# Applied Mathematics

John T. Rettaliata Engineering Center, Suite 208

10 W. 32nd St.

Chicago, IL 60616

312.567.8980

amath@iit.edu

science.iit.edu/applied-mathematics

**Chair**

Fred J. Hickernell

**Associate Chair and Director of Undergraduate Studies**

Michael J. Pelsmajer

**Faculty with Research Interests**

For information regarding faculty visit the Department of Applied Mathematics website.

Applied mathematics is the mathematics that is created in response to problems in science, engineering, and society. Applied mathematicians work on a wide variety of topics such as how to construct methods for multi-criteria decision making (requiring discrete mathematics and statistics), predicting how the financial markets will behave (requiring probability/statistics, analysis, optimization), and analyzing how liquid flows around solids (requiring expertise in computational methods and analysis). Students with an applied mathematics background are prepared for careers in the insurance industry, electronics and computer manufacturers, logistics companies, pharmaceutical firms, and more. Students will also be prepared to continue in graduate school.

Our graduates work in financial and insurance companies as analysts, computer companies as programmers and hardware developers, and in many different fields as researchers, as well as in academia. They have gone to excellent graduate schools in mathematics (pure, applied, and financial), physics, design, accounting, and M.B.A. programs. Students have the flexibility to assemble a portfolio of courses that will satisfy both intellectual needs and career preparation. There is a wide variety of courses offered, with strengths in contemporary topics in applied mathematics: stochastic analysis (including mathematical finance), applied analysis, computational mathematics, discrete mathematics, and statistics.

A minor is required, which gives students an area of focus where mathematics may be applied. It consists of five or more related courses in an area outside of applied mathematics. With a minor in computer science, business, or one of the engineering areas, for example, the student will be well prepared to enter the job market in business or government.

If desired, a student can choose a specialization, which selects electives appropriate for different career paths. Another popular option is to double major in both applied mathematics and another subject, such as computer science or physics. There is also the option of a co-terminal degree, where a student graduates with a B.S. and a master's degree at the same time, in as little as five years.

## Degree Programs

## Co-Terminal Options

The Department of Applied Mathematics also offers the following co-terminal degrees, which enables a student to simultaneously complete both an undergraduate and graduate degree in as few as five years:

- Bachelor of Science in Applied Mathematics/Master of Science in Applied Mathematics
- Bachelor of Science in Applied Mathematics/Master of Computer Science
- Bachelor of Science in Applied Mathematics/Master of Science in Computer Science
- Bachelor of Science in Applied Mathematics/Master of Data Science
- Bachelor of Science in Applied Mathematics/Master of Mathematical Finance

These co-terminal degrees allow students to gain greater knowledge in specialized areas while, in most cases, completing a smaller number of credit hours with increased scheduling flexibility. For more information, please visit the Department of Applied Mathematics website (science.iit.edu/applied-mathematics)**.**

## Course Descriptions

**Introduction to the Profession**

Introduces the student to the scope of mathematics as a profession, develops a sense of mathematical curiosity and problem solving skills, identifies and reinforces the student's career choices, and provides a mechanism for regular academic advising. Provides integration with other first-year courses. Introduces applications of mathematics to areas such as engineering, physics, computer science, and finance. Emphasis is placed on the development of teamwork skills.

**Lecture:**3

**Lab:**0

**Credits:**3

**Satisfies:**Communications (C)

**Geometry for Architects**

Basic Euclidean and analytic geometry in two and three dimensions; trigonometry. Equations of lines, circles and conic sections; resolution of triangles; polar coordinates. Equations of planes, lines, quadratic surfaces. Applications. This course does not count toward any mathematics requirements in business, computer science, engineering, mathematics, or natural science degree programs.

**Lecture:**3

**Lab:**0

**Credits:**3

**Satisfies:**Communications (C)

**Introduction to Calculus**

Basic concepts of calculus of a single variable; limits, continuity, derivatives, and integrals. Applications. This course does not count toward any mathematics requirements in business, computer science, engineering, mathematics, or natural science degree programs.

**Lecture:**3

**Lab:**0

**Credits:**3

**Thinking Mathematically**

This course allows students to discover, explore, and apply modern mathematical ideas. Emphasis is placed on using sound reasoning skills, visualizing mathematical concepts, and communicating mathematical ideas effectively. Classroom discussion and group work on challenging problems are central to the course. Topics from probability, statistics, logic, number theory, graph theory, combinatorics, chaos theory, the concept of infinity, and geometry may be included. This course does not count toward any mathematics requirements in business, computer science, engineering, mathematics, or natural science degree programs.

