# Course Descriptions

## West Lafayette Campus

### Mathematics Courses

Click on name of course to view prerequisites and additional information.

**MA 10800 - Mathematics As A Profession And A Discipline**.

A seminar course for undergraduate students interested in majoring in an area of mathematics at Purdue. The purpose is to build prospective mathematics majors' awareness of opportunities to enhance their experiences at Purdue and of career paths available for graduates with a good mathematical background. The format of most classes is a presentation and discussion with an invited speaker/guest, including experts on a different aspect of mathematics in our world today. This course is recommended for undergraduates in their first or second year at Purdue. 1 credit hour

**MA 13700 - Mathematics For Elementary Teachers I**.

Designed for prospective elementary school teachers. Problem solving. Numerical reasoning including self-generated and conventional algorithms. Whole and fractional number systems, elementary number theory. (Not available for credit toward graduation in the College of Science.). 3 credit hours

**MA 13800 - Mathematics For Elementary Teachers II**.

Elementary school teachers must understand how multiplication gives rise to exponents and how to represent, interpret, and compute exponents from problem situations. They must also understand how to represent practical situations using algebraic and fractional expressions, and verbally interpret graphs of functions. They have to know basic concepts of probability theory. This course covers conceptual and practical notions of exponents and radicals; algebraic and rational functions, algebraic equations and inequalities, systems of linear equations, polynomial, exponential, and logarithmic functions. Notions of probability. 3 credit hours

**MA 13900 - Mathematics For Elementary Teachers III**.

Geometric, measurement and spatial reasoning in one, two and three dimensions as the basis for elementary school geometry. Metric and non-metric geometry, transformation geometry. (Not available for credit toward graduation in the College of Science.) 3 credit hours

Exponents and radicals; algebraic and fractional expressions. Equations and inequalities, systems of linear equations. Polynomial, exponential, and logarithmic functions. Not open to students with credit in MA 15900. Not available for credit toward graduation in the School of Science. CTL:IMA 1601 College Algebra 0 OR 3 credit hours

**MA 15555 - Quantitative Reasoning**.

This course will cover important mathematical ideas, including proportion, weighted averages, linear models, exponential models, basic probability and statistics, and some algebra, by using concrete real-world problems. It will not be a prerequisite for any other mathematics course. CTL: Quantitative Reasoning 0 OR 3 credit hours

**MA 15800 - Precalculus - Functions And Trigonometry**.

Functions, Trigonometry, and Algebra of calculus topics designed to fully prepare students for all first semester calculus courses. Functions topics include Quadratic, Higher Order Polynomials, Rational, Exponential, Logarithmic, and Trigonometric. Other focuses include graphing of functions and solving application problems. Not Available for credit toward graduation in the College of Science. Students may not receive credit for both MA 15400 and MA 15800. Students may not receive credit for both MA 15900 and MA 15800. 0 OR 3 credit hours

**MA 16010 - Applied Calculus I**.

Topics include trigonometric and exponential functions; limits and differentiation, rules of differentiation, maxima, minima and optimization; curve sketching, integration, anti-derivatives, fundamental theorem of calculus. Properties of definite integrals and numerical methods. Applications to life, managerial and social sciences. CTL:IMA 1604 Calculus - Short I 0 OR 3 credit hours

**MA 16020 - Applied Calculus II**.

This course covers techniques of integration; infinite series, convergence tests; differentiation and integration of functions of several variables; maxima and minima, optimization; differential equations and initial value problems; matrices, determinants, eigenvalues and eigenvectors. Applications. CTL:IMA 1605 Calculus - Short II 0 OR 3 credit hours

**MA 16100 - Plane Analytic Geometry And Calculus I**.

Introduction to differential and integral calculus of one variable, with applications. Some schools or departments may allow only 4 credit hours toward graduation for this course. Designed for students who have not had at least a one-semester calculus course in high school, with a grade of "A" or "B". Not open to students with credit in MA 16500. Demonstrated competence in college algebra and trigonometry. 0 OR 5 credit hours

**MA 16200 - Plane Analytic Geometry And Calculus II**.

Continuation of MA 16100. Vectors in two and three dimensions, techniques of integration, infinite series, conic sections, polar coordinates, surfaces in three dimensions. Some schools or departments may allow only 4 credit hours toward graduation for this course. 0 OR 5 credit hours

