# Mathematical Sciences

## Undergraduate Courses

### BCB 4004. Statistical Methods in Genetics and Bioinformatics

Cat II (offered at least every other Year).

This course provides students with knowledge and understanding of the applications of statistics in modern genetics and bioinformatics. The course generally covers population genetics, genetic epidemiology, and statistical models in bioinformatics. Specific topics include meiosis modeling, stochastic models for recombination, linkage and association studies (parametric vs. nonparametric models, family-based vs. population-based models) for mapping genes of qualitative and quantitative traits, gene expression data analysis, DNA and protein sequence analysis, and molecular evolution. Statistical approaches include log-likelihood ratio tests, score tests, generalized linear models, EM algorithm, Markov chain Monte Carlo, hidden Markov model, and classification and regression trees. This course will be offered in 2021-22, and in alternating years thereafter.

### CS 2022. Discrete Mathematics

Cat I (offered at least 1x per Year).

This course serves as an introduction to some of the more important concepts, techniques, and structures of discrete mathematics providing a bridge between computer science and mathematics. Topics include sets, functions and relations, propositional and predicate calculus, mathematical induction, properties of integers, counting techniques, and graph theory. Students will be expected to develop simple proofs for problems drawn primarily from computer science and applied mathematics.

### CS 4032. Numerical Methods for Linear and Nonlinear Systems

Cat I (offered at least 1x per Year).

This course provides an introduction to modern computational methods for linear and nonlinear equations and systems and their applications. Topics covered include solution of nonlinear scalar equations, direct and iterative algorithms for the solution of systems of linear equations, solution of nonlinear systems, and the eigenvalue problem for matrices. Error analysis will be emphasized throughout.

### CS 4033. Numerical Methods for Calculus and Differential Equations

Cat I (offered at least 1x per Year).

This course provides an introduction to modern computational methods for differential and integral calculus and differential equations. Topics covered include interpolation and polynomial approximation, approximation theory, numerical differentiation and integration, and numerical solutions of ordinary differential equations. Error analysis will be emphasized throughout.

### DS 4635. Data Analytics and Statistical Learning

Cat I (offered at least 1x per Year).

The focus of this class will be on statistical learning - the intersection of applied statistics and modeling techniques used to analyze and to make predictions and inferences from complex real-world data. Topics covered include: regression; classification/clustering; sampling methods (bootstrap and cross validation); and decision tree learning. Students may not receive credit for both MA 463X and MA 4635.

### MA 1020. Calculus I with Preliminary Topics

This course is a 14-week alternative to the 7-week MA 1021. This course offers 1/3 unit of credit. It is designed for students looking to develop essential skills in algebra and trigonometry or strengthen their mathematical background. It provides a review of selected topics from algebra, trigonometry, and analytic geometry, then continues on to provide an introduction to differentiation and its applications. Topics covered include: trigonometry, conic sections; functions, their graphs, and inverses; limits, continuity, and differentiation; linear approximation; chain rule; and applications of derivatives such as min/max problems. Although the course will make use of computers, no programming experience is assumed. Students may not receive credit for both MA 1020 and MA 1021.

### MA 1021. Calculus I

Cat I (offered at least 1x per Year).

This course provides an introduction to differentiation and its applications. Topics covered include: functions and their graphs, limits, continuity, differentiation, linear approximation, chain rule, min/max problems, and applications of derivatives. Students may not receive credit for both MA 1021 and MA 1020.

### MA 1022. Calculus II

Cat I (offered at least 1x per Year).

This course provides an introduction to integration and its applications. Topics covered include: inverse trigonometric functions, Riemann sums, fundamental theorem of calculus, basic techniques of integration, volumes of revolution, arc length, exponential and logarithmic functions, and applications of integration to engineering.

### MA 1023. Calculus III

Cat I (offered at least 1x per Year).

This course provides an introduction to series, parametric curves and vector algebra. Topics covered include: numerical methods, indeterminate forms, improper integrals, sequences, Taylors theorem with remainder, convergence of series and power series, polar coordinates, parametric curves and vector algebra.

### MA 1024. Calculus IV

Cat I (offered at least 1x per Year).

This course provides an introduction to multivariable calculus. Topics covered include: vector functions, partial derivatives and gradient, multivariable optimization, double and triple integrals, polar coordinates, other coordinate systems and applications.

### MA 1033. Theoretical Calculus III

Cat I (offered at least 1x per Year).

This course will cover the same material as MA 1023 Calculus III but from a different perspective. A more rigorous study of sequences and series will be undertaken: starting from the least upper bound property in R, the fundamental theorems for convergent series will be proved. Convergence criteria for series will be rigorously justified and LHospitals rule will be introduced and proved. Homework problems will include a blend of computational exercises as usually assigned in MA 1023 Calculus III and problems with a stronger theoretical flavor. Note: Students can receive credit for this class and MA 1023 Calculus III.

### MA 1034. Theoretical Calculus IV

Cat I (offered at least 1x per Year).

This course will cover the same material as MA 1024 Calculus IV from a more mathematically rigorous perspective. The course gives a rigorous introduction of differentiation and integration for functions of one variable. After introducing vector functions, differentiation and integration will be extended to functions of several variables. Note: Students can receive credit for this class and MA 1024 Calculus IV.

### MA 1120. Calculus II (Semester Version)

Cat I (offered at least 1x per Year).

This course is a 14-week alternative to the 7-week MA 1022. This course offers 1/3 unit of credit. It is designed for students who would benefit from additional contact hours and wish to strengthen their mathematical background. Topics covered include: inverse trigonometric functions, Riemann sums, fundamental theorem of calculus, basic techniques of integration, volumes of revolution, arc length, exponential and logarithmic functions, and applications of integration to engineering. The 14-week framework allows for an in-depth study of many of these topics as well as an introduction to some MA 1023 topics such as arithmetic and geometric sequences and series.

### MA 1801. Denksport

Cat I (offered at least 1x per Year).

Problem solving is a fundamental mathematical skill. In this course students will be exposed to problems coming from a wide range of mathematical disciplines; and will work together in a collaborative environment to explore potential solutions. Discussion problems may be inspired by the research of faculty leading the discussion, by past mathematical competitions (such as the Putnam Competition) or elsewhere. This course meets once per week, with an emphasis on discussion and exploration of problems. There will be no exam and no assigned homework. Grading is by participation only. This course may be taken multiple times; content will vary depending on the speakers. Grading for this course will be on a Pass/NR basis.

### MA 1971. Bridge to Higher Mathematics

Cat I (offered at least 1x per Year).

The principal aim of this course is to practice mathematical problem interpretation, proof techniques, and question formulation. The course is intended not only for beginning students in the mathematical sciences, but also for all students interested in mathematical art and rigor. Students in the course will be expected to explain, justify, defend, disprove, conjecture and verify mathematical statements, both orally and in writing, in order to develop proof-writing skills. (These skills should prove useful in more advanced mathematics courses). Topics covered include basic logic; basic set theory; definitions and properties of functions; definitions and properties of binary relations; fundamental proof techniques, including proof by induction. Depending on student background and instructor preferences, the course objectives may be conveyed through a selection of problems from various mathematical sub-disciplines, through discussions of current events in the mathematical sciences, including recently solved problems and open challenges facing todays scientists, or through discussions of applications of mathematics.

### MA 2051. Ordinary Differential Equations

Cat I (offered at least 1x per Year).

This course develops techniques for solving ordinary differential equations. Topics covered include: introduction to modeling using first-order differential equations, solution methods for linear higher-order equations, qualitative behavior of nonlinear first-order equations, oscillatory phenomena including spring-mass system and RLC-circuits and Laplace transform. Additional topics may be chosen from power series method, methods for solving systems of equations and numerical methods for solving ordinary differential equations.

### MA 2071. Matrices and Linear Algebra I

Cat I (offered at least 1x per Year).

This course provides an introduction to the theory and techniques of matrix algebra and linear algebra. Topics covered include: operations on matrices, systems of linear equations, linear transformations, determinants, eigenvalues and eigenvectors, least squares, vector spaces, inner products, introduction to numerical techniques, and applications of linear algebra. Credit may not be earned for this course and MA 2072.