**Lecture:**3

**Lab:**0

**Credits:**3

**Satisfies:**Communications (C)

**Preparation for Calculus**

Review of algebra and analytic geometry. Functions, limits, derivatives. Trigonometry, trigonometric functions and their derivatives. Inverse functions, inverse trigonometric functions and their derivatives. Exponential and logarithmic functions. This course does not count toward any mathematics requirements in business, computer science, engineering, mathematics, or natural science degree programs.

**Lecture:**4

**Lab:**0

**Credits:**4

**Calculus I**

Analytic geometry. Functions and their graphs. Limits and continuity. Derivatives of algebraic and trigonometric functions. Applications of the derivative. Introduction to integrals and their applications.

**Prerequisite(s):**IIT Mathematics Placement score of 151 or MATH 145 with min. grade of C or MATH 148 with min. grade of C

**Lecture:**4

**Lab:**1

**Credits:**5

**Satisfies:**Communications (C)

**Calculus II**

Transcendental functions and their calculus. Integration techniques. Applications of the integral. Indeterminate forms and improper integrals. Polar coordinates. Numerical series and power series expansions.

**Prerequisite(s):**MATH 149 with min. grade of C or MATH 151 with min. grade of C

**Lecture:**4

**Lab:**1

**Credits:**5

**Satisfies:**Communications (C)

**Introduction to Discrete Math**

Sets, statements, and elementary symbolic logic; relations and digraphs; functions and sequences; mathematical induction; basic counting techniques and recurrence. Credit will not be granted for both CS 330 and MATH 230.

**Lecture:**3

**Lab:**0

**Credits:**3

**Satisfies:**Communications (C)

**Multivariate and Vector Calculus**

Analytic geometry in three-dimensional space. Partial derivatives. Multiple integrals. Vector analysis. Applications.

**Prerequisite(s):**MATH 152

**Lecture:**4

**Lab:**0

**Credits:**4

**Introduction to Differential Equations**

Linear differential equations of order one. Linear differential equations of higher order. Series solutions of linear DE. Laplace transforms and their use in solving linear DE. Introduction to matrices. Systems of linear differential equations.

**Prerequisite(s):**MATH 152

**Lecture:**4

**Lab:**0

**Credits:**4

**Perspectives in Analysis**

The course is focused on selected topics related to fundamental concepts and methods of classic analysis and their applications with emphasis on various problem-solving strategies, visualization, mathematical modeling, and interrelation of different areas of mathematics.

**Lecture:**3

**Lab:**0

**Credits:**3

**Elementary Linear Algebra**

Systems of linear equations; matrix algebra, inverses, determinants, eigenvalues, and eigenvectors, diagonalization; vector spaces, basis, dimension, rank and nullity; inner product spaces, orthonormal bases; quadratic forms.

**Prerequisite(s):**MATH 251*, An asterisk (*) designates a course which may be taken concurrently.

**Lecture:**3

**Lab:**0

**Credits:**3

**Matrix Algebra and Complex Variables**

Vectors and matrices; matrix operations, transpose, rank, inverse; determinants; solution of linear systems; eigenvalues and eigenvectors. The complex plane; analytic functions; contour integrals; Laurent series expansions; singularities and residues.

**Prerequisite(s):**MATH 251

**Lecture:**3

**Lab:**0

**Credits:**3

**Introduction to Computational Mathematics**

Study and design of mathematical models for the numerical solution of scientific problems. This includes numerical methods for the solution on linear and nonlinear systems, basic data fitting problems, and ordinary differential equations. Robustness, accuracy, and speed of convergence of algorithms will be investigated including the basics of computer arithmetic and round-off errors. Same as MMAE 350.

**Lecture:**3

**Lab:**0

**Credits:**3

**Satisfies:**Communications (C)

**Probability and Statistics for Electrical and Computer Engineers**

This course focuses on the introductory treatment of probability theory including: axioms of probability, discrete and continuous random variables, random vectors, marginal, joint, conditional and cumulative probability distributions, moment generating functions, expectations, and correlations. Also covered are sums of random variables, central limit theorem, sample means, and parameter estimation. Furthermore, random processes and random signals are covered. Examples and applications are drawn from problems of importance to electrical and computer engineers. Credit only granted for one of MATH 374, MATH 474, and MATH 475.

**Prerequisite(s):**MATH 251

**Lecture:**3

**Lab:**0

**Credits:**3

**Introduction to Mathematical Modeling**

This course provides an introduction to problem-driven (as opposed to method-driven) applications of mathematics with a focus on design and analysis of models using tools from all parts of mathematics.

**Prerequisite(s):**(CS 104 or CS 105 or CS 115) and MATH 251 and MATH 252* and MATH 332, An asterisk (*) designates a course which may be taken concurrently.

**Lecture:**3

**Lab:**0

**Credits:**3

**Satisfies:**Communications (C)

**Real Analysis**

Real numbers, continuous functions; differentiation and Riemann integration. Functions defined by series.