**MA 16290 - Data Science Labs: Calculus**.

This course consists of weekly computer laboratories which apply concepts learned in Calculus I and II to data science problems. Main topics covered include function sampling and approximation, numerical differentiation, numerical integration, Jupyter notebooks, introductory Python programming, object oriented programming, and data acquisition with microcontrollers and sensors. 1 credit hour

**MA 16500 - Analytic Geometry And Calculus I**.

Introduction to differential and integral calculus of one variable, with applications. Conic sections. Designed for students who have had at least a one-semester calculus course in high school, with a grade of "A" or "B", but are not qualified to enter MA 16200 or 16600, or the advanced placement courses MA 27100. Demonstrated competence in college algebra and trigonometry. CTL:IMA 1602 Calculus - Long I 0 OR 4 credit hours

**MA 16600 - Analytic Geometry And Calculus II**.

Continuation of MA 16500. Vectors in two and three dimensions. Techniques of integration, infinite series, polar coordinates, surfaces in three dimensions. Not open to students with credit in MA 16200. CTL:IMA 1603 Calculus - Long II 0 OR 4 credit hours

**MA 17000 - Introduction To Actuarial Science**.

(STAT 17000) An introduction to actuarial science from the point of view of practicing actuaries from life insurance, casualty insurance and consulting; introduction to insurance and the mathematical theory of interest; application of spreadsheets to problems related to actuarial science. 0 OR 2 credit hours

**MA 18300 - Professional Practicum I**.

Professional Practicum. For Cooperative Education students only; must be accepted for the program by the cooperative program coordinator. Permission of department required. 0 credit hours

**MA 19000 - Topics In Mathematics For Undergraduates**.

Supervised reading courses as well as special topics courses for undergraduates are given under this number. Permission of instructor required. 0 to 5 credit hours

**MA 25000 - Problem Solving In Probability**.

(STAT 25000) This course is designed to teach techniques for solving problems in probability theory which are relevant to the actuarial sciences. It is intended to help actuarial students prepare for the Society of Actuaries and Casualty Actuarial Society Exam P/1. Credit of Examination is not available for this course. 2 credit hours

**MA 26100 - Multivariate Calculus**.

Planes, lines, and curves in three dimensions. Differential calculus of several variables; multiple integrals. Introduction to vector calculus. Not open to students with credit in MA 27100. 0 OR 4 credit hours

**MA 26190 - Data Science Labs: Multivariate Calculus**.

This course consists of weekly computer laboratories which apply concepts learned in Multivariate Calculus (Calculus III) to data science problems. The students will also practice programming in Python and use sensors and microprocessors to acquire data. Topics covered include representation and perception of color, motion detection in videos, and construction of a planimeter based on Green's theorem. 1 credit hour

**MA 26200 - Linear Algebra And Differential Equations**.

Linear algebra, elements of differential equations. Not open to students with credit in MA 26500 or 26600. 0 OR 4 credit hours

Introduction to linear algebra. Systems of linear equations, matrix algebra, vector spaces, determinants, eigenvalues and eigenvectors, diagonalization of matrices, applications. Not open to students with credit in MA 26200, 27200, 35000 or 35100. 3 credit hours

**MA 26600 - Ordinary Differential Equations**.

First order equations, second and nth order linear equations, series solutions, solution by Laplace transform, systems of linear equations. It is preferable but not required to take MA 26500 either first or concurrently. Not open to students with credit in MA 26200, 27200, 36000, 36100, or 36600. 3 credit hours

**MA 27101 - Honors Multivariate Calculus**.

This course is the Honors version of MA 26100, Multivariate Calculus; it will also include a review of infinite series. The course is intended for first-year students who have credit for Calculus I and II. There will be a significant emphasis on conceptual explanation, but not on formal proof. Permission of department is required. 0 OR 5 credit hours

**MA 27900 - Modern Mathematics In Science And Society**.

The course covers topics in combinatorics and probability applied to real life situations such as the paradoxes of democracy, weighted voting, fair division, apportionment, traveling salesmen, the mathematics of networks, Fibonacci numbers, golden ratio, growth patterns in nature, mathematics of money, symmetry, fractals, censuses and surveys, random sampling, sample spaces, permutations and uniform probability spaces. 3 credit hours

