bases, factorization of matrices. Compared with MA 305 Introductory Linear Algebra, more emphasis is placed on theory and proofs. MA 225 is recommended as a prerequisite. Credit is not allowed for both MA 305 and MA 405

*Typically offered in Fall, Spring, and Summer*

**MA 407**

**Introduction to Modern Algebra for Mathematics Majors**(3 credit hours)

Elementary number theory, equivalence relations, groups, homomorphisms, cosets, Cayley's Theorem, symmetric groups, rings, polynomial rings, quotient fields, principal ideal domains, Euclidean domains. Credit is not allowed for both MA 403 and MA 407

*Typically offered in Fall and Spring*

**MA 408**

**Foundations of Euclidean Geometry**(3 credit hours)

An examination of Euclidean geometry from a modern perspective. The axiomatic approach with alternative possibilities explored using models.

Prerequisite: MA 225

*Typically offered in Fall and Spring*

**MA 410**

**Theory of Numbers**(3 credit hours)

Arithmetic properties of integers. Congruences, arithmetic functions, diophantine equations. Other topics chosen from quadratic residues, the quadratic reciprocity Law of Gauss, primitive roots, and algebraic number fields.

Prerequisite: One year of calculus

*Typically offered in Spring only*

Long-term probability models for risk management systems. Theory and applications of compound interest, probability distributions of failure time random variables, present value models of future contingent cash flows, applications to insurance, health care, credit risk, environmental risk, consumer behavior and warranties.

Prerequisite: MA 241 or MA 231, Corequisite: MA 421, BUS(ST) 350, ST 301, ST 305, ST 311, ST 361, ST 370, ST 371, ST 380 or equivalent

*Typically offered in Fall and Summer*

Short-term probability models for risk management systems. Frequency distributions, loss distributions, the individual risk model, the collective risk model, stochastic process models of solvency requirements, applications to insurance and businessdecisions.

*Typically offered in Summer only*

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 Spring only*

Basic principles of counting: addition and multiplication principles, generating functions, recursive methods, inclusion-exclusion, pigeonhole principle; basic concepts of graph theory: graphs, digraphs, connectedness, trees; additional topics from:Polya theory of counting, Ramsey theory; combinatorial optimization - matching and covering, minimum spanning trees, minimum distance, maximum flow; sieves; mobius inversion; partitions; Gaussian numbers and q-analogues; bijections and involutions; partially ordered sets.

*Typically offered in Spring only*

*This course is offered alternate years*

**MA 421**

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

Axioms of probability, conditional probability and independence, basic combinatorics, discrete and continuous random variables, joint densities and mass functions, expectation, central, limit theorem, simple stochastic processes.

Prerequisite: MA 242

*Typically offered in Fall, Spring, and Summer*

**MA 425**

**Mathematical Analysis I**(3 credit hours)

Real number system, functions and limits, topology on the real line, continuity, differential and integral calculus for functions of one variable. Infinite series, uniform convergence. Credit is not allowed for both MA 425 and MA 511.

*Typically offered in Fall and Spring*

**MA 426**

**Mathematical Analysis II**(3 credit hours)

Calculus of several variables, topology in n-dimensions, limits, continuity, differentiability, implicit functions, integration. Credit is not allowed for both MA 426 and MA 512.

*Typically offered in Fall and Spring*

Theory and practice of computational procedures including approximation of functions by interpolating polynomials, numerical differentiation and integration, and solution of ordinary differential equations including both initial value and boundary value problems. Computer applications and techniques.

Prerequisite: (MA 341 or MA 301) and ( CSC 111 or CSC 112 or CSC 113 or CSC 114 or CSC 116 or MA 116 or PY 251 or ST 114 or ECE 209)

*Typically offered in Fall only*

Computational procedures including direct and iterative solution of linear and nonlinear equations, matrices and eigenvalue calculations, function approximation by least squares, smoothing functions, and minimax approximations.

Prerequisite: (MA 305 or MA 405) and (CSC 111 or CSC 112 or CSC 113 or CSC 114 or CSC 116 or MA 116 or PY 251 or ST 114 or ECE 209)

*Typically offered in Spring only*

**MA 430**

**Mathematical Models in the Physical Sciences**(3 credit hours)

Application of mathematical techniques to topics in the physical sciences. Problems from such areas as conservative and dissipative dynamics, calculus of variations, control theory, and crystallography.

*Typically offered in Fall only*

**MA 432**

**Mathematical Models in Life Sciences**(3 credit hours)

Topics from differential and difference equations, and matrix algebra applied to formulation and analysis of mathematical models in biological science (e.g., population growth or disease models).

Prerequisite: (MA 331 or MA 341) and (MA 305 or MA 405). Programming proficiency and some experience with basic statistics is recommended.

*Typically offered in Spring only*

**MA 437**

**Applications of Algebra**(3 credit hours)

Error correcting codes, cryptography, crystallography, enumeration techniques, exact solutions of linear equations, and block designs.

*Typically offered in Fall and Spring*

**MA 444**

**Problem Solving Strategies for Competitions**(1 credit hours)

Analyze the most common problem-solving techniques and illustrate their use by interesting examples from past Putnam and Virginia Tech math competitions. Problem solving methods are divided into groups and taught by professors of the math department. After the lecture, students practice writing the solutions for the assignment and have informal discussions in the next class.