**Prerequisite(s):**MATH 251

**Lecture:**3

**Lab:**0

**Credits:**3

**Complex Analysis**

Analytic functions, conformal mapping, contour integration, series expansions, singularities and residues, and applications. Intended as a first course in the subject for students in the physical sciences and engineering.

**Prerequisite(s):**MATH 251

**Lecture:**3

**Lab:**0

**Credits:**3

**Introduction to Iteration and Chaos**

Functional iteration and orbits, periodic points and Sharkovsky's cycle theorem, chaos and dynamical systems of dimensions one and two. Julia sets and fractals, physical implications.

**Lecture:**3

**Lab:**0

**Credits:**3

**Number Theory**

Divisibility, congruencies, distribution of prime numbers, functions of number theory, diophantine equations, applications to encryption methods.

**Prerequisite(s):**MATH 230

**Lecture:**3

**Lab:**0

**Credits:**3

**Geometry**

The course is focused on selected topics related to fundamental ideas and methods of Euclidean geometry, non-Euclidean geometry, and differential geometry in two and three dimensions and their applications with emphasis on various problem-solving strategies, geometric proof, visualization, and interrelation of different areas of mathematics. Permission of the instructor is required.

**Lecture:**3

**Lab:**0

**Credits:**3

**Statistical Methods**

Concepts and methods of gathering, describing and analyzing data including basic statistical reasoning, basic probability, sampling, hypothesis testing, confidence intervals, correlation, regression, forecasting, and nonparametric statistics. No knowledge of calculus is assumed. This course is useful for students in education or the social sciences. This course does not count for graduation in any mathematics programs. Credit not given for both MATH 425 and MATH 476.

**Lecture:**3

**Lab:**0

**Credits:**3

**Statistical Tools for Engineers**

Descriptive statistics and graphs, probability distributions, random sampling, independence, significance tests, design of experiments, regression, time-series analysis, statistical process control, introduction to multivariate analysis. Same as CHE 426. Credit not given for both Math 426 and CHE 426.

**Lecture:**3

**Lab:**0

**Credits:**3

**Applied Algebra**

Introduction to groups, homomorphisms, group actions, rings, field theory. Applications, including constructions with ruler and compass, solvability by radicals, error correcting codes.

**Prerequisite(s):**MATH 230 or MATH 332*, An asterisk (*) designates a course which may be taken concurrently.

**Lecture:**3

**Lab:**0

**Credits:**3

**Satisfies:**Communications (C)

**Computational Algebraic Geometry**

Systems of polynomial equations and ideals in polynomial rings; solution sets of systems of equations and algebraic varieties in affine n-space; effective manipulation of ideals and varieties, algorithms for basic algebraic computations; Groebner bases; applications. Credit may not be granted for both MATH 431 and MATH 530.

**Lecture:**3

**Lab:**0

**Credits:**3

**Satisfies:**Communications (C)

**Linear Optimization**

Introduction to both theoretical and algorithmic aspects of linear optimization: geometry of linear programs, simplex method, anticycling, duality theory and dual simplex method, sensitivity analysis, large scale optimization via Dantzig-Wolfe decomposition and Benders decomposition, interior point methods, network flow problems, integer programming. Credit may not be granted for both MATH 435 and MATH 535.

**Prerequisite(s):**MATH 332

**Lecture:**3

**Lab:**0

**Credits:**3

**Combinatorics**

Permutations and combinations; pigeonhole principle; inclusion-exclusion principle; recurrence relations and generating functions; enumeration under group action.

**Prerequisite(s):**MATH 230

**Lecture:**3

**Lab:**0

**Credits:**3

**Graph Theory and Applications**

Directed and undirected graphs; paths, cycles, trees, Eulerian cycles, matchings and coverings, connectivity, Menger's Theorem, network flow, coloring, planarity, with applications to the sciences (computer, life, physical, social) and engineering.

**Lecture:**3

**Lab:**0

**Credits:**3

**Satisfies:**Communications (C)

**Fourier Series and Boundary-Value Problems**

Fourier series and integrals. The Laplace, heat, and wave equations: Solutions by separation of variables. D'Alembert's solution of the wave equation. Boundary-value problems.

**Lecture:**3

**Lab:**0

**Credits:**3

**Probability and Statistics**

Elementary probability theory including discrete and continuous distributions, sampling, estimation, confidence intervals, hypothesis testing, and linear regression. Credit not granted for both MATH 474 and MATH 475.

**Prerequisite(s):**MATH 251

**Lecture:**3

**Lab:**0

**Credits:**3

**Probability**

Elementary probability theory; combinatorics; random variables; discrete and continuous distributions; joint distributions and moments; transformations and convolution; basic theorems; simulation. Credit not granted for both MATH 474 and MATH 475.