**MA 29000 - Topics In Mathematics For Undergraduates**.

Supervised reading courses as well as special topics courses for undergraduates are given under this number. Permission of instructor required. 1 to 5 credit hours

**MA 29199 - Cooperative Experience I**.

Professional experience in mathematics. Program coordinated by school with cooperation of participating employers. Students submit summary report and company evaluation. Professional Practice students only. Permission of department required. 0 credit hours

**MA 29299 - Cooperative Experience II**.

Professional experience in mathematics. Program coordinated by school with cooperation of participating employers. Students submit summary report and company evaluation. Professional Practice students only. Permission of department required. 0 credit hours

**MA 30100 - An Introduction To Proof Through Real Analysis**.

An introduction to abstract reasoning in the context of real analysis. Topics may include axioms for the real numbers, mathematical induction, formal definition of limits, density, decimal representations, convergence of sequences and series, continuity, differentiability, the extreme value, mean value and intermediate value theorems, and cardinality. The emphasis, however, is more on the concept of proof than on any one given topic. 3 credit hours

This is a methods course for juniors in any branch of engineering and science, designed to follow MA 26200 or MA 26600. Materials to be covered are: linear systems of ordinary differential equations, nonlinear systems, Fourier series, separation of variables for partial differential equations, and Sturm-Liouville theory. 3 credit hours

**MA 34100 - Foundations Of Analysis**.

An introductory course in rigorous analysis, covering real numbers, sequences, series, continuous functions, differentiation, and Riemann integration. MA 30100 is helpful but not required. 3 credit hours

**MA 34900 - Signals And Systems For Mathematicians**.

This course introduces the mathematical framework for the description, analysis and processing of signals such as music, speech and images. Main topics covered include signal representations in different bases; continuous-time signal sampling; and signal processing by linear and time-invariant systems. 3 credit hours

**MA 35100 - Elementary Linear Algebra**.

Systems of linear equations, finite dimensional vector spaces, matrices, determinants, eigenvalues and eigenvector applications to analytical geometry. Not open to students with credit in MA 26500. 3 credit hours

Theoretical background for methods and results that appear in MA 35100. Inner products, orthogonality, and applications including least squares. 3 credit hours

**MA 36200 - Topics In Vector Calculus**.

Multivariate calculus; partial differentiation; implicit function theorems and transformations; line and surface integrals; vector fields; theorems of Gauss, Green, and Stokes. Credit granted for only one of MA 36200 and 51000. 3 credit hours

**MA 36600 - Ordinary Differential Equations**.

An introduction to ordinary differential equations with emphasis on problem solving and applications. The one-hour computer lab will give students an opportunity for hands-on experience with both the theory and applications of the subject. 0 OR 4 credit hours

**MA 37300 - Financial Mathematics**.

A mathematical treatment of some fundamental concepts of financial mathematics and their application to real world business situations and basic risk management. Includes discussions of valuing investments, capital budgeting, valuing contingent cash flows, yield curves, spot rates, forward rates, short sales, Macaulay duration, modified duration, convexity, and immunization. Provides preparation for the SOA/CAS Actuarial Exam FM/2. 3 credit hours

**MA 37400 - Mathematical Foundations For Machine Learning**.

This course combines data, computation, and inferential thinking to solve challenging problems. In this class, we explore key areas of machine learning including question formulation, statistical inference, predictive modeling, and decision making. Through a strong emphasis on data-centric computing, and quantitative critical thinking, this class covers key principles and techniques of machine learning. These include algorithms for machine learning methods including regression, classification, and clustering; and statistical concepts of measurement error and prediction. 3 credit hours

**MA 37500 - Introduction To Discrete Mathematics**.

Induction, permutations, combinations, finite probability, relations, graphs, trees, graph algorithms, recurrence relations, generating functions. Problem solving in all these areas. Credit granted for only one of MA 27600 and 37500. 3 credit hours

**MA 38500 - Introduction To Logic**.

Propositional calculus and predicate calculus with applications to mathematical proofs, valid arguments, switching theory, and formal languages. 3 credit hours

