¸£Àû±ÆÕ¾

Preparation for ²Ñ´¡Ìý103, ²Ñ´¡Ìý105, ²Ñ´¡Ìý107, ²Ñ´¡Ìý111, and ²Ñ´¡Ìý114. 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. ²Ñ´¡Ìý101 may not be counted as credit toward meeting graduation. Credit for ²Ñ´¡Ìý101 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: ²Ñ´¡Ìý101 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: ²Ñ´¡Ìý101 or equivalent completed in high school

GEP Mathematical Sciences

Typically offered in Fall and Spring

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: ²Ñ´¡Ìý101 or equivalent completed in high school

GEP Mathematical Sciences

Typically offered in Fall, Spring, and Summer

Algebra and basic trigonometry; polynomial, rational, exponential, logarithmic and trigonometric functions and their graphs. Credit for ²Ñ´¡Ìý107 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 ²Ñ´¡Ìý107 and ²Ñ´¡Ìý111

Prerequisite: C- or better in ²Ñ´¡Ìý101, 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

Algebra, analytic geometry and trigonometry; inequalities, conic sections, complex numbers, sequences and series, solving triangles, polar coordinates, and applications.Credit for ²Ñ´¡Ìý108 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 ²Ñ´¡Ìý108 and ²Ñ´¡Ìý111. Also, ²Ñ´¡Ìý108 should not be counted toward the GER mathematical sciences.

Prerequisite: C- or better in ²Ñ´¡Ìý107

Typically offered in Spring only

The study of real numbers, functions and their graphs with an emphasis on absolute value, polynomial, piecewise, rational, exponential, logarithmic, and trigonometric functions. Credit is not allowed for both ²Ñ´¡Ìý111 and either ²Ñ´¡Ìý107 or ²Ñ´¡Ìý108. Note that this course does not count towards graduation for all programs: please check with your program.

GEP Mathematical Sciences

Typically offered in Fall, Spring, and Summer

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: ²Ñ´¡Ìý101 or equivalent completed in high school.

GEP Mathematical Sciences

Typically offered in Fall, Spring, and Summer

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.

Prerequisite: ²Ñ´¡Ìý141, and either °ä°¿³§Ìý100 or ·¡Ìý115; Corequisite: ²Ñ´¡Ìý241

Typically offered in Fall and Spring

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 business and management.

Prerequisite: C- or better in ²Ñ´¡Ìý107 or ²Ñ´¡Ìý111, or qualifying score on math placement exam. Credit is not allowed in more than one of ²Ñ´¡Ìý121, 131, 141.

GEP Mathematical Sciences

Typically offered in Fall, Spring, and Summer

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.

Prerequisite: C- or better in ²Ñ´¡Ìý107 or ²Ñ´¡Ìý111, or qualifying score on math placement exam. Credit is not allowed in more than one of ²Ñ´¡Ìý121, ²Ñ´¡Ìý131, or ²Ñ´¡Ìý141.

GEP Mathematical Sciences

Typically offered in Fall, Spring, and Summer

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.

Prerequisite: C- or better in ²Ñ´¡Ìý121 or ²Ñ´¡Ìý131

Typically offered in Fall and Spring

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: ²Ñ´¡Ìý111 or ²Ñ´¡Ìý108 with a grade C- or better or qualifying score on math placement exam. Credit is not allowed for both ²Ñ´¡Ìý141 and ²Ñ´¡Ìý121 or ²Ñ´¡Ìý131.

GEP Mathematical Sciences

Typically offered in Fall, Spring, and Summer

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 ²Ñ´¡Ìý151, ²Ñ´¡Ìý121, ²Ñ´¡Ìý131, or ²Ñ´¡Ìý141. ²Ñ´¡Ìý151 is not an accepted prerequisite for ²Ñ´¡Ìý231 and ²Ñ´¡Ìý241. This course is restricted to Elementary Education majors only.

Prerequisite: C- or better in ²Ñ´¡Ìý107 or ²Ñ´¡Ìý111, or qualifying score on math placement exam. Credit is not allowed in more than one of ²Ñ´¡Ìý121, ²Ñ´¡Ìý131, or ²Ñ´¡Ìý141.

GEP Mathematical Sciences

Typically offered in Spring only

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 ²Ñ´¡Ìý152 and ²Ñ´¡Ìý121, ²Ñ´¡Ìý131, or ²Ñ´¡Ìý141. ²Ñ´¡Ìý152 is not an accepted prerequisite for ²Ñ´¡Ìý241.