### MA 2072. Accelerated Matrices and Linear Algebra I

Cat I (offered at least 1x per Year).

This course provides an accelerated introduction to the theory and techniques of matrix algebra and linear algebra, aimed at Mathematical Sciences majors and others interested in advanced concepts of linear algebra. Topics covered include: matrix algebra, systems of linear equations, linear transformations, determinants, eigenvalues and eigenvectors, the method of least squares, vector spaces, inner products, non-square matrices and singular value decompositions. Students will be exposed to computational and numerical techniques, and to applications of linear algebra, particularly to Data Science. Credit may not be earned for this course and MA 2071.

### MA 2073. Matrices and Linear Algebra II

Cat I (offered at least 1x per Year).

This course provides a deeper understanding of topics introduced in MA 2071, and continues the development of linear algebra. Topics covered include: abstract vector spaces, linear transformations, matrix representations of a linear transformation, determinants, characteristic and minimal polynomials, diagonalization, eigenvalues and eigenvectors, the matrix exponential, inner product spaces. This course is designed primarily for Mathematical Science majors and those interested in the deeper mathematical issues underlying linear algebra.

### MA 2210. Mathematical Methods in Decision Making

Cat I (offered at least 1x per Year).

This course introduces students to the principles of decision theory as applied to the planning, design and management of complex projects. It will be useful to students in all areas of engineering, actuarial mathematics as well as those in such interdisciplinary areas as environmental studies. It emphasizes quantitative, analytic approaches to decision making using the tools of applied mathematics, operations research, probability and computations. Topics covered include: the systems approach, mathematical modeling, optimization and decision analyses. Case studies from various areas of engineering or actuarial mathematics are used to illustrate applications of the materials covered in this course.

### MA 2211. Theory of Interest I

Cat I (offered at least 1x per Year).

An introduction to actuarial mathematics is provided for those who may be interested in the actuarial profession. Topics usually included are: measurement of interest, including accumulated and present value factors; annuities certain; amortization schedules and sinking funds; and bonds.

### MA 2212. Theory of Interest II

Cat I (offered at least 1x per Year).

This course covers topics in fixed income securities. Topics are chosen to cover the mechanics and pricing of modern-day fixed income products and can include: yield curve theories; forward rates; interest rate swaps; credit-default swaps; bonds with credit risk and options; bond duration and convexity; bond portfolio construction; asset-backed securities, including collateralized debt obligations and mortgage-backed securities with prepayment risk; asset-liability hedging; applications of binomial interest rate trees.

### MA 2251. Vector and Tensor Calculus

Cat I (offered at least 1x per Year).

This course provides an introduction to tensor and vector calculus, an essential tool for applied mathematicians, scientists, and engineers. Topics covered include: scalar and vector functions and fields, tensors, basic differential operations for vectors and tensors, line and surface integrals, change of variable theorem in integration, integral theorems of vector and tensor calculus. The theory will be illustrated by applications to areas such as electrostatics, theory of heat, electromagnetics, elasticity and fluid mechanics.

### MA 2271. Graph Theory

Cat II (offered at least every other Year).

This course introduces the concepts and techniques of graph theorya part of mathematics finding increasing application to diverse areas such as management, computer science and electrical engineering. Topics covered include: graphs and digraphs, paths and circuits, graph and digraph algorithms, trees, cliques, planarity, duality and colorability. This course is designed primarily for Mathematical Science majors and those interested in the deeper mathematical issues underlying graph theory. Undergraduate credit may not be earned both for this course and for MA 3271. This course will be offered in 2022-23, and in alternating years thereafter.

### MA 2273. Combinatorics

Cat II (offered at least every other Year).

This course introduces the concepts and techniques of combinatorics a part of mathematics with applications in computer science and in the social, biological, and physical sciences. Emphasis will be given to problem solving. Topics will be selected from: basic counting methods, inclusion-exclusion principle, generating functions, recurrence relations, systems of distinct representatives, combinatorial designs, combinatorial algorithms and applications of combinatorics. This course is designed primarily for Mathematical Sciences majors and those interested in the deeper mathematical issues underlying combinatorics. Undergraduate credit may not be earned both for this course and for MA 3273. This course will be offered in 2021-22, and in alternating years thereafter.

### MA 2431. Mathematical Modeling with Ordinary Differential Equations

Cat I (offered at least 1x per Year).

This course focuses on the principles of building mathematical models from a physical, chemical or biological system and interpreting the results. Students will learn how to construct a mathematical model and will be able to interpret solutions of this model in terms of the context of the application. Mathematical topics focus on solving systems of ordinary differential equations, and may include the use of stability theory and phase-plane analysis. Applications will be chosen from electrical and mechanical oscillations, control theory, ecological or epidemiological models and reaction kinetics. This course is designed primarily for students interested in the deeper mathematical issues underlying mathematical modeling. Students may be required to use programming languages such as Matlab or Maple to further investigate different models.

### MA 2610. Applied Statistics for the Life Sciences

Cat I (offered at least 1x per Year).

This course is designed to introduce the student to statistical methods and concepts commonly used in the life sciences. Emphasis will be on the practical aspects of statistical design and analysis with examples drawn exclusively from the life sciences, and students will collect and analyze data. Topics covered include analytic and graphical and numerical summary measures, probability models for sampling distributions, the central limit theorem, and one and two sample point and interval estimation, parametric and non-parametric hypothesis testing, principles of experimental design, comparisons of paired samples and categorical data analysis. Undergraduate credit may not be earned for both this course and for MA 2611.

### MA 2611. Applied Statistics I

Cat I (offered at least 1x per Year).

This course is designed to introduce the student to data analytic and applied statistical methods commonly used in industrial and scientific applications as well as in course and project work at WPI. Emphasis will be on the practical aspects of statistics with students analyzing real data sets on an interactive computer package. Topics covered include analytic and graphical representation of data, exploratory data analysis, basic issues in the design and conduct of experimental and observational studies, the central limit theorem, one and two sample point and interval estimation and tests of hypotheses.

### MA 2621. Probability for Applications

Cat I (offered at least 1x per Year).

This course is designed to introduce the student to probability. Topics to be covered are: basic probability theory including Bayes theorem; discrete and continuous random variables; special distributions including the Bernoulli, Binomial, Geometric, Poisson, Uniform, Normal, Exponential, Chi-square, Gamma, Weibull, and Beta distributions; multivariate distributions; conditional and marginal distributions; independence; expectation; transformations of univariate random variables.

### MA 2631. Probability Theory

Cat I (offered at least 1x per Year).

The purpose of this course is twofold: To introduce fundamental ideas and methods of mathematics using the study of probability as the vehicle. These ideas and methods may include systematic theorem-proof development starting with basic axioms; mathematical induction; set theory; applications of univariate and multivariate calculus. To introduce the student to probability. Topics to be covered will be chosen from: axiomatic development of probability; independence; Bayes theorem; discrete and continuous random variables; expectation; special distributions including the binomial and normal; moment generating functions; multi-variate distributions; conditional and marginal distributions; independence of random variables; transformations of random variables; limit theorems. This course is designed primarily for Mathematical Sciences majors and those interested in the deeper mathematical issues underlying probability theory. A more applications-oriented course with similar content is MA 2621 Probability for Applications which is primarily designed for students in departments other than Mathematical Sciences. Undergraduate credit may not be earned both for this course and for MA 2621 Probability for Applications.

### MA 3212. Actuarial Mathematics I

Cat I (offered at least 1x per Year).

A study of actuarial mathematics with emphasis on the theory and application of contingency mathematics in various areas of insurance. Topics usually included are: survival functions and life tables; life insurance; property insurance; annuities; net premiums; and premium reserves.

### MA 3213. Actuarial Mathematics II

Cat I (offered at least 1x per Year).

A continuation of the study of actuarial mathematics with emphasis on calculations in various areas of insurance, based on multiple insureds, multiple decrements, and multiple state models. Topics usually included are: survival functions; life insurance; property insurance; common shock; Poisson processes and their application to insurance settings; gross premiums; and reserves.