**MA 38600 - Professional Practicum IV**.

Professional Practicum. Permission of department required. 0 credit hours

**MA 39000 - Topics In Mathematics For Undergraduates**.

Supervised reading courses as well as special topics courses for undergraduates are given under this number. Permission of instructor required. 1 to 5 credit hours

**MA 39399 - Cooperative Experience III**.

Professional experience in mathematics. Program coordinated by school with cooperation of participating employers. Students submit summary report and company evaluation. Professional Practice students only. Permission of department required. 0 credit hours

**MA 39499 - Extensive Cooperative Experience IV**.

**MA 39599 - Extensive Cooperative Experience V**.

(STAT 41600) An introduction to mathematical probability suitable as a preparation for actuarial science, statistical theory, and mathematical modeling. General probability rules, conditional probability and Bayes theorem, discrete and continuous random variables, moments and moment generating functions, joint and conditional distributions, standard discrete and continuous distributions and their properties, law of large numbers and central limit theorem. 3 credit hours

**MA 42100 - Linear Programming And Optimization Techniques**.

Solution of linear programming problems by the simplex method, duality theory, transportation problems, assignment problems, network analysis, dynamic programming. 3 credit hours

**MA 42500 - Elements Of Complex Analysis**.

Complex numbers and complex-valued functions; differentiation of complex functions; power series, uniform convergence; integration, contour integrals; elementary conformal mapping. 3 credit hours

**MA 42800 - Introduction To Fourier Analysis**.

Topics include: Fourier series, convolutions, kernels, summation methods, Fourier transforms, and applications to the wave, heat, and Laplace equations. 3 credit hours

**MA 43200 - Elementary Stochastic Processes**.

An introduction to some classes of stochastic processes that arise in probabilistic models of time-dependent random processes. The main stochastic processes studied will be discrete time Markov chains and Poisson processes. Other possible topics covered may include continuous time Markov chains, renewal processes, queueing networks, and martingales. 3 credit hours

**MA 44000 - Honors Real Analysis I**.

Real analysis in one and n-dimensional Euclidean spaces. Topics include the completeness property of real numbers, topology of Euclidean spaces, Heine-Borel theorem, convergence of sequences and series in Euclidean spaces, limit superior and limit inferior, Bolzano-Weierstrass theorem, continuity, uniform continuity, limits and uniform convergence of functions, Riemann or Riemann-Stieltjes integrals. 3 credit hours

**MA 44200 - Honors Real Analysis II**.

Real analysis in one and n-dimensional Euclidean spaces--continued from MA 44000. Topics include mappings of Euclidean spaces and their derivatives, multivariable chain rule, inverse function theorem and implicit function theorem, sets with content and integration in n dimensions, the integrability theorem, Jacobian and change of variables theorem, related topics. 3 credit hours

This course, which is essentially the first half of MA 55300, is recommended for students wanting a more substantial background in algebra than is afforded by MA 45300, in particular students intending to do graduate work in science or engineering. Topics include the elements of number theory and group theory; unique factorization in polynomial rings and in principal ideal domains. 3 credit hours

**MA 45300 - Elements Of Algebra I**.

Fundamental properties of integers, polynomials, groups, rings, and fields, with emphasis on problem solving and applications. Not open to students with credit in MA 45000. 3 credit hours

**MA 45401 - Galois Theory Honors**.

This course will give a thorough introduction to Galois theory. Galois theory is a fundamental tool in many areas of mathematics, including number theory and algebraic geometry. This course will increase students' mathematical maturity and prepare them for graduate school. Topics include finite extension fields and their symmetries, ruler and compass constructions, complex roots of unity, solvable groups, and the solvability of polynomial equations by arithmetic and radical operations. This course is intended for third- or fourth-year students who have taken MA 45000 (Algebra Honors) or MA 45300 (Elements of Algebra I). 3 credit hours

This course begins at the high-school level and then moves quickly to intermediate and advanced topics including an introduction to non-Euclidean geometry. Emphasis on proofs. 3 credit hours

**MA 46200 - Elementary Differential Geometry**.

The geometry of curves and surfaces based on familiar parts of calculus and linear algebra. An introduction to the study of differentiable manifolds and Riemannian geometry. 3 credit hours