Prerequisite: ²Ñ´¡Ìý151

GEP Mathematical Sciences

Typically offered in Fall only

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: ²Ñ´¡Ìý241

Typically offered in Fall, Spring, and Summer

Functions of several variables - partial derivatives, optimization, least squares, Lagrange multiplier method; differential equations - population growth, finance and investment models, systems, numerical methods; ²Ñ´¡Ìý121 is not an accepted prerequisite for ²Ñ´¡Ìý231.

Prerequisite: ²Ñ´¡Ìý131 or ²Ñ´¡Ìý141; Credit is not allowed for both ²Ñ´¡Ìý231 and ²Ñ´¡Ìý241.

GEP Mathematical Sciences

Typically offered in Fall, Spring, and Summer

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: ²Ñ´¡Ìý141 with grade of C- or better or AP Calculus credit. Credit is not allowed for both ²Ñ´¡Ìý241 and ²Ñ´¡Ìý231.

GEP Mathematical Sciences

Typically offered in Fall, Spring, and Summer

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: ²Ñ´¡Ìý241 with grade of C- or better or AP Calculus credit, or Higher Level IB credit.

Typically offered in Fall, Spring, and Summer

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: ²Ñ´¡Ìý241

Typically offered in Fall and Spring

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 ²Ñ´¡Ìý341 or ²Ñ´¡Ìý405

Prerequisite: ²Ñ´¡Ìý241

Typically offered in Fall and Spring

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 ²Ñ´¡Ìý405 Introductory Linear Algebra, more emphasis is placed on methods and calculations,. Credit is not allowed for both ²Ñ´¡Ìý305 and ²Ñ´¡Ìý405.

Prerequisite: ²Ñ´¡Ìý241 or ²Ñ´¡Ìý231 with ²Ñ´¡Ìý132

Typically offered in Fall, Spring, and Summer

For sophomore meteorology and marine science students. A complement to ²Ñ´¡Ìý242 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.

Prerequisite: ²Ñ·¡´¡Ìý217 or ²Ñ´¡Ìý116 or °ä³§°äÌý113; Corequisite: ²Ñ´¡Ìý242

Typically offered in Spring only

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.

Prerequisite: (²Ñ´¡Ìý231 or ²Ñ´¡Ìý242) and (²Ñ´¡Ìý116 or °ä³§°äÌý112 or CSC 114 or °ä³§°äÌý116)

Typically offered in Spring only

The course covers foundational mathematical concepts fundamental to data science and data-driven mathematical modeling. The course includes the following topics: introductory probability and vector calculus, theory for classification algorithms, linear and parametric classifiers, unsupervised and clustering methods, decision trees and ensemble methods. The focus is on applying mathematical concepts to data science methods. The course includes an introduction to Python, but some familiarity with programming is strongly recommended. Basic programming proficiency (Python preferred).

Prerequisite: ²Ñ´¡Ìý242 and (²Ñ´¡Ìý303 or ²Ñ´¡Ìý305 or ²Ñ´¡Ìý405)

Typically offered in Fall and Spring

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 ²Ñ´¡Ìý341 and ²Ñ´¡Ìý331.

P: ²Ñ´¡Ìý231 or ²Ñ´¡Ìý241; X: Credit cannot be given for both ²Ñ´¡Ìý341 and ²Ñ´¡Ìý331

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.

Prerequisite: ³¢°¿³ÒÌý201 or ²Ñ´¡Ìý225 or °ä³§°äÌý226

GEP Mathematical Sciences

Typically offered in Fall only

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 ²Ñ´¡Ìý341

Prerequisite: ²Ñ´¡Ìý242 or (²Ñ´¡Ìý132 and ²Ñ´¡Ìý231)

Typically offered in Fall, Spring, and Summer

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

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 ²Ñ´¡Ìý401 and ²Ñ´¡Ìý501 will not be given

Prerequisite: ²Ñ´¡Ìý341; Credit not allowed for both ²Ñ´¡Ìý401 and ²Ñ´¡Ìý501

Typically offered in Fall, Spring, and Summer

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.

P: (²Ñ´¡Ìý341 or ²Ñ´¡Ìý405) and programming proficiency (MATLAB, C++, Java, Fortran, or other language)

Typically offered in Fall and Spring

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 ²Ñ´¡Ìý403 and ²Ñ´¡Ìý407

Prerequisite: ²Ñ´¡Ìý225

Typically offered in Fall only

The course covers foundational mathematical concepts fundamental to data science. It builds upon the basic concepts in ²Ñ´¡Ìý326 and develops theory for a range of central data science techniques. The course includes the following topics: Optimization algorithms, neural networks, graph-based models and generative learning. These algorithms will be explored computationally using Python and practical data sets.