### MA 3231. Linear Programming

Cat I (offered at least 1x per Year).

The mathematical subject of linear programming deals with those problems in optimal resource allocation which can be modeled by a linear profit (or cost) function together with feasibility constraints expressible as linear inequalities. Such problems arise regularly in many industries, ranging from manufacturing to transportation, from the design of livestock diets to the construction of investment portfolios. This course considers the formulation of such real-world optimization problems as linear programming problems, the most important algorithms for their solution, and techniques for their analysis. The core material includes problem formulation, the primal and dual simplex algorithms, and duality theory. Further topics may include: sensitivity analysis; applications such as matrix games or network flow models; bounded variable linear programs; interior point methods.

### MA 3233. Discrete Optimization

Cat II (offered at least every other Year).

Discrete optimization is a lively field of applied mathematics in which techniques from combinatorics, linear programming, and the theory of algorithms are used to solve optimization problems over discrete structures, such as networks or graphs. The course will emphasize algorithmic solutions to general problems, their complexity, and their application to real-world problems drawn from such areas as VLSI design, telecommunications, airline crew scheduling, and product distribution. Topics will be selected from: Network flow, optimal matching, integrality of polyhedra, matroids, and NP-completeness. This course will be offered in 2022-23, and in alternating years thereafter.

### MA 3457. Numerical Methods for Calculus and Differential Equations

Cat I (offered at least 1x per Year).

This course provides an introduction to modern computational methods for differential and integral calculus and differential equations. Topics covered include: interpolation and polynomial approximation, approximation theory, numerical differentiation and integration, numerical solutions of ordinary differential equations. Error analysis will be emphasized throughout.

### MA 3471. Advanced Ordinary Differential Equations

Cat II (offered at least every other Year).

The first part of the course will cover existence and uniqueness of solutions, continuous dependence of solutions on parameters and initial conditions, maximal interval of existence of solutions, Gronwalls inequality, linear systems and the variation of constants formula, Floquet theory, stability of linear and perturbed linear systems. The second part of the course will cover material selected by the instructor. Possible topics include: Introduction to dynamical systems, stability by Lyapunovs direct method, study of periodic solutions, singular perturbation theory and nonlinear oscillation theory. This course will be offered in 2021-22, and in alternating years thereafter.

### MA 3475. Calculus of Variations

Cat II (offered at least every other Year).

This course covers the calculus of variations and select topics from optimal control theory. The purpose of the course is to expose students to mathematical concepts and techniques needed to handle various problems of design encountered in many fields, e. g. electrical engineering, structural mechanics and manufacturing. Topics covered will include: derivation of the necessary conditions of a minimum for simple variational problems and problems with constraints, variational principles of mechanics and physics, direct methods of minimization of functions, Pontryagins maximum principle in the theory of optimal control and elements of dynamic programming. This course will be offered in 2022-23, and in alternating years thereafter.

### MA 3627. Introduction to the Design and Analysis of Experiments

Cat II (offered at least every other Year).

This course will teach students how to design experiments in order to collect meaningful data for analysis and decision making. This course continues the exploration of statistics for scientific and industrial applications begun in MA 2611 and MA 2612. The course offers comprehensive coverage of the key elements of experimental design used by applied researchers to solve problems in the field, such as random assignment, replication, blocking, and confounding. Topics covered include the design and analysis of general factorial experiments; two-level factorial and fractional factorial experiments; principles of design; completely randomized designs and one-way analysis of variance (ANOVA); complete block designs and two-way analysis of variance; complete factorial experiments; fixed, random, and mixed models; split-plot designs; nested designs. This course will be offered in 2022-23, and in alternating years thereafter.

### MA 3631. Mathematical Statistics

Cat I (offered at least 1x per Year).

This course introduces students to the mathematical principles of statistics. Topics will be chosen from: Sampling distributions, limit theorems, point and interval estimation, sufficiency, completeness, efficiency, consistency; the Rao-Blackwell theorem and the Cramer-Rao bound; minimum variance unbiased estimators and maximum likelihood estimators; tests of hypotheses including the Neyman-Pearson lemma, uniformly most powerful and likelihood radio tests.

### MA 3823. Group Theory

Cat I (offered at least 1x per Year).

This course provides an introduction to one of the major areas of modern algebra. Topics covered include: groups, subgroups, permutation groups, normal subgroups, factor groups, homomorphisms, isomorphisms and the fundamental homomorphism theorem.

### MA 3825. Rings and Fields

Cat II (offered at least every other Year).

This course provides an introduction to one of the major areas of modern algebra. Topics covered include: rings, integral domains, ideals, quotient rings, ring homomorphisms, polynomial rings, polynomial factorization, extension fields and properties of finite fields. This course will be offered in 2021-22, and in alternating years thereafter.

### MA 3831. Principles of Real Analysis I

Cat I (offered at least 1x per Year).

Principles of Real Analysis is a two-part course giving a rigorous presentation of the important concepts of classical real analysis. Topics covered in the sequence include: basic set theory, elementary topology of Euclidean spaces, metric spaces, compactness, limits and continuity, differentiation, Riemann-Stieltjes integration, infinite series, sequences of functions, and topics in multivariate calculus.

### MA 4213. Loss Models I - Risk Theory

Cat II (offered at least every other Year).

This course covers topics in loss models and risk theory as it is applied, under specified assumptions, to insurance. Topics covered include: economics of insurance, short term individual risk models, single period and extended period collective loss models, and applications.

### MA 4214. Loss Models II - Survival Models

Cat II (offered at least every other Year).

Survival models are statistical models of times to occurrence of some event. They are widely used in areas such as the life sciences and actuarial science (where they model such events as time to death, or to the development or recurrence of a disease), and engineering (where they model the reliability or useful life of products or processes). This course introduces the nature and properties of survival models, and considers techniques for estimation and testing of such models using realistic data. Topics covered will be chosen from: parametric and nonparametric survival models, censoring and truncation, nonparametric estimation (including confidence intervals and hypothesis testing) using right-, left-, and otherwise censored or truncated data.

### MA 4216. Actuarial Seminar

Cat I (offered at least 1x per Year).

This pass/fail graduation requirement will be offered every term, under the supervision of the actuarial professors. In order to receive a passing grade, students will need to complete some or all of the following: attend speaker talks, attend company visits to campus, take part and help out with Actuarial Club activities, prepare for actuarial exams, or complete other activities as approved by the instructor(s).

### MA 4222. Top Algorithms in Applied Mathematics

Cat II (offered at least every other Year).

This course will introduce students to the top algorithms in applied mathematics. These algorithms have tremendous impact on the development and practice of modern science and engineering. Class discussions will focus on introducing students to the mathematical theory behind the algorithms as well as their applications. In particular, the course will address issues of computational efficiency, implementation, and error analysis. Algorithms to be considered may include the Krylov Subspace Methods, Fast Multipole Method, Monte Carlo Methods, Fast Fourier Transform, Kalman Filters and Singular Value Decomposition. Students will be expected to apply these algorithms to real-world problems; e.g., image processing and audio compression (Fast Fourier Transform), recommendation systems (Singular Value Decomposition), electromagnetics or fluid dynamics (Fast Multipole Method, Krylov Subspace Methods, and Fast Fourier Transform), and the tracking and prediction of an objects position (Kalman Filters). In addition to studying these algorithms, students will learn about high performance computing and will have access to a machine with parallel and GPU capabilities to run code for applications with large data sets. This course will be offered in 2021-22, and in alternating years thereafter.

### MA 4235. Mathematical Optimization

Cat II (offered at least every other Year).

This course explores theoretical conditions for the existence of solutions and effective computational procedures to find these solutions for optimization problems involving nonlinear functions. Topics covered include: classical optimization techniques, Lagrange multipliers and Kuhn-Tucker theory, duality in nonlinear programming, and algorithms for constrained and unconstrained problems. This course will be offered in 2021-22, and in alternating years thereafter.

### MA 4237. Probabilistic Methods in Operations Research

Cat II (offered at least every other Year).

