## Course Catalogue

Herein are courses that we offer regularly. For undergraduate courses, we indicate when these courses will usually be offered; occasional exceptions may occur.

### Undergraduate Courses

**APMA 0160. Introduction to Computing Sciences (offered every spring)**

For students in any discipline that may involve numerical computations. Includes instruction for programming in MATLAB. Applications include solution of linear equations (with vectors and matrices) and nonlinear equations (by bisection, iteration, and Newton's method), interpolation, and curve-fitting, difference equations, iterated maps, numerical differentiation and integration, and differential equations. Prerequisite: MATH 0100 or its equivalent.

**APMA 0070. Introduction to Applied Complex Variables**

Applications of complex analysis that do not require calculus as a prerequisite. Topics include algebra of complex numbers, plane geometry by means of complex coordinates, complex exponentials, and logarithms and their relation to trigonometry, polynomials, and roots of polynomials, conformal mappings, rational functions and their applications, finite Fourier series and the FFT, iterations and fractals. Uses MATLAB, which has easy and comprehensive complex variable capabilities.

**APMA 0090. Introduction to Mathematical Modeling**

We will explore issues of mathematical modeling and analysis. Five to six self-contained topics will be discussed and developed. The course will include seminars in which modeling issues are discussed, lectures to provide mathematical background, and computational experiments. Required mathematical background is knowledge of one-variable calculus, and no prior computing experience will be assumed.

**APMA 0100. Elementary Probability for Applications**

This course serves as an introduction to probability and stochastic processes with applications to practical problems. It will cover basic probability and stochastic processes such as basic concepts of probability and conditional probability, simple random walk, Markov chains, continuous distributions, Brownian motion and option pricing. Enrollment limited to 20 first year students.

**APMA 0120. Mathematics of Finance**

The current volatility in international financial markets makes it imperative for us to become competent in financial calculations early in our liberal arts and scientific career paths. This course is designed to prepare the student with those elements of mathematics of finance appropriate for the calculations necessary in financial transactions.

**APMA 0200. Introduction to Modelling **

This course provides an introduction to the mathematical modeling of selected biological, chemical, engineering, and physical processes. The goal is to illustrate the typical way in which applied mathematicians approach practical applications, from understanding the underlying problem, creating a model, analyzing the model using mathematical techniques, and interpreting the findings in terms of the original problem. Single-variable calculus is the only requirement; all other techniques from differential equations, linear algebra, and numerical methods, to probability and statistics will be introduced in class. Prerequisites: Math 0100 or equivalent.

**APMA 0330, 0340. Methods of Applied Mathematics I,II (each offered every semester)**

Mathematical techniques involving differential equations used in the analysis of physical, biological and economic phenomena. Emphasis on the use of established methods, rather than rigorous foundations. I: First and second order differential equations. II: Applications of linear algebra to systems of equations; numerical methods; nonlinear problems and stability; introduction to partial differential equations; introduction to statistics. Prerequisite: MATH 0100.

**APMA 0350, 0360. Methods of Applied Mathematics I,II (each offered every semester)**

Follows APMA 0330, APMA 0340. Intended primarily for students who desire a rigorous development of the mathematical foundations of the methods used, for those students considering one of the applied mathematics concentrations, and for all students in the sciences who will be taking advanced courses in applied mathematics, mathematics, physics, engineering, etc. Three hours lecture and one hour recitation. MATH 0180 is desirable as a corequisite. Prerequisite: MATH 0100.

**APMA 0410. Mathematical Methods in the Brain Sciences (usually offered every Fall)**

Basic mathematical methods commonly used in the cognitive and neural sciences. Topics include: introduction to differential equations, emphasizing qualitative behavior; introduction to probability and statistics, emphasizing hypothesis testing and modern nonparametric methods; and some elementary information theory. Examples from biology, psychology, and linguistics. Prerequisites: MATH 0100 or equivalent.

**APMA 0650. Essential Statistics (offered ****every**** Spring)**

A first course in statistics emphasizing statistical reasoning and basic concepts. Comprehensive treatment of most commonly used statistical methods through linear regression. Elementary probability and the role of randomness. Data analysis and statistical computing using Excel. Examples and applications from the popular press and the life, social and physical sciences. No mathematical prerequisites beyond high school algebra.