**Prerequisite(s):**MATH 251

**Lecture:**3

**Lab:**0

**Credits:**3

**Statistics**

Estimation theory; hypothesis tests; confidence intervals; goodness-of-fit tests; correlation and linear regression; analysis of variance; nonparametric methods.

**Prerequisite(s):**MATH 475

**Lecture:**3

**Lab:**0

**Credits:**3

**Satisfies:**Communications (C)

**Numerical Linear Algebra**

Fundamentals of matrix theory; least squares problems; computer arithmetic; conditioning and stability; direct and iterative methods for linear systems; eigenvalue problems. Credit may not be granted for both MATH 477 and MATH 577.

**Lecture:**3

**Lab:**0

**Credits:**3

**Numerical Methods for Differential Equations**

Polynomial interpolation; numerical integration; numerical solution of initial value problems for ordinary differential equations by single and multi-step methods, Runge-Kutta, Predictor-Corrector; numerical solution of boundary value problems for ordinary differential equations by shooting method, finite differences and spectral methods. Credit may not be granted for both MATH 478 and MATH 578.

**Lecture:**3

**Lab:**0

**Credits:**3

**Introduction to Stochastic Processes**

This is an introductory, undergraduate course in stochastic processes. Its purpose is to introduce students to a range of stochastic processes which are used as modeling tools in diverse fields of applications, especially in risk management applications for finance and insurance. The course covers basic classes of stochastic processes: Markov chains and martingales in discrete time; Brownian motion; and Poisson process. It also presents some aspects of stochastic calculus.

**Lecture:**3

**Lab:**0

**Credits:**3

**Design and Analysis of Experiments**

Review of elementary probability and statistics; analysis of variance for design of experiments; estimation of parameters; confidence intervals for various linear combinations of the parameters; selection of sample sizes; various plots of residuals; block designs; Latin squares; one, two, and 2^k factorial designs; nested and cross factor designs; regression; nonparametric techniques.

**Prerequisite(s):**MATH 476

**Lecture:**3

**Lab:**0

**Credits:**3

**Regression and Forecasting**

Simple linear regression; multiple linear regression; least squares estimates of parameters; hypothesis testing and confidence intervals in linear regression models; testing of models, data analysis, and appropriateness of models; linear time series models; moving average, autoregressive and/or ARIMA models; estimation, data analysis, and forecasting with time series models; forecasting errors and confidence intervals. Credit may not be granted for both MATH 484 and MATH 564.

**Lecture:**3

**Lab:**0

**Credits:**3

**Satisfies:**Communications (C)

**Introduction to Mathematical Finance**

This is an introductory course in mathematical finance. Technical difficulty of the subject is kept at a minimum while the major ideas and concepts underlying modern mathematical finance and financial engineering are explained and illustrated. The course covers the binomial model for stock prices and touches on continuous time models and the Black-Scholes formula.

**Prerequisite(s):**MATH 475

**Lecture:**3

**Lab:**0

**Credits:**3

**Mathematical Modeling I**

The course provides a systematic approach to modeling applications from areas such as physics and chemistry, engineering, biology, and business (operations research). The mathematical models lead to discrete or continuous processes that may be deterministic or stochastic. Dimensional analysis and scaling are introduced to prepare a model for study. Analytic and computational tools from a broad range of applied mathematics will be used to obtain information about the models. The mathematical results will be compared to physical data to assess the usefulness of the models. Credit may not be granted for both MATH 486 and MATH 522.

**Lecture:**3

**Lab:**0

**Credits:**3

**Satisfies:**Communications (C)

**Mathematical Modeling II**

The formulation of mathematical models, solution of mathematical equations, interpretation of results. Selected topics from queuing theory and financial derivatives.

**Prerequisite(s):**MATH 252

**Lecture:**3

**Lab:**0

**Credits:**3

**Ordinary Differential Equations and Dynamical Systems**

Boundary-value problems and Sturm-Liouville theory; linear system theory via eigenvalues and eigenvectors; Floquet theory; nonlinear systems: critical points, linearization, stability concepts, index theory, phase portrait analysis, limit cycles, and stable and unstable manifolds; bifurcation; and chaotic dynamics.

**Lecture:**3

**Lab:**0

**Credits:**3

**Partial Differential Equations**

First-order equations, characteristics. Classification of second-order equations. Laplace's equation; potential theory. Green's function, maximum principles. The wave equation: characteristics, general solution. The heat equation: use of integral transforms.

**Prerequisite(s):**MATH 461

**Lecture:**3

**Lab:**0

**Credits:**3

**Reading and Research**

Independent reading and research. **Instructor permission required.**

**Credit:**Variable

**Satisfies:**Communications (C)

**Special Problems**

Special problems.

**Credit:**Variable

**Satisfies:**Communications (C)