This course develops probabilistic methods useful to planners and decision makers in such areas as strategic planning, service facilities design, and failure of complex systems. Topics covered include: decisions theory, inventory theory, queuing theory, reliability theory, and simulation. This course will be offered in 2021-22, and in alternating years thereafter.

### MA 4291. Applied Complex Variables

Cat I (offered at least 1x per Year).

This course provides an introduction to the ideas and techniques of complex analysis that are frequently used by scientists and engineers. The presentation will follow a middle ground between rigor and intuition. Topics covered include: complex numbers, analytic functions, Taylor and Laurent expansions, Cauchy integral theorem, residue theory, and conformal mappings.

### MA 4411. Numerical Analysis of Differential Equations

Cat II (offered at least every other Year).

This course is concerned with the development and analysis of numerical methods for differential equations. Topics covered include: well-posedness of initial value problems, analysis of Eulers method, local and global truncation error, Runge-Kutta methods, higher order equations and systems of equations, convergence and stability analysis of one-step methods, multistep methods, methods for stiff differential equations and absolute stability, introduction to methods for partial differential equations. This course will be offered in 2022-23, and in alternating years thereafter.

### MA 4451. Boundary Value Problems

Cat I (offered at least 1x per Year).

Science and engineering majors often encounter partial differential equations in the study of heat flow, vibrations, electric circuits and similar areas. Solution techniques for these types of problems will be emphasized in this course. Topics covered include: derivation of partial differential equations as models of prototype problems in the areas mentioned above, Fourier Series, solution of linear partial differential equations by separation of variables, Fourier integrals and a study of Bessel functions.

### MA 4473. Partial Differential Equations

Cat II (offered at least every other Year).

The first part of the course will cover the following topics: classification of partial differential equations, solving single first order equations by the method of characteristics, solutions of Laplaces and Poissons equations including the construction of Greens function, solutions of the heat equation including the construction of the fundamental solution, maximum principles for elliptic and parabolic equations. For the second part of the course, the instructor may choose to expand on any one of the above topics. This course will be offered in 2022-23, and in alternating years thereafter.

### MA 4631. Probability and Mathematical Statistics I

Cat I (offered at least 1x per Year).

(14 week course) Intended for advanced undergraduates and beginning graduate students in the mathematical sciences and for others intending to pursue the mathematical study of probability and statistics, this course begins by covering the material of MA 3613 at a more advanced level. Additional topics covered are: one-to-one and many-to-one transformations of random variables; sampling distributions; order statistics, limit theorems.

### MA 4632. Probability and Mathematical Statistics II

Cat I (offered at least 1x per Year).

(14 week course) This course is designed to complement MA 4631 and provide background in principles of statistics. Topics covered include: point and interval estimation; sufficiency, completeness, efficiency, consistency; the Rao-Blackwell theorem and the Cramer-Rao bound; minimum variance unbiased estimators, maximum likelihood estimators and Bayes estimators; tests of hypothesis including uniformly most powerful, likelihood ratio, minimax and bayesian tests.

### MA 4892. Topics in Actuarial Mathematics

Cat II (offered at least every other Year).

Topics covered in this course would vary from one offering to the next. The purpose of this course will be to introduce actuarial topics that typically arise in the professional actuarial organizations curriculum beyond the point where aspiring actuaries are still in college. Topics might include ratemaking, estimation of unpaid claims, equity linked insurance products, simulation, or stochastic modeling of insurance products. This course will be offered in 2022-23, and in alternating years thereafter.

### MA 4895. Differential Geometry

Cat II (offered at least every other Year).

The course gives an introduction to differential geometry with a focus on Riemannian geometry. Starting with the geometry of curves and surfaces in the three-dimensional Euclidean space and Riemannian metrics in 2 and higher dimensions, the course introduces the first fundamental form, tangent bundles, vector fields, distance functions and geodesics, followed by covariant derivatives and second fundamental form. The proof of Gausss Theorema Egregium is highlighted. Additional topics are by instructors discretion. Students may not receive credit for both MA 489X and MA 4895.

## Graduate Courses

### BCB 504. Statistical Methods in Genetics and Bioinformatics

This course provides students with knowledge and understanding of the applications of statistics in modern genetics and bioinformatics. The course generally covers population genetics, genetic epidemiology, and statistical models in bioinformatics. Specific topics include meiosis modeling, stochastic models for recombination, linkage and association studies (parametric vs. nonparametric models, family-based vs. population-based models) for mapping genes of qualitative and quantitative traits, gene expression data analysis, DNA and protein sequence analysis, and molecular evolution. Statistical approaches include log-likelihood ratio tests, score tests, generalized linear models, EM algorithm, Markov chain Monte Carlo, hidden Markov model, and classification and regression trees. Students may not receive credit for both BCB 4004 and BCB 504.

### DS 517. Mathematical Foundations for Data Science

The foci of this class are the essential statistics and linear algebra skills required for Data Science students. The class builds the foundation for theoretical and computational abilities of the students to analyze high dimensional data sets. Topics covered include Bayes theorem, the central limit theorem, hypothesis testing, linear equations, linear transformations, matrix algebra, eigenvalues and eigenvectors, and sampling techniques, including Bootstrap and Markov chain Monte Carlo. Students will use these techniques while engaging in hands-on projects with real data.

### MA 500. Basic Real Analysis

This course covers basic set theory, topology of Rn, continuous functions, uniform convergence, compactness, infinite series, theory of differentiation and integration. Other topics covered may include the inverse and implicit function theorems and Riemann-Stieltjes integration. Students may not count both MA 3831 and MA 300 toward their undergraduate degree requirements.

### MA 501. Engineering Mathematics

This course develops mathematical techniques used in the engineering disciplines. Preliminary concepts will be reviewed as necessary, including vector spaces, matrices and eigenvalues. The principal topics covered will include vector calculus, Fourier transforms, fast Fourier transforms and Laplace transformations. Applications of these techniques for the solution of boundary value and initial value problems will be given. The problems treated and solved in this course are typical of those seen in applications and include problems of heat conduction, mechanical vibrations and wave propagation.

### MA 502. Linear Algebra

This course provides an introduction to the theory and methods of applicable linear algebra. The goal is to bring out the fundamental concepts and techniques that underlie and unify the many ways in which linear algebra is used in applications. The course is suitable for students in mathematics and other disciplines who wish to obtain deeper insights into this very important subject than are normally offered in undergraduate courses. It is also intended to provide a foundation for further study in subjects such as numerical linear algebra and functional analysis.

### MA 503. Lebesgue Measure and Integration

This course begins with a review of topics normally covered in undergraduate analysis courses: open, closed and compact sets; liminf and limsup; continuity and uniform convergence. Next the course covers Lebesgue measure in Rn including the Cantor set, the concept of a sigma-algebra, the construction of a nonmeasurable set, measurable functions, semicontinuity, Egorovs and Lusins theorems, and convergence in measure. Next we cover Lebesgue integration, integral convergence theorems (monotone and dominated), Tchebyshevs inequality and Tonellis and Fubinis theorems. Finally Lp spaces are introduced with emphasis on L2 as a Hilbert space. Other related topics will be covered at the instructors discretion.

### MA 504. Functional Analysis

This course will give a comprehensive presentation of fundamental concepts and theorems in Banach and Hilbert spaces. Whenever possible, the theory will be illustrated by examples in Lebesgue spaces. Topics include: The Hahn-Banach theorems, the Uniform Boundedness principle (Banach-Steinhaus Theorem), the Open Mapping and Closed Graph theorems, and weak topologies and convergence. Additional topics will be covered at the instructors discretion.

### MA 505. Complex Analysis

This course will provide a rigorous and thorough-treatment of the theory of functions of one complex variable. The topics to be covered include complex numbers, complex differentiation, the Cauchy-Riemann equations, analytic functions, Cauchys theorem, complex integration, the Cauchy integral formula, Liouvilles theorem, the Gauss mean value theorem, the maximum modulus theorem, Rouches theorem, the Poisson integral formula, Taylor-Laurent expansions, singularity theory, conformal mapping with applications, analytic continuation, Schwarzs reflection principle and elliptic functions.