**APMA 1070. Quantitative Models of Biological Systems (offered ****every**** Fall)**

An introduction to the use of quantitative modeling techniques in solving problems in biology. Each year one major biological area is explored in detail from a modeling perspective. The particular topic will vary from year to year. Mathematical techniques will be discussed as they arrive in the context of biological problems. Prerequisites: Introductory Level Biology, APMA 0330, APMA 0340, or APMA 0350, APMA 0360, or written permission.

**APMA 1080. Inference in Genomics and Molecular Biology (offered every Spring)**

Sequencing of genomes has generated a massive quantity of fundamental biological data. We focus on drawing traditional and Bayesian statistical inferences from these data, including: alignment of biopolymer sequences; prediction of their structures, regulatory signals; significances in database searches; and functional genomics. Emphasis is on inferences in the discrete high dimensional spaces. Statistical topics: Bayesian inference, estimation, hypothesis testing and false discovery rates, statistical decision theory. APMA 1650, 1655 or MATH 1610; BIOL 0200 or CSCI 1450 recommended; Matlab or programming experience. Programming experience and good mathematical background are essential. For graduate level, probability theory is strongly suggested. Prerequisite: One course on programming from the following list: APMA 0090, 0160, CSCI 0040, 0150 or 0170.

**APMA 1160. An Inroduction to Numerical Optimization****(offered every Spring) **

This course provides a thorough introduction to numerical methods and algorithms for solving non-linear continuous optimization problems. A particular attention will be given to the mathematical underpinnings to understand the theoretical properties of the optimization problems and the algorithms designed to solve them. Topics will include: line search methods, trust-region methods, nonlinear conjugate gradient methods, an introduction to constrained optimization (Karush-Kuhn-Tucker conditions, mini-maximization, saddle-points of Lagrangians). Some applications in signal and image processing will be explored. Prereq: functions of several variables (MATH 0180) or equivalent of this. Computational Linear Algebra (APMA 1170) or a similar course is recommended.

**APMA 1170. Introduction to Computational Linear Algebra (offered**

**every**

**Fall)**

Focuses on fundamental algorithms in computational linear algebra with relevance to all science concentrators. Basic linear algebra and matrix decompositions (Cholesky, LU, QR, etc.), round-off errors and numerical analysis of errors and convergence. Iterative methods and conjugate gradient techniques. Computation of eigenvalues and eigenvectors, and an introduction to least squares methods. A brief introduction to Matlab is given. Prerequisites: MATH 0520 is recommended, but not required.

**APMA 1180. Introduction to the Numerical Solution of Partial Differential Equations (offered every other Spring)**

Fundamental numerical techniques for solving ordinary and partial differential equations. Overview of techniques for approximation and integration of functions. Development of multistep and multistage methods, error analysis, step-size control for ordinary differential equations. Solution of two-point boundary value problems, introduction to methods for solving linear partial differential equations. Introduction to Matlab is given but some programming experience is expected. Prerequisites: APMA 0330, 0340 or 0350, 0360. APMA 1170 is recommended.

**APMA 1200. Operational Analysis: Probabilistic models (offered every Spring)**

Basic probabilistic problems and methods in operations research and management science. Methods of problem formulation and solution. Markov chains, birth-death processes, stochastic service and queueing systems, the theory of sequential decisions under uncertainty, dynamic programming. Applications. Prerequisite: APMA 1650 or MATH 1610, or equivalent.

**APMA 1210. Operations Research: Deterministic Methods (ENGN 1310, offered every Fall)**

An introduction to the basic mathematical ideas and computational methods of optimizing allocation of effort or resources, with or without constraints. Linear programming, network models, dynamic programming, and integer programming.

**APMA 1330. Methods of Applied Mathematics III, IV (usually offered every Fall)**

Review of vector calculus and curvilinear coordinates. Partial differential equations. Heat conduction and diffusion equations, the wave equation, Laplace and Poisson equations. Separation of variables, special functions, Fourier series and power series solution of differential equations. Sturm-Liouville problem and eigenfunction expansions.

**APMA 1340. Methods of Applied Mathematics III, IV**

See Methods Of Applied Mathematics III, IV (APMA 1330) for course description.

**APMA 1360. Topics in Chaotic Dynamics **

Overview and introduction to dynamical systems. Local and global theory of maps. Attractors and limit sets. Lyapunov exponents and dimensions. Fractals: definition and examples. Lorentz attractor, Hamiltonian systems, homoclinic orbits and Smale horseshoe orbits. Chaos in finite dimensions and in PDEs. Can be used to fulfill the senior seminar requirement in applied mathematics. Prerequisites: Differential equations and linear algebra.