**MA 47201 - Actuarial Models-Life Contingencies**.

Mathematical foundation of actuarial science, emphasizing probability models for life contingencies as the basis for analyzing life insurance and life annuities and determining premiums and reserves. This course provides the background for Course MLC of the Society of Actuaries and Course 3L of the Casualty Actuarial Society. 0 OR 4 credit hours

**MA 48100 - Advanced Problem-Solving Seminar**.

Seminar intended to prepare students for the national Putnam examination in mathematics. 3 credit hours

**MA 48400 - Seminar On Teaching College Algebra And Trigonometry**.

This course is a seminar on the teaching of mathematics for our best undergraduate mathematics education students. It provides supervised teaching experience along with a chance for the students to perfect their knowledge of algebra before going on to be high school teachers. Students who take this class will also teach a section of MA 15300. Permission of instructor required. 3 credit hours

**MA 48700 - Professional Practicum V**.

r 1.00. Professional Practicum. Permission of department required. 0 OR 1 credit hour

**MA 49000 - Topics In Mathematics For Undergraduates**.

Supervised reading courses as well as special topics courses for undergraduates are given under this number. Permission of instructor required. 1 to 6 credit hours

**MA 49500 - Advanced Topics In Mathematics For Undergraduates**.

Advanced topics courses in mathematics for undergraduates are given under this number. Permission of instructor required. 1 to 5 credit hours

Group theory: definitions, examples, subgroups, quotient groups, homomorphisms, and isomorphism theorems. Ring theory: definitions, examples, homomorphisms, ideals, quotient rings, fraction fields, polynomial rings, Euclidean domains, and unique factorization domains. Field theory: algebraic field extensions, straightedge and compass constructions. 3 credit hours

Completeness of the real number system, basic topological properties, compactness, sequences and series, absolute convergence of series, rearrangement of series, properties of continuous functions, the Riemann-Stieltjes integral, sequences and series of functions, uniform convergence, the Stone-Weierstrass theorem, equicontinuity, and the Arzela-Ascoli theorem. 3 credit hours

Calculus of functions of several variables and of vector fields in orthogonal coordinate systems. Optimization problems, implicit function theorem, Green's theorem, Stokes' theorem, divergence theorems. Applications to engineering and the physical sciences. Not open to students with credit in MA 36200 or 41000. 3 credit hours

**MA 51100 - Linear Algebra With Applications**.

Real and complex vector spaces; linear transformations; Gram-Schmidt process and projections; least squares; QR and LU factorization; diagonalization, real and complex spectral theorem; Schur triangular form; Jordan canonical form; quadratic forms. 3 credit hours

**MA 51400 - Numerical Analysis**.

(CS 51400) Iterative methods for solving nonlinear; linear difference equations, applications to solution of polynomial equations; differentiation and integration formulas; numerical solution of ordinary differential equations; roundoff error bounds. 3 credit hours

**MA 51800 - Advanced Discrete Mathematics**.

The course covers mathematics useful in analyzing computer algorithms. Topics include recurrence relations, evaluation of sums, integer functions, elementary number theory, binomial coefficients, generating functions, discrete probability, and asymptotic methods. 3 credit hours

**MA 51900 - Introduction To Probability**.

(STAT 51900) Algebra of sets, sample spaces, combinatorial problems, independence, random variables, distribution functions, moment generating functions, special continuous and discrete distributions, distribution of a function of a random variable, limit theorems. 3 credit hours

**MA 52000 - Boundary Value Problems Of Differential Equations**.

Separation of variables; Fourier series; boundary value problems; Fourier transforms; Bessel functions; Legendre polynomials. 3 credit hours

**MA 52100 - Introduction To Optimization Problems**.

Necessary and sufficient conditions for local extrema in programming problems and in the calculus of variations. Control problems; statement of maximum principles and applications. Discrete control problems. 3 credit hours

**MA 52300 - Introduction To Partial Differential Equations**.

First order quasi-linear equations and their applications to physical and social sciences; the Cauchy-Kovalevsky theorem; characteristics, classification and canonical forms of linear equations; equations of mathematical physics; study of Laplace, wave and heat equations; methods of solution. 3 credit hours