### MA 508. Mathematical Modeling

This course introduces mathematical modelbuilding using dimensional analysis, perturbation theory and variational principles. Models are selected from the natural and social sciences according to the interests of the instructor and students. Examples are: planetary orbits, spring-mass systems, fluid flow, isomers in organic chemistry, biological competition, biochemical kinetics and physiological flow. Computer simulation of these models will also be considered.

### MA 509. Stochastic Modeling

This course gives students a background in the theory and methods of probability, stochastic processes and statistics for applications. The course begins with a brief review of basic probability, discrete and continuous random variables, expectations, conditional probability and basic statistical inference. Topics covered in greater depth include generating functions, limit theorems, basic stochastic processes, discrete and continuous time Markov chains, and basic queuing theory including M/M/1 and M/G/l queues. This course is offered by special arrangement only, based on expressed student interest.

### MA 510. Numerical Methods

This course provides an introduction to a broad range of modern numerical techniques that are widely used in computational mathematics, science, and engineering. It is suitable for both mathematics majors and students from other departments. It covers introductory-level material for subjects treated in greater depth in MA 512 and MA 514 and also topics not addressed in either of those courses. Subject areas include numerical methods for systems of linear and nonlinear equations, interpolation and approximation, differentiation and integration, and differential equations. Specific topics include basic direct and iterative methods for linear systems; classical rootfinding methods; Newtons method and related methods for nonlinear systems; fixed-point iteration; polynomial, piecewise polynomial, and spline interpolation methods; least-squares approximation; orthogonal functions and approximation; basic techniques for numerical differentiation; numerical integration, including adaptive quadrature; and methods for initial-value problems for ordinary differential equations. Additional topics may be included at the instructors discretion as time permits. Both theory and practice are examined. Error estimates, rates of convergence, and the consequences of finite precision arithmetic are also discussed. Topics from linear algebra and elementary functional analysis will be introduced as needed. These may include norms and inner products, orthogonality and orthogonalization, operators and projections, and the concept of a function space.

### MA 511. Applied Statistics for Engineers and Scientists

This course is an introduction to statistics for graduate students in engineering and the sciences. Topics covered include basic data analysis, issues in the design of studies, an introduction to probability, point and interval estimation and hypothesis testing for means and proportions from one and two samples, simple and multiple regression, analysis of one and two-way tables, one-way analysis of variance. As time permits, additional topics, such as distribution-free methods and the design and analysis of factorial studies will be considered.

### MA 512. Numerical Differential Equations

This course begins where MA 510 ends in the study of the theory and practice of the numerical solution of differential equations. Central topics include a review of initial value problems, including Eulers method, Runge-Kutta methods, multi-step methods, implicit methods and predictor-corrector methods; the solution of two-point boundary value problems by shooting methods and by the discretization of the original problem to form systems of nonlinear equations; numerical stability; existence and uniqueness of solutions; and an introduction to the solution of partial differential equations by finite differences. Other topics might include finite element or boundary element methods, Galerkin methods, collocation, or variational methods.

### MA 514. Numerical Linear Algebra

This course provides students with the skills necessary to develop, analyze and implement computational methods in linear algebra. The central topics include vector and matrix algebra, vector and matrix norms, the singular value decomposition, the LU and QR decompositions, Householder transformations and Givens rotations, and iterative methods for solving linear systems including Jacobi, Gauss-Seidel, SOR and conjugate gradient methods; and eigenvalue problems. Applications to such problem areas as least squares and optimization will be discussed. Other topics might include: special linear systems, such as symmetric, positive definite, banded or sparse systems; preconditioning; the Cholesky decomposition; sparse tableau and other least-square methods; or algorithms for parallel architectures.

### MA 517. Mathematical Foundations for Data Science

The foci of this class are the essential statistics and linear algebra skills required for Data Science students. The class builds the foundation for theoretical and computational abilities of the students to analyze high dimensional data sets. Topics covered include Bayes theorem, the central limit theorem, hypothesis testing, linear equations, linear transformations, matrix algebra, eigenvalues and eigenvectors, and sampling techniques, including Bootstrap and Markov chain Monte Carlo. Students will use these techniques while engaging in hands-on projects with real data.

### MA 520. Fourier Transforms and Distributions

The course will cover L1, L2, L and basic facts from Hilbert space theory (Hilbert basis, projection theorems, Riesz theory). The first part of the course will introduce Fourier series: the L2 theory, the C theory: rate of convergence, Fourier series of real analytic functions, application to the trapezoidal rule, Fourier transforms in L1, Fourier integrals of Gaussians, the Schwartz class S, Fourier transforms and derivatives, translations, convolution, Fourier transforms in L2, and characteristic functions of probability distribution functions. The second part of the course will cover tempered distributions and applications to partial differential equations. Other related topics will be covered at the instructors discretion.

### MA 521. Partial Differential Equations

This course considers a variety of material in partial differential equations (PDE). Topics covered will be chosen from the following: classical linear elliptic, parabolic and hyperbolic equations and systems, characteristics, fundamental/Greens solutions, potential theory, the Fredholm alternative, maximum principles, Cauchy problems, Dirichlet/Neumann/Robin problems, weak solutions and variational methods, viscosity solutions, nonlinear equations and systems, wave propagation, free and moving boundary problems, homogenization. Other topics may also be covered.

### MA 522. Hilbert Spaces and Applications to PDE

The course covers Hilbert space theory with special emphasis on applications to linear ODs and PDEs. Topics include spectral theory for linear operators in n-dimensional and infinite dimensional Hilbert spaces, spectral theory for symmetric compact operatos, linear and bilinear forms, Riesz and Lax-Milgram theorems, weak derivatives, Sobolev spaces H1, H2, Rellich compactness theorem, weak and classical solutions for Dirichlet and Neumann problems in one variable and in Rn, Dirichlet variational principle, eigenvalues and eigenvectors. Other related topics will be covered at the instructors discretion.

### MA 524. Convex Analysis and Optimization

This course covers topics in functional analysis that are critical to the study of convex optimization problems. The first part of the course will include the minimization theory for quadratic and convex functionals on convex sets and cones, the Legendre-Fenchel duality, variational inequalities and complementarity systems. The second part will include optimal stopping time problems in deterministic control, value functions and Hamilton-Jacobi inequalities and linear and quadratic programming, duality and Kuhn-Tucker multipliers. Other related topics will be covered at the instructors discretion.

### MA 525. Optimal Control and Design with Composite Materials I

Modern technology involves a wide application of materials with internal structure adapted to environmental demands. This, the first course in a two-semester sequence, will establish a theoretical basis for identifying structures that provide optimal response to prescribed external factors. Material covered will include basics of the calculus of variations: Euler equations; transversality conditions; Weierstrass-Erdmann conditions for corner points; Legendre, Jacobi and Weierstrass conditions; Hamiltonian form of the necessary conditions; and Noethers theorem. Pontryagins maximum principle in its original lumped parameter form will be put forth as well as its distributed parameter extension. Chattering regimes of control and relaxation through composites will be introduced at this point. May be offered by special arrangement.

### MA 526. Optimal Control and Design with Composite Materials II

Topics presented will include basics of homogenization theory. Bounds on the effective properties of composites will be established using the translation method and Hashin-Shtrikman variational principles. The course concludes with a number of examples demonstrating the use of the theory in producing optimal structural designs. The methodology given in this course turns the problem of optimal design into a problem of rigorous mathematics. This course can be taken independently or as the sequel to MA 525.

### MA 528. Measure Theoretic Probability Theory

This course is designed to give graduate students interested in financial mathematics and stochastic analysis the necessary background in measure-theoretic probability and provide a theoretical foundation for Ph.D. students with research interests in analysis and mathematical statistics. Besides classical topics such as the axiomatic foundations of probability, conditional probabif ities and independence, random variables and their distributions, and limit theorems, this course focuses on concepts crucial for the understanding of stochastic processes and quantitative finance: conditional expectations, filtrations and martingales as well as change of measure techniques and the Radon-Nikodym theorem. A wide range of illustrative examples from a topic chosen by the instructors discretion (e.g financial mathematics, signal processing, actuarial mathematics) will be presented.