**APMA 1650. Statistical Inference I (offered every Fall)**

APMA 1650 begins an integrated first course in mathematical statistics. The first half of APMA 1650 covers probability and the last half is statistics, integrated with its probabilistic foundation. Specific topics include probability spaces, discrete and continuous random variables, methods for parameter estimation, confidence intervals, and hypothesis testing. Prerequisite: MATH 0100 or its equivalent.

**APMA 1660. Statistical Inference II (offered every Spring)**

APMA 1660 is designed as a sequel to APMA 1650 to form one of the alternative tracks for an integrated year's course in mathematical statistics. The main topic is linear models in statistics. Specific topics include likelihood-ratio tests, nonparametric tests introduction to statistical computing, matrix approach to simple-linear and multiple regression, analysis of variance, and design of experiments. Prerequisite: APMA 1650 or equivalent, basic linear algebra.

**APMA 1655. Statistical Inference I (offered every Fall and Spring)**A more challenging and in-depth version of APMA 1650. The class lectures for APMA 1650/1655 will be the same: they are taught at the same time by the same instructor, but the homework and other assignments will be different. APMA 1655 can be used as a replacement for APMA 1650. Prerequisite: MATH 0100 or its equivalent.

**APMA 1690. Computational Probability and Statistics (offered every Fall)**

Examination of probability theory and statistical inference from the point of view of modern computing. Random number generation, Monte Carlo methods, simulation, and other special topics. Prerequisites: calculus, linear algebra, APMA 1650, or equivalent. Some experience with programming desirable.

**APMA 1710. Information Theory (CSCI 1850, ENGN 1510) (usually offered every Fall) **

Information theory is the study of the fundamental limits of information transmission and storage. This course, intended primarily for advanced undergraduates, and beginning graduate students, offers a broad introduction to information theory and its applications: Entropy and information; lossless data compression, communication in the presence of noise, capacity, channel coding; source-channel separation; lossy data compression.

**APMA 1720. Monte Carlo Simulation with Applications to Finance (offered every other Spring)**

The course will cover the basics of Monte Carlo and its applications to financial engineering: generating random variables and simulating stochastic processes; analysis of simulated data; variance reduction techniques; binomial trees and option pricing; Black-Scholes formula; portfolio optimization; interest rate models. The course will use MATLAB as the standard simulation tool. Prerequisites: APMA 1650 or MATH 1610. Offered in alternate years.

**APMA 1740. Recent Applications of Probability and Statistics (offered every Spring)**

This course covers a selection of modern applications of probability and statistics in the computational, cognitive, engineering, and neural sciences. The course will be rigorous, but the emphasis will be on application. Topics will include: Markov chains and their applications to MCMC computing and hidden Markov models; Dependency graphs and Bayesian networks; parameter estimation and the EM algorithm; Kalman and particle filtering; Nonparametric statistics ("learning theory"), including consistency, bias/variance tradeoff, and regularization; the Bayesian approach to nonparametrics, including the Dirichlet and other conjugate priors; principle and independent component analysis; Gibbs distributions, maximum entropy, and their connections to large deviations. Each topic will be introduced with several lectures on the mathematical underpinnings, and concluded with a computer project, carried out by each student individually, demonstrating the mathematics and the utility of the approach.

**APMA 1850. Introduction to High Performance Parallel Computing**There is no description available.

**APMA 1880. Advanced Matrix Theory**Canonical forms of orthogonal, Hermitian and normal matrices: Rayleigh quotients. Norms, eigenvalues, matrix equations, generalized inverses. Banded, sparse, non-negative and circulant matrices. Prerequisite: APMA 0340 or 0360, or MATH 0520 or 0540, or permission of the instructor.

**APMA 1910. Race and Gender in the Scientific Community** This course examines the (1) disparities in representation in the scientific community, (2) issues facing different groups in the sciences, and (3) paths towards a more inclusive scientific environment. We will delve into the current statistics on racial and gender demographics in the sciences and explore their background through texts dealing with the history, philosophy, and sociology of science. We will also explore the specific problems faced by underrepresented and well-represented racial minorities, women, and LGBTQ community members. The course is reading intensive and discussion based.

**APMA 1930, APMA 1940. Senior Seminars (usually offered every semester)