**MA 52500 - Introduction To Complex Analysis**.

Complex numbers and complex-valued functions of one complex variable; differentiation and contour integration; Cauchy's theorem; Taylor and Laurent series; residues; conformal mapping; applications. Not open to students with credit in MA 42500. 3 credit hours

**MA 52700 - Advanced Mathematics For Engineers And Physicists I**.

MA 52700 is not a prerequisite for MA 52800; these courses can be taken independently. Topics in MA 52700 include linear algebra, systems of ordinary differential equations, Laplace transforms, Fourier series and transforms, and partial differential equations. MA 51100 is recommended. 3 credit hours

**MA 52800 - Advanced Mathematics For Engineers And Physicists II**.

MA 52700 is not a prerequisite for MA 52800; these courses can be taken independently. Topics in MA 52800 include divergence theorem, Stokes theorem, complex variables, contour integration, calculus of residues and applications, conformal mapping, and potential theory. MA 51000 is recommended. 3 credit hours

**MA 53000 - Functions Of A Complex Variable I**.

Complex numbers and complex-valued functions of one complex variable; differentiation and contour integration; Cauchy's theorem; Taylor and Laurent series; residues; conformal mapping; special topics. More mathematically rigorous than MA 52500. 3 credit hours

**MA 53100 - Functions Of A Complex Variable II**.

Advanced topics. 3 credit hours

**MA 53200 - Elements Of Stochastic Processes**.

(STAT 53200) A basic course in stochastic models, including discrete and continuous time Markov chains and Brownian motion, as well as an introduction to topics such as Gaussian processes, queues, epidemic models, branching processes, renewal processes, replacement, and reliability problems. 3 credit hours

**MA 53800 - Probability Theory I**.

(STAT 53800) Mathematically rigorous, measure-theoretic introduction to probability spaces, random variables, independence, weak and strong laws of large numbers, conditional expectations, and martingales. 3 credit hours

**MA 53900 - Probability Theory II**.

(STAT 53900) Convergence of probability laws; characteristic functions; convergence to the normal law; infinitely divisible and stable laws; Brownian motion and the invariance principle. 3 credit hours

**MA 54200 - Theory Of Distributions And Applications**.

Definition and basic properties of distributions; convolution and Fourier transforms; applications to partial differential equations; Sobolev spaces. 3 credit hours

**MA 54300 - Ordinary Differential Equations And Dynamical Systems**.

This course focuses on the theory of ordinary differential equations and methods of proof for developing this theory. Topics include basic results for linear systems, the local theory for nonlinear systems (existence and uniqueness, dependence on parameters, flows and linearization, stable manifold theorem) and the global theory for nonlinear systems (global existence, limit sets and periodic orbits, Poincare maps). Permission of instructor required. 3 credit hours

**MA 54400 - Real Analysis And Measure Theory**.

Metric space topology; continuity, convergence; equicontinuity; compactness; bounded variation, Helly selection theorem; Riemann-Stieltjes integral; Lebesgue measure; abstract measure spaces; LP-spaces; Holder and Minkowski inequalities; Riesz-Fischer theorem. 3 credit hours

**MA 54500 - Functions Of Several Variables And Related Topics**.

Differentiation of functions; Besicovitch covering theorem; differentiation of one measure with respect to another; Hardy-Littlewood maximal function; functions of several variables; Sobolev spaces. 3 credit hours

**MA 54600 - Introduction To Functional Analysis**.

Fundamentals of functional analysis. Banach spaces, Hahn-Banach theorem. Principle of uniform boundedness. Closed graph and open mapping theorems. Applications. Hilbert spaces. Orthonormal sets. Spectral theorem for Hermitian operators and compact operators. 3 credit hours

**MA 55300 - Introduction To Abstract Algebra**.

Group theory: Sylow theorems, Jordan-Holder theorem, solvable groups. Ring theory: unique factorization in polynomial rings and principal ideal domains. Field theory: ruler and compass constructions, roots of unity, finite fields, Galois theory, solvability of equations by radicals. 3 credit hours

Review of basics: vector spaces, dimension, linear maps, matrices determinants, linear equations. Bilinear forms; inner product spaces; spectral theory; eigenvalues. Modules over a principal ideal domain; finitely generated abelian groups; Jordan and rational canonical forms for a linear transformation. 3 credit hours