### MA 529. Stochastic Processes

This course is designed to introduce students to continuous-time stochastic processes. Stochastic processes play a central role in a wide range of applications from signal processing to finance and also offer an alternative novel viewpoint to several areas of mathematical analysis, such as partial differential equations and potential theory. The main topics for this course are martingales, maximal inequalities and applications, optimal stopping and martingale convergence theorems, the strong Markov property, stochastic integration, Itos formula and applications, martingale representation theorems, Girsanovs theorem and applications, and an introduction to stochastic differential equations, the Feynman-Kac formula, and connections to partial differential equations. Optional topics (at the instructors discretion) include Markov processes and Poisson-and jump-processes.

### MA 530. Discrete Mathematics

This course provides the student of mathematics or computer science with an overview of discrete structures and their applications, as well as the basic methods and proof techniques in combinatorics. Topics covered include sets, relations, posets, enumeration, graphs, digraphs, monoids, groups, discrete probability theory and propositional calculus.

### MA 533. Discrete Mathematics II

This course is designed to provide an in-depth study of some topics in combinatorial mathematics and discrete optimization. Topics may vary from year to year. Topics covered include, as time permits, partially ordered sets, lattices, matroids, matching theory, Ramsey theory, discrete programming problems, computational complexity of algorithms, branch and bound methods.

### MA 535. Algebra

Fundamentals of group theory: homomorphisms and the isomorphism theorems, finite groups, structure of finitely generated Abelian groups. Structure of rings: homomorphisms, ideals, factor rings and the isomorphism theorems, integral domains, factorization. Field theory: extension fields, finite fields, theory of equations. Selected topics from: Galois theory, Sylow theory, Jordan-Holder theory, Polya theory, group presentations, basic representation theory and group characters, modules. Applications chosen from mathematical physics, Grobner bases, symmetry, cryptography, error-correcting codes, number theory.

### MA 540. Probability and Mathematical Statistics I

Intended for advanced undergraduates and beginning graduate students in the mathematical sciences, and for others intending to pursue the mathematical study of probability and statistics. Topics covered include axiomatic foundations, the calculus of probability, conditional probability and independence, Bayes Theorem, random variables, discrete and continuous distributions, joint, marginal and conditional distributions, covariance and correlation, expectation, generating functions, exponential families, transformations of random variables, types of convergence, laws of large numbers the Central Limit Theorem, Taylor series expansion, the delta method.

### MA 541. Probability and Mathematical Statistics II

This course is designed to provide background in principles of statistics. Topics covered include estimation criteria: method of moments, maximum likelihood, least squares, Bayes, point and interval estimation, Fishers information, Cramer-Rao lower bound, sufficiency, unbiasedness, and completeness, Rao-Blackwell Theorem, efficiency, consistency, interval estimation pivotal quantities, Neyman-Person Lemma, uniformly most powerful tests, unbiased, invariant and similar tests, likelihood ratio tests, convex loss functions, risk functions, admissibility and minimaxity, Bayes decision rules.

### MA 542. Regression Analysis

Regression analysis is a statistical tool that utilizes the relation between a response variable and one or more predictor variables for the purposes of description, prediction and/or control. Successful use of regression analysis requires an appreciation of both the theory and the practical problems that often arise when the technique is employed with real-world data. Topics covered include the theory and application of the general linear regression model, model fitting, estimation and prediction, hypothesis testing, the analysis of variance and related distribution theory, model diagnostics and remedial measures, model building and validation, and generalizations such as logistic response models and Poisson regression. Additional topics may be covered as time permits. Application of theory to real-world problems will be emphasized using statistical computer packages.

### MA 543. Statistical Methods for Data Science

Statistical Methods for Data Science surveys the statistical methods most useful in data science applications. Topics covered include predictive modeling methods, including multiple linear regression, and time series, data dimension reduction, discrimination and classification methods, clustering methods, and committee methods. Students will implement these methods using statistical software.

### MA 546. Design and Analysis of Experiments

Controlled experimentsstudies in which treatments are assigned to observational units are the gold standard of scientific investigation. The goal of the statistical design and analysis of experiments is to (1) identify the factors which most affect a given process or phenomenon; (2) identify the ways in which these factors affect the process or phenomenon, both individually and in combination; (3) accomplish goals 1 and 2 with minimum cost and maximum efficiency while maintaining the validity of the results. Topics covered in this course include the design, implementation and analysis of completely randomized complete block, nested, split plot, Latin square and repeated measures designs. Emphasis will be on the application of the theory to real data using statistical computer packages.

### MA 547. Design and Analysis of Observational and Sampling Studies

Like controlled experiments, observational studies seek to establish cause-effect relationships, but unlike controlled experiments, they lack the ability to assign treatments to observational units. Sampling studies, such as sample surveys, seek to characterize aspects of populations by obtaining and analyzing samples from those populations. Topics from observational studies include: prospective and retrospective studies; overt and hidden bias; adjustments by stratification and matching. Topics from sampling studies include: simple random sampling and associated estimates for means, totals, and proportions; estimates for subpopulations; unequal probability sampling; ratio and regression estimation; stratified, cluster, systematic, multistage, double sampling designs, and, time permitting, topics such as model-based sampling, spatial and adaptive sampling.

### MA 548. Quality Control

This course provides the student with the basic statistical tools needed to evaluate the quality of products and processes. Topics covered include the philosophy and implementation of continuous quality improvement methods, Shewhart control charts for variables and attributes, EWMA and Cusum control charts, process capability analysis, factorial and fractional factorial experiments for process design and improvement, and response surface methods for process optimization. Additional topics will be covered as time permits. Special emphasis will be placed on realistic applications of the theory using statistical computer packages.

### MA 549. Analysis of Lifetime Data

Lifetime data occurs frequently in engineering, where it is known as reliability or failure time data, and in the biomedical sciences, where it is known as survival data. This course covers the basic methods for analyzing such data. Topics include: probability models for lifetime data, censoring, graphical methods of model selection and analysis, parametric and distribution-free inference, parametric and distribution-free regression methods. As time permits, additional topics such as frailty models and accelerated life models will be considered. Special emphasis will be placed on realistic applications of the theory using statistical computer packages.

### MA 550. Time Series Analysis

Time series are collections of observations made sequentially in time. Examples of this type of data abound in many fields ranging from finance to engineering. Special techniques are called for in order to analyze and model these data. This course introduces the student to time and frequency domain techniques, including topics such as autocorrelation, spectral analysis, and ARMA and ARIMA models, Box-Jenkins methodology, fitting, forecasting, and seasonal adjustments. Time permitting, additional topics will be chosen from: Kalman filter, smoothing techniques, Holt-Winters procedures, FARIMA and GARCH models, and joint time-frequency methods such as wavelets. The emphasis will be in application to real data situations using statistical computer packages.

### MA 551. Computational Statistics

Computational statistics is an essential component of modern statistics that often requires efficient algorithms and programing strategies for statistical learning and data analysis. This course will introduce principles and techniques of statistical computing and data management necessary for computationally intensive statistical analysis especially for big data. Topics covered include management of large data (data structure, data query), parallelized data analyses, stochastic simulations (Monte Carlo methods, permutation-based inference), numerical optimization in statistical inference (deterministic and stochastic convex analysis, EM algorithm, etc.), randomization methods (bootstrap methods), etc. Students will use these techniques while engaging in hands-on projects with real data. Students who have taken the MA590 version of this course cannot also earn credit for MA 551.

### MA 552. Distribution-Free and Robust Statistical Methods

Distribution-free statistical methods relax the usual distributional modeling assumptions of classical statistical methods. Robust methods are statistical procedures that are relatively insensitive to departures from typical assumptions, while retaining the expected behavior when assumptions are satisfied. Topics covered include, time permitting, order statistics and ranks; classical distribution-free tests such as the sign, Wilcoxon signed rank, and Wilcoxon rank sum tests, and associated point estimators and confidence intervals; tests pertaining to one and two-way layouts; the Kolmogorov-Smirnov test; permutation methods; bootstrap and Monte Carlo methods; M, L, and R estimators, regression, kernel density estimation and other smoothing methods. Comparisons will be made to standard parametric methods.