Prerequisite: ²Ñ´¡Ìý326

Typically offered in Fall only

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 ²Ñ´¡Ìý305 Introductory Linear Algebra, more emphasis is placed on theory and proofs. ²Ñ´¡Ìý225 is recommended as a prerequisite. Credit is not allowed for both ²Ñ´¡Ìý305 and ²Ñ´¡Ìý405

Prerequisite: ²Ñ´¡Ìý241 (²Ñ´¡Ìý225 recommended); Corequisite: ²Ñ´¡Ìý341 is recommended

Typically offered in Fall, Spring, and Summer

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 ²Ñ´¡Ìý403 and ²Ñ´¡Ìý407

Prerequisite: ²Ñ´¡Ìý225 and ²Ñ´¡Ìý405

Typically offered in Fall and Spring

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

Prerequisite: ²Ñ´¡Ìý225

Typically offered in Fall and Spring

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: ²Ñ´¡Ìý241 or ²Ñ´¡Ìý231, Corequisite: ²Ñ´¡Ìý421, BUS(ST) 350, ST 301, ST 305, ³§°ÕÌý311, ST 361, ³§°ÕÌý370, ³§°ÕÌý371, 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.

Prerequisite: ²Ñ´¡Ìý241 or ²Ñ´¡Ìý231, and one of ²Ñ´¡Ìý421, ST 301, ST 305, ³§°ÕÌý370, ³§°ÕÌý371, ST 380, ³§°ÕÌý421.

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.

Prerequisite: (°ä³§°äÌý226 AND °ä³§°äÌý333) OR ²Ñ´¡Ìý225

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.

Prerequisite: Grade of C or better in either ²Ñ´¡Ìý225 or °ä³§°äÌý226

Typically offered in Spring only

This course is offered alternate years

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: ²Ñ´¡Ìý242

Typically offered in Fall, Spring, and Summer

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 ²Ñ´¡Ìý425 and ²Ñ´¡Ìý511.

Prerequisite: ²Ñ´¡Ìý225 (²Ñ´¡Ìý407 desirable)

Typically offered in Fall and Spring

Calculus of several variables, topology in n-dimensions, limits, continuity, differentiability, implicit functions, integration. Credit is not allowed for both ²Ñ´¡Ìý426 and MA 512.

Prerequisite: ²Ñ´¡Ìý425 and 405

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: (²Ñ´¡Ìý341 or MA 301) and ( °ä³§°äÌý111 or °ä³§°äÌý112 or °ä³§°äÌý113 or CSC 114 or °ä³§°äÌý116 or ²Ñ´¡Ìý116 or ±Ê³ÛÌý251 or ³§°ÕÌý114 or ·¡°ä·¡Ìý209)

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: (²Ñ´¡Ìý305 or ²Ñ´¡Ìý405) and (°ä³§°äÌý111 or °ä³§°äÌý112 or °ä³§°äÌý113 or CSC 114 or °ä³§°äÌý116 or ²Ñ´¡Ìý116 or ±Ê³ÛÌý251 or ³§°ÕÌý114 or ·¡°ä·¡Ìý209)

Typically offered in Spring only

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.

Prerequisite: ²Ñ´¡Ìý341 and ²Ñ´¡Ìý405

Typically offered in Fall only

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: (²Ñ´¡Ìý331 or ²Ñ´¡Ìý341) and (²Ñ´¡Ìý305 or ²Ñ´¡Ìý405). Programming proficiency and some experience with basic statistics is recommended.

Typically offered in Spring only

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

Prerequisite: ²Ñ´¡Ìý403 or 407, ²Ñ´¡Ìý405

Typically offered in Fall and Spring

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

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 ²Ñ´¡Ìý451 Methods of Applied Mathematics II.

Prerequisite: ²Ñ´¡Ìý341

Typically offered in Fall only

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 ²Ñ´¡Ìý450 Methods of Applied Mathematics I.

Prerequisite: ²Ñ´¡Ìý341

Typically offered in Spring only

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

Directed individual study or experimental course offerings.

Typically offered in Fall and Spring

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

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 ²Ñ´¡Ìý499 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

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 ²Ñ´¡Ìý401 is not allowed.