*Typically offered in Fall only*

**MA 450**

**Methods of Applied Mathematics I**(3 credit hours)

Mathematical methods covered include dimensional analysis, asymptotics, continuum modeling and traffic flow analysis. These topics are discussed in the context of applications and real data. This course is independent of MA 451 Methods of Applied Mathematics II.

Prerequisite: MA 341

*Typically offered in Fall only*

**MA 451**

**Methods of Applied Mathematics II**(3 credit hours)

The mathematical methods of this course give insight into physical continuum processes such as fluid flow and the deformation of solid elastic materials. Techniques include the modeling and formulation of equations of motion, the use of Lagrangian and Eulerian variables; further topics are: examples of incompressible fluid flow, calculus of variations and applications to optimal control problems. This course is independent of MA 450 Methods of Applied Mathematics I.

Prerequisite: MA 341

*Typically offered in Spring only*

**MA 491**

**Reading in Honors Mathematics**(1-6 credit hours)

A reading (independent study) course available as an elective for students participating in the mathematics honors program.

Prerequisite: Membership in honors program

*Typically offered in Fall and Spring*

**MA 493**

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

Directed individual study or experimental course offerings.

*Typically offered in Fall and Spring*

**MA 494**

**Major Paper in Math**(1 credit hours)

Introduces students to one or more forms of writing used in scientific and research environments. Students are required to take a companion math course at the 400-level or above, and adapt writing assignment(s) to the topics in the companion course.Instruction covers all phases of the writing process (planning, drafting, revising, and critiquing other people's work). Emphasis is placed on organizing for needs of a variety of readers; concise, clear expression.

Corequisite: MA class at the 400-level or above

*Typically offered in Fall and Spring*

**MA 499**

**Independent Research in Mathematics**(1-6 credit hours)

Study and research in mathematics. Topics for theoretical, modeling or computational investigation. Consent of Department Head. Honors Program should enroll in MA 491H. At most 6 hours total of MA 499 and 491H credit can be applied towards an undergraduate degree. Individualized/Independent Study and Research courses require a Course Agreement for Students Enrolled in Non-Standard Courses be completed by the student and faculty member prior to registration by the department.

*Typically offered in Fall, Spring, and Summer*

**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 Spring only*

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*

*This course is offered alternate years*

**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*

*This course is offered alternate years*

**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*

*This course is offered alternate years*

**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*

*This course is offered alternate years*

**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 only*

**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*

*This course is offered alternate even years*

**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*

*This course is offered alternate even years*

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 630**

**Independent Study in Mathematics**(1-6 credit hours)

Independent study of an advanced mathematics topic under the direction of mathematics faculty member on tutorial basis. Requires a faculty sponsor and departmental approval.

R: Graduate Standing

*Typically offered in Fall, Spring, and Summer*

**MA 635**

**Readings in Mathematics**(1-6 credit hours)

Readings in advanced topics in mathematics

R: Graduate Standing

*Typically offered in Fall, Spring, and Summer*

**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*

*This course is offered alternate years*

**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*

*This course is offered alternate years*

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*

*This course is offered alternate odd years*

**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*

*This course is offered alternate years*

**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*

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 754**

**Advanced Algebraic Topology**(3 credit hours)

Properties of cohomology, homotopy groups, fiber bundles, characteristic classes, and homological algebra. Additional topics may include spectra, spectral sequences, K-theory, group cohomology, and connections with smooth manifold topology.

Prerequisite: MA 753

*Typically offered in Fall only*

*This course is offered alternate odd years*

**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*

*This course is offered alternate years*

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*

*This course is offered alternate years*

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*

*This course is offered alternate years*

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 782**

**Computational Methods for Variational Inverse Problems**(3 credit hours)

Computational methods for inverse problems that are governed by partial differential equations. Topics will include variational formulations, ill-posedness, regularization, discretization methods, and optimization algorithms, statistical formulations, and numerical implementations.

*Typically offered in Spring only*

**MA 784**

**Numerical Methods for Nonlinear Equations and Optimization**(3 credit hours)

The course provides a graduate-level introduction to the numerical methods of solving linear and nonlinear optimization problems and nonlinear equations, along with the fundamental mathematical theory necessary to develop these algorithms. Topics selected from: 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, conic programming, first-order methods for large-scale optimization problems.

*Typically offered in Spring only*

**MA 788**

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

A review of modern numerical techniques for time-dependent nonlinear partial differential equations. Topics include Finite Difference, Finite Volume, Particle and Hybrid Eulerian- Lagrangian Methods; Splitting Methods and Implicit-Explicit Discretization; Spectral and Pseudo-Spectral Methods including Stochastic Galerkin and Stochastic Collocation Methods, and others. Applications including problems in fluid and gas dynamics, geophysics, meteorology, astrophysics, biology, and other fields.

*Typically offered in Spring only*

*This course is offered alternate years*

**MA 790**

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

Special advanced topics in mathematics.

R: Graduate Standing

*Typically offered in Fall, Spring, and Summer*

**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*

*This course is offered alternate years*

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 830**

**Independent Study in Mathematics**(1-6 credit hours)

Independent study of an advanced mathematics topic under the direction of mathematics faculty member on tutorial basis. Requires a faculty sponsor and departmental approval.

R: Graduate Standing

*Typically offered in Fall, Spring, and Summer*

**MA 835**

**Readings in Mathematics**(1-6 credit hours)

Readings in advanced topics in mathematics

R: Graduate Standing

*Typically offered in Fall, Spring, and Summer*

**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*