### MA 554. Applied Multivariate Analysis

This course is an introduction to statistical methods for analyzing multivariate data. Topics covered are multivariate sampling distributions, tests and estimation of multivariate normal parameters, multivariate ANOVA, regression, discriminant analysis, cluster analysis, factor analysis and principal components. Additional topics will be covered as time permits. Students will be required to analyze real data using one of the standard packages available.

### MA 556. Applied Bayesian Statistics

Bayesian statistics makes use of an inferential process that models data summarizing the results in terms of probability distributions for the model parameters. A key feature is that in the Bayesian approach, past information can be updated with new data in an elegant way in order to aid in decision making. Topics included in the courses: statistical decision theory, the Bayesian inferential framework (model specification, model fitting and model checking); computational methods for posterior simulation integration; regression models, hierarchical models, and ANOVA; time permitting, additional topics will include generalized linear models, multivariate models, missing data problems, and time series analysis.

### MA 557. Graduate Seminar in Applied Mathematics

This seminar introduces students to modern issues in Applied Mathematics. In the seminar, students and faculty will read and discuss survey and research papers, make and attend presentations, and participate in brainstorming sessions toward the solution of advanced mathematical problems.

### MA 559. Statistics Graduate Seminar

This seminar introduces students to issues and trends in modern statistics. In the seminar, students and faculty will read and discuss survey and research papers, make and attend presentations, and participate in brainstorming sessions toward the solution of advanced statistical problems.

### MA 562. Professional Master's Seminar

This seminar will introduce professional masters students to topics related to general writing, presentation, group communication and interviewing skills, and will provide the foundations to successful cooperation within interdisciplinary team environments. All full-time students will be required to take both components A and B of the seminar during their professional masters studies.

### MA 571. Financial Mathematics I

This course provides an introduction to many of the central concepts in mathematical finance. The focus of the course is on arbitrage-based pricing of derivative securities. Topics include stochastic calculus, securities markets, arbitrage-based pricing of options and their uses for hedging and risk management, forward and futures contracts, European options, American options, exotic options, binomial stock price models, the Black-Scholes-Merton partial differential equation, risk-neutral option pricing, the fundamental theorems of asset pricing, sensitivity measures (Greeks), and Mertons credit risk model.

### MA 572. Financial Mathematics II

The course is devoted to the mathematics of fixed income securities and to the financial instruments and methods used to manage interest rate risk. The first topics covered are the term-structure of interest rates, bonds, futures, interest rate swaps and their uses as investment or hedging tools and in asset-liability management. The second part of the course is devoted to dynamic term-structure models, including risk-neutral interest rate trees, the Heath-Jarrow-Morton model, Libor market models, and forward measures. Applications of these models are also covered, including the pricing of non-linear interest rate derivatives such as caps, floors, collars, swaptions and the dynamic hedging of interest rate risk. The course concludes with the coverage of mortgage-backed and asset-backed securities.

### MA 573. Computational Methods of Financial Mathematics

Most realistic quantitative finance models are too complex to allow explicit analytic solutions and are solved by numerical computational methods. The first part of the course covers the application of finite difference methods to the partial differential equations and interest rate models arising in finance. Topics included are explicit, implicit and Crank-Nicholson finite difference schemes for fixed and free boundary value problems, their convergence and stability. The second part of the course covers Monte Carlo simulation methods, including random number generation, variance reduction techniques and the use of low discrepancy sequences.

### MA 574. Portfolio Valuation and Risk Management

Balancing financial risks vs returns by the use of asset diversification is one of the fundamental tasks of quantitative financial management. This course is devoted to the use of mathematical optimization and statistics to allocate assets, to construct and manage portfolios and to measure and manage the resulting risks. The fist part of the course covers Markowitzs mean-variance optimization and efficient frontiers, Sharpes single index and capital asset pricing models, arbitrage pricing theory, structural and statistical multifactor models, risk allocation and risk budgeting. The second part of the course is devoted to the intertwining of optimization and statistical methodologies in modern portfolio management, including resampled efficiency, robust and Bayesian statistical methods, the Black-Litterman model and robust portfolio optimization.

### MA 575. Market and Credit Risk Models and Management

The objective of the course is to familiarize students with the most important quantitative models and methods used to measure and manage financial risk, with special emphasis on market and credit risk. The course starts with the introduction of metrics of risk such as volatility, value-at-risk and expected shortfall and with the fundamental quantitative techniques used in financial risk evaluation and management. The next section is devoted to market risk including volatility modeling, time series, non-normal heavy tailed phenomena and multivariate notions of codependence such as copulas, correlations and tail-dependence. The final section concentrates on credit risk including structural and dynamic models and default contagion and applies the mathematical tools to the valuation of default contingent claims including credit default swaps, structured credit portfolios and collateralized debt obligations.

### MA 579. Financial Programming Workshop

The objective is to elevate the students computer programming skills to the semi-professional level required in quantitative finance. Participants learn through hands-on experience by working on a structured set of mini projects from computational finance under the guidance of an experienced trainer and the faculty in charge. The programming language used may be C++, MATLAB, R/S, VB or another language widely used in quantitative finance and may alternate from year to year.

### MA 584. Statistical Methods in Genetics and Bioinformatics

This course provides students with knowledge and understanding of the applications of statistics in modern genetics and bioinformatics. The course generally covers population genetics, genetic epidemiology, and statistical models in bioinformatics. Specific topics include meiosis modeling, stochastic models for recombination, linkage and association studies (parametric vs. nonparametric models, family-based vs. population-based models) for mapping genes of qualitative and quantitative traits, gene expression data analysis, DNA and protein sequence analysis, and molecular evolution. Statistical approaches include log-likelihood ratio tests, score tests, generalized linear models, EM algorithm, Markov chain Monte Carlo, hidden Markov model, and classification and regression trees. Students may not receive credit for both MA 584 and MA 4603.

### MA 598. Professional Master's Project

This project will provide the opportunity to apply and extend the material studied in the coursework to the study of a real-world problem originating in the industry. The project will be a capstone integrating industrial experience with the previously acquired academic knowledge and skills. The topic of the project will come from a problem generated in industry, and could originate from prior internship or industry experience of the student. The student will prepare a written project report and make a presentation before a committee including the faculty advisor, at least one additional WPI faculty member and representatives of a possible industrial sponsor. The advisor of record must be a faculty member of the WPI Mathematical Sciences Department. The student must submit a written project proposal for approval by the Graduate Committee prior to registering for the project.

### MME 518. Geometrical Concepts

This course focuses primarily on the foundations and applications of Euclidean and non-Euclidean geometries. The rich and diverse nature of the subject also implies the need to explore other topics, for example, chaos and fractals. The course incorporates collaborative learning and the investigation of ideas through group projects. Possible topics include geometrical software and computer graphics, tiling and tessellations, two- and three-dimensional geometry, inversive geometry, graphical representations of functions, model construction, fundamental relationship between algebra and geometry, applications of geometry, geometry transformations and projective geometry, and convexity.

### MME 522. Applications of Calculus

There are three major goals for this course: to establish the underlying principles of calculus, to reinforce students calculus skills through investigation of applications involving those skills, and to give students the opportunity to develop projects and laboratory assignments for use by first-year calculus students. The course will focus heavily on the use of technology to solve problems involving applications of calculus concepts. In addition, MME students will be expected to master the mathematical rigor of these calculus concepts so that they will be better prepared to develop their own projects and laboratory assignments. For example, if an MME student chose to develop a lab on convergence of sequence, he/she would be expected to understand the rigorous definition of convergence and how to apply it to gain sufficient and/or necessary conditions for convergence. The process of developing these first-year calculus assignments will enable the MME students to increase their own mathematical understanding of concepts while learning to handle mathematical and computer issues which will be encountered by their own calculus students. Their understanding of the concepts and applications of calculus will be further reinforced through computer laboratory assignments and group projects. Applications might include exponential decay of drugs in the body, optimal crankshaft design, population growth, or development of cruise control systems.