Prerequisite: ²Ñ´¡Ìý341; credit not allowed for both ²Ñ´¡Ìý501 and ²Ñ´¡Ìý401

Typically offered in Fall, Spring, and Summer

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 ²Ñ´¡Ìý502 may receive credit for, atmost, one of the following: ²Ñ´¡Ìý405, MA 512, ²Ñ´¡Ìý513

Prerequisite: ²Ñ´¡Ìý341.

Typically offered in Spring only

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

Prerequisite: ²Ñ´¡Ìý242, ²Ñ´¡Ìý405

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: An introductory linear algebra course similar to ²Ñ´¡Ìý405

Typically offered in Fall only

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

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

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

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

Fundamental theorems on continuous functions; convergence theory of sequences, series and integrals; the Riemann integral. Credit for both ²Ñ´¡Ìý425 and ²Ñ´¡Ìý511 is not allowed

Prerequisite: ²Ñ´¡Ìý341

Typically offered in Fall and Spring

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: ²Ñ´¡Ìý242

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.

Prerequisite: (°ä³§°äÌý226 AND °ä³§°äÌý333) OR ²Ñ´¡Ìý225

Typically offered in Fall only

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: ²Ñ´¡Ìý426

Typically offered in Fall only

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.

Prerequisite: ²Ñ´¡Ìý242 and ²Ñ´¡Ìý405

Typically offered in Spring only

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: ²Ñ´¡Ìý405

Typically offered in Fall and Spring

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

Prerequisite: ²Ñ´¡Ìý405 and ²Ñ´¡Ìý407

Typically offered in Fall only

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.

Prerequisite: ²Ñ´¡Ìý405 and ²Ñ´¡Ìý407

Typically offered in Fall only

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: ²Ñ´¡Ìý405

Typically offered in Fall and Spring

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.

Prerequisite: ²Ñ´¡Ìý405 and ²Ñ´¡Ìý407

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.

Prerequisites: ²Ñ´¡Ìý341 and ²Ñ´¡Ìý405 and ²Ñ´¡Ìý421

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.

Prerequisite: ²Ñ´¡Ìý341, ²Ñ´¡Ìý405

Typically offered in Fall only

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

Prerequisite: ²Ñ´¡Ìý341, 405, 425 or 511, Corequisite: ²Ñ´¡Ìý426 or 512

Typically offered in Fall only

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.

Prerequisite: ²Ñ´¡Ìý425 or ²Ñ´¡Ìý511, ²Ñ´¡Ìý341, Corequisite: ²Ñ´¡Ìý426 or 512

Typically offered in Fall only

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.

Prerequisite: ²Ñ´¡Ìý341 and ²Ñ´¡Ìý405

Typically offered in Spring only

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: ²Ñ´¡Ìý341 and basic knowledge of probability, linear algebra, and scientific computation

Typically offered in Fall and Spring

This course is offered alternate even years

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: ²Ñ´¡Ìý141, 241, 242, or equivalent and ²Ñ´¡Ìý405 or equivalent; Some notions of elementary convex analysis are an asset but are neither required nor assumed known.

Typically offered in Fall only

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: ²Ñ´¡Ìý421

Typically offered in Spring only

Mathematical foundations of probability theory. Probabilistic measure theory, random variables and their distributions, construction of expectation. Notions of convergence: almost sure, in probability, in L^p, weak convergence, vague convergence. Conditioning, independence, Borel-Cantelli lemmas, weak and strong laws of large numbers, characteristic functions, central limit theorem, and related concentration inequalities.

Prerequisite: ²Ñ´¡Ìý421 and ²Ñ´¡Ìý425 or ²Ñ´¡Ìý511

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: ¹ó±õ²ÑÌý528 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.

Prerequisites: (²Ñ´¡Ìý421 or ³§°ÕÌý421), ²Ñ´¡Ìý341, and ²Ñ´¡Ìý405

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.

Prerequisites: ²Ñ´¡Ìý405 and (²Ñ´¡Ìý421 or ³§°ÕÌý421) and (MA/³§°ÕÌý412 or MA/³§°ÕÌý413)

Typically offered in Spring only

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

Prerequisite: ²Ñ´¡Ìý426

Typically offered in Fall only

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.

Prerequisite: ²Ñ´¡Ìý405 and ²Ñ´¡Ìý426

Typically offered in Fall only

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: ²Ñ´¡Ìý407

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.

Prerequisite: °ä³§°äÌý226 or ²Ñ´¡Ìý351.

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: ²Ñ´¡Ìý341 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/µþ²Ñ´¡Ìý573

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.