**MA 55600 - Introduction To The Theory Of Numbers**.

Divisibility, congruences, quadratic residues, Diophantine equations, the sequence of primes. 3 credit hours

**MA 55700 - Abstract Algebra I**.

Review of fundamental structures of algebra (groups, rings, fields, modules, algebras); Jordan-Holder and Sylow theorems; Galois theory; bilinear forms; modules over principal ideal domains; Artinian rings and semisimple modules. Polynomial and power series rings; Noetherian rings and modules; localization; integral dependence; rudiments of algebraic geometry and algebraic number theory; ramification theory. 3 credit hours

**MA 55800 - Abstract Algebra II**.

A continuation of MA 55700. 3 credit hours

**MA 56000 - Fundamental Concepts Of Geometry**.

Foundations of Euclidean geometry, including a critique of Euclid's "Elements" and a detailed study of an axiom system such as that of Hilbert. Independence of the parallel axiom and introduction to non-Euclidean geometry. 3 credit hours

**MA 56200 - Introduction To Differential Geometry And Topology**.

Smooth manifolds; tangent vectors; inverse and implicit function theorems; submanifolds; vector fields; integral curves; differential forms; the exterior derivative; DeRham cohomology groups; surfaces in E3., Gaussian curvature; two dimensional Riemannian geometry; Gauss-Bonnet and Poincare theorems on vector fields. 3 credit hours

**MA 57100 - Elementary Topology**.

Fundamentals of point set topology with a brief introduction to the fundamental group and related topics, topological and metric spaces, compactness, connectedness, separation properties, local compactness, introduction to function spaces, basic notions involving deformations of continuous paths. 3 credit hours

**MA 57200 - Introduction In Algebraic Topology**.

Singular homology theory; Eilenberg-Steenrod axioms; simplicial and cell complexes; elementary homotopy theory; Lefschetz fixed point theorem. 3 credit hours

**MA 57300 - Numerical Solution Of Ordinary Differential Equations**.

Numerical solution of initial-value problems by Runge-Kutta methods, general one-step methods, and multistep methods; analysis of truncation error, discretization error, and rounding error; stability of multistep methods; numerical solution of boundary- and eigen-value problems by initial-value techniques and finite difference methods. 3 credit hours

**MA 57400 - Numerical Optimization**.

Convex optimization algorithms using modern large-scale algorithms for convex optimization, with a heavy emphasis on analysis including monotone operator, fixed point iteration and duality in splitting methods. The course will cover and focus on the following three parts: smooth optimization algorithms, nonsmooth convex optimization algorithms, and stochastic and randomized algorithms. Permission of department required. Prerequisites: MA 51100 and MA 50400. 3 credit hours

Introduction to graph theory with applications. 3 credit hours

**MA 58400 - Algebraic Number Theory**.

Dedekind domains, norm, discriminant, different, finiteness of class number, Dirichlet unit theorem, quadratic and cyclotomic extensions, quadratic reciprocity, decomposition and inertia groups, completions and local fields. 3 credit hours

**MA 58500 - Mathematical Logic I**.

Propositional and predicate calculus; the Godel completeness and compactness theorem, primitive recursive and recursive functions; the Godel incompleteness theorem; Tarski's theorem; Church's theorem; recursive undecidability; special topics such as nonstandard analysis. 3 credit hours

**MA 59000 - Topics In Mathematics**.

Supervised reading courses as well as dual-level special topics courses are given under this number. Permission of instructor required. 0 to 5 credit hours

**MA 59500 - Topics In Mathematics**.

Special topics courses including dual-level special topics. Permission of instructor required. 1 to 5 credit hours

**MA 59800 - Topics In Mathematics**.

Supervised reading courses as well as dual-level special topics courses are given under this number. Permission of instructor required. 1 to 5 credit hours

**MA 61100 - Methods Of Applied Mathematics I**.

Banach and Hilbert spaces; linear operators; spectral theory of compact linear operators; applications to linear integral equations and to regular Sturm-Liouville problems for ordinary differential equations. Prerequisite: MA 51100, 54400. 3 credit hours