### MME 523. Analysis with Applications

This course introduces students to mathematical analysis and its use in modeling. It will emphasize topics of calculus (including multidimensional) in a rigorous way. These topics will be motivated by their usefulness for understanding concepts of the calculus and for facilitating the solutions of engineering and science problems. Projects involving applications and appropriate use of technology will be an essential part of the course. Topics covered may include dynamical systems and differential equations; growth and decay; equilibrium; probabilistic dynamics; optimal decisions and reward; applying, building and validating models; functions on n-vectors; properties of functions; parametric equations; series; applications such as pendulum problems; electromagnetism; vibrations; electronics; transportation; gravitational fields; and heat loss.

### MME 524-25. Probability, Statistics and Data Analysis I, II

This course introduces students to probability, the mathematical description of random phenomena, and to statistics, the science of data. Students in this course will acquire the following knowledge and skills: Probability models-mathematical models used to describe and predict random phenomena. Students will learn several basic probability models and their uses, and will obtain experience in modeling random phenomena. Data analysis-the art/science of finding patterns in data and using those patterns to explain the process which produced the data. Students will be able to explore and draw conclusions about data using computational and graphical methods. The iterative nature of statistical exploration will be emphasized. Statistical inference and modeling-the use of data sampled from a process and the probability model of that process to draw conclusions about the process. Students will attain proficiency in selecting, fitting and criticizing models, and in drawing inference from data. Design of experiments and sampling studies - the proper way to design experiments and sampling studies so that statistically valid inferences can be drawn. Special attention will be given to the role of experiments and sampling studies in scientific investigation. Through lab and project work, students will obtain practical skills in designing and analyzing studies and experiments. Course topics will be motivated whenever possible by applications and reinforced by experimental and computer lab experiences. One in-depth project per semester involving design, data collection, and statistical or probabilistic analysis will serve to integrate and consolidate student skills and understanding. Students will be expected to learn and use a statistical computer package such as MINITAB.

### MME 526-27. Linear Models I, II

This two-course sequence imparts computational skills, particularly those involving matrices, to deepen understanding of mathematical structure and methods of proof; it also includes discussion on a variety of applications of the material developed, including linear optimization. Topics in this sequence may include systems of linear equations, vector spaces, linear independence, bases, linear transformations, determinants, eigenvalues and eigenvectors, systems of linear inequalities, linear programming problems, basic solutions, duality and game theory. Applications may include economic models, computer graphics, least squares approximation, systems of differential equations, graphs and networks, and Markov processes.

### MME 528. Mathematical Modeling and Problem Solving

This course introduces students to the process of developing mathematical models as a means for solving real problems. The course will encompass several different modeling situations that utilize a variety of mathematical topics. The mathematical fundamentals of these topics will be discussed, but with continued reference to their use in finding the solutions to problems. Problems to be covered include balance in small group behavior, traffic flow, air pollution flow, group decision making, transportation, assignment, project planning and the critical path method, genetics, inventory control and queueing.

### MME 529. Numbers, Polynomials and Algebraic Structures

This course enables secondary mathematics teachers to see how commonly taught topics such as number systems and polynomials fit into the broader context of algebra. The course will begin with treatment of arithmetic, working through Euclids algorithm and its applications, the fundamental theorem of arithmetic and its applications, multiplicative functions, the Chinese remainder theorem and the arithmetic of Z/n. This information will be carried over to polynomials in one variable over the rational and real numbers, culminating in the construction of root fields for polynomials via quotients of polynomial rings. Arithmetic in the Gaussian integers and the integers in various other quadratic fields (especially the field of cube roots of unity) will be explored through applications such as the generation of Pythagorean triples and solutions to other Diophantine equations (like finding integersided triangles with a 60 degree angle). The course will then explore cyclotomy, and the arithmetic in rings of cyclotomic integers. This will culminate in Gausss construction of the regular 5-gon and 17-gon and the impossibility of constructing a 9-gon or trisecting a 60-degree angle. Finally, solutions of cubics and quartics by radicals will be studied. All topics will be based on the analysis of explicit calculations with (generalized) numbers. The proposed curriculum covers topics that are part of the folklore for high school mathematics (the impossibility of certain ruler and compass constructions), but that many teachers know only as facts. There are also many applications of the ideas that will allow the teachers to use results and ideas from abstract algebra to construct for their students problems that have manageable solutions.

### MME 531. Discrete Mathematics

This course deals with concepts and methods which emphasize the discrete nature in many problems and structures. The rapid growth of this branch of mathematics has been inspired by its wide range of applicability to diverse fields such as computer science, management, and biology. The essential ingredients of the course are: Combinatorics -The Art of Counting. Topics include basic counting principles and methods such as recurrence relations, generating functions, the inclusion-exclusion principle and the pigeonhole principle. Applications may include block designs, latin squares, finite projective planes, coding theory, optimization and algorithmic analysis. Graph Theory. This includes direct graphs and networks. Among the parameters to be examined are traversibility, connectivity, planarity, duality and colorability.

### MME 532. Differential Equations

This course would have concepts and techniques for both Ordinary and Partial Differential Equations. Topics from ordinary differential equations include existence and uniqueness for first order, single variable problems as well as separation of variables and linear methods for first order problems. Second order, linear equations would be solved for both the homogeneous and non homogeneous cases. The phenomena of beats and resonance would be analyzed. The Laplace Transform would be introduced for appropriate second order nonhomogeneous problems. Partial Differential Equations would focus on boundary value problems arising from the Heat and Wave equations in one variable. Fourier Series expansions would be used to satisfy initial conditions and the concepts of orthogonality and convergence addressed.

### MME 592. Project Preparation (Part of a 3-Course Sequence with Mme 594 and Mme 596)

Students will research and develop a mathematical topic or pedagogical technique. The project will typically lead to classroom implementation; however, a project involving mathematical research at an appropriate level of rigor will also be acceptable. Preparation will be completed in conjunction with at least one faculty member from the Mathematical Sciences Department and will include exhaustive research on the proposed topic. The course will result in a detailed proposal that will be presented to the MME Project Committee for approval; continuation with the project is contingent upon this approval.

### MME 594. Project Implementation

Students will implement and carry out the project developed during the project preparation course. Periodic contact and/or observations will be made by the project advisor (see MME 592 Project Preparation) in order to provide feedback and to ensure completion of the proposed task. Data for the purpose of evaluation will be collected by the students throughout the term, when appropriate. If the project includes classroom implementation, the experiment will last for the duration of a semester.

### MME 596. Project Analysis and Report

Students will complete a detailed statistical analysis of any data collected during the project implementation using techniques from MME 524-525 Probability, Statistics, and Data Analysis. The final report will be a comprehensive review of the relevant literature, project description, project implementation, any statistical results and conclusions. Project reports will be subject to approval by the MME Project committee and all students will be required to present their project to the mathematical sciences faculty. Course completion is contingent upon approval of the report and satisfactory completion of the presentation.

### SEME 524-25. Probability, Statistics and Data Analysis I, II

This course introduces students to probability, the mathematical description of random phenomena, and to statistics, the science of data. Students in this course will acquire the following knowledge and skills: Probability models-mathematical models used to describe and predict random phenomena. Students will learn several basic probability models and their uses, and will obtain experience in modeling random phenomena. Data analysis-the art/science of finding patterns in data and using those patterns to explain the process which produced the data. Students will be able to explore and draw conclusions about data using computational and graphical methods. The iterative nature of statistical exploration will be emphasized. Statistical inference and modeling-the use of data sampled from a process and the probability model of that process to draw conclusions about the process. Students will attain proficiency in selecting, fitting and criticizing models, and in drawing inference from data. Design of experiments and sampling studies the proper way to design experiments and sampling studies so that statistically valid inferences can be drawn. Special attention will be given to the role of experiments and sampling studies in scientific investigation. Through lab and project work, students will obtain practical skills in designing and analyzing studies and experiments. Course topics will be motivated whenever possible by applications and reinforced by experimental and computer lab experiences. One in-depth project per semester involving design, data collection, and statistical or probabilistic analysis will serve to integrate and consolidate student skills and understanding. Students will be expected to learn and use a statistical computer package such as MINITAB.