Prerequisite: ²Ñ´¡Ìý405; ²Ñ´¡Ìý425 or ²Ñ´¡Ìý511; high-level computer language

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.

Prerequisite: °ä³§°äÌý302 or ²Ñ´¡Ìý402 or MA/°ä³§°äÌý428 or MA/°ä³§°äÌý580

Typically offered in Spring only

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: ²Ñ´¡Ìý501; knowledge of a high level programming language

Typically offered in Fall only

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: ²Ñ´¡Ìý501; knowledge of a high level programming language

Typically offered in Spring only

Typically offered in Fall and Spring

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

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

Readings in advanced topics in mathematics

R: Graduate Standing

Typically offered in Fall, Spring, and Summer

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

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

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

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

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

Thesis Research

Prerequisite: Master's student

Typically offered in Fall and Spring

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

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 ²Ñ´¡Ìý425

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.

Prerequisite: ²Ñ´¡Ìý405, OR (MA,IE) 505, Corequisite: Some familiarity with computers (e.g., °ä³§°äÌý112)

Typically offered in Spring only

This course is offered alternate years

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: ²Ñ´¡Ìý515

Typically offered in Spring only

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

Prerequisite: ²Ñ´¡Ìý715

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.

Prerequisite: ²Ñ´¡Ìý405, 511

Typically offered in Fall only

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: ²Ñ´¡Ìý520

Typically offered in Spring only

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: ²Ñ´¡Ìý521

Typically offered in Spring only

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: ²Ñ´¡Ìý522

Typically offered in Spring only

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

Prerequisite: ²Ñ´¡Ìý520 or 523

Typically offered in Spring only

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: ²Ñ´¡Ìý524

Typically offered in Spring only

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: ²Ñ´¡Ìý720

Typically offered in Fall only

This course is offered alternate odd years

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: ²Ñ´¡Ìý521

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

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

Prerequisite: ²Ñ´¡Ìý532, Corequisite: ²Ñ´¡Ìý515

Typically offered in Spring only

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.

Prerequisite: ²Ñ´¡Ìý515 and ²Ñ´¡Ìý534

Typically offered in Spring only

Advanced development of stochastic processes. Conditional expectation, filtrations of sigma-algebras, stopping times. Martingales, associated convergence theorems and inequalities, martingale decomposition, optional stopping. Markov chains including random walks, recurrence versus transience, asymptotic behavior. General Markov processes and the related semigroup operators. Construction and properties of Brownian motion, Donsker's invariance principle. Other potential topics include stationary processes, Birkhoff's ergodic theorem, branching processes, Poisson processes.

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

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

Prerequisite: ²Ñ´¡Ìý551 or ²Ñ´¡Ìý555

Typically offered in Spring only

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: ²Ñ´¡Ìý753

Typically offered in Fall only

This course is offered alternate odd years

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: ²Ñ´¡Ìý555

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

Theory of stochastic processes and its application to contemporary problems in biology. Discrete- and continuous-time Markov chains, branching processes, birth-and-death processes, diffusion approximations, and elementary stochastic differential equations. Survey of applications in areas such as population genetics, infectious disease dynamics, neurobiology, and community ecology.

Prerequisite: 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: µþ²Ñ´¡Ìý772 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.

Prerequisite: µþ²Ñ´¡Ìý771 or MA/°¿¸éÌý731; µþ²Ñ´¡Ìý772 or ²Ñ´¡Ìý401 or ²Ñ´¡Ìý501

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: ²Ñ´¡Ìý580

Typically offered in Spring only

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.

P: ²Ñ´¡Ìý401 and ²Ñ´¡Ìý580 or equivalent.

Typically offered in Spring only

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.

P: ²Ñ´¡Ìý580 or ²Ñ´¡Ìý523

Typically offered in Spring only

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.

Prerequisite: ²Ñ´¡Ìý401 or ²Ñ´¡Ìý427 or ²Ñ´¡Ìý428; knowledge of a high level programming language

Typically offered in Spring only

This course is offered alternate years

Special advanced topics in mathematics.

R: Graduate Standing

Typically offered in Fall, Spring, and Summer

Typically offered in Fall and Spring

Typically offered in Fall and Spring

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

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

Readings in advanced topics in mathematics

R: Graduate Standing

Typically offered in Fall, Spring, and Summer

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

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

Prerequisite: Doctoral student

Typically offered in Fall and Spring

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

Dissertation Research

Prerequisite: Doctoral student

Typically offered in Fall and Spring

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

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