**MA 61500 - Numerical Methods For Partial Differential Equations I**.

(CS 615) Finite element method for elliptic partial differential equations; weak formulation; finite-dimensional approximations; error bounds; algorithmic issues; solving sparse linear systems; finite element method for parabolic partial differential equations; backward difference and Crank-Nicholson time-stepping; introduction to finite difference methods for elliptic, parabolic, and hyperbolic equations; stability, consistency, and convergence; discrete maximum principles. Prerequisite: MA 51400, 52300. 3 credit hours

**MA 62000 - Mathematical Theory Of Optimal Control**.

Existence theorems; the maximum principle; relationship to the calculus of variations; linear systems with quadratic criteria; applications. Offered in alternate years. Prerequisite: MA 54400. 3 credit hours

**MA 63100 - Several Complex Variables**.

Power series, holomorphic functions, representation by integrals, extension of functions, holomorphically convex domains. Local theory of analytic sets (Weierstrass preparation theorem and consequences). Functions and sets in the projective space Pn (theorems of Weierstrass and Chow and their extensions). Prerequisite: MA 53000. 3 credit hours

**MA 63800 - Stochastic Processes I**.

(STAT 638) Advanced topics in probability theory which may include stationary processes, independent increment processes, Gaussian processes; martingales, Markov processes, ergodic theory. Prerequisite: MA 53900. 3 credit hours

**MA 63900 - Stochastic Process II**.

(STAT 63900) Continuation of MA 63800. 3 credit hours

**MA 64200 - Methods Of Linear And Nonlinear Partial Differential Equations I**.

Second order elliptic equations including maximum principles, Harnack inequality, Schauder estimates, and Sobolev estimates. Applications of linear theory to nonlinear equations. Prerequisite: MA 52300. 3 credit hours

**MA 64300 - Methods Of Partial Differential Equations II**.

Continuation of MA 642. Topics to be covered are Lp theory for solutions of elliptic equations, including Moser's estimates, Aleksandrov maximum principle, and the Calderon-Zygmund theory. Introduction to evolution problems for parabolic and hyperbolic equations, including Galerkin approximation and semigroup methods. Applications to nonlinear problems. Prerequisite: MA 64200. 3 credit hours

**MA 64400 - Calculus Of Variations**.

Direct methods; necessary and sufficient conditions for lower semicontinuity of multiple integrals; existence theorems and connections with optimal control theory. Prerequisite: MA 54400. 3 credit hours

**MA 65000 - Commutative Algebra**.

The study of those rings of importance in algebraic and analytic geometry and algebraic number theory. Prerequisite: MA 55800. 3 credit hours

**MA 66100 - Modern Differential Geometry**.

Topics chosen by the instructor. Prerequisite: MA 54400, 55400. 3 credit hours

**MA 66300 - Algebraic Curves And Functions I**.

Algebraic functions of one variable from the geometric, algebraic, or function-theoretic points of view. Riemann-Roch theorem, differentials. Prerequisite: MA 55800. 3 credit hours

**MA 66400 - Algebraic Curves And Functions II**.

Continuation of MA 663. Topics chosen by the instructor. Prerequisite: MA 66300. 3 credit hours

**MA 66500 - Algebraic Geometry**.

Topics of current interest will be chosen by the instructor. Prerequisite: MA 65000 or 66300. 3 credit hours

**MA 68400 - Class Field Theory**.

Ideles, adeles, L-functions, Artin symbol, reciprocity, local and global class fields, Kronecker-Weber Theorem. Prerequisite: MA 58400. 3 credit hours

Topics vary. Permission of instructor required. 1 to 3 credit hours

**MA 69200 - Topics Applied Math**.

Topics in applied math. Permission of instructor required. 1 to 3 credit hours

**MA 69300 - Topics In Analysis**.

Topics in analysis. Permission of instructor required. 1 to 3 credit hours

**MA 69400 - Topics In Differential Equations**.

Topics In Differential Equations. Permission of instructor required. 1 to 3 credit hours

**MA 69600 - Topics In Geometry**.

Topics in geometry. Permission of instructor required. 1 to 3 credit hours

**MA 69700 - Topics In Topology**.

Topics in topology. Permission of instructor required. 1 to 3 credit hours

**MA 69900 - Research PhD Thesis**.

Research PhD Thesis. Permission of instructor required. 1 to 18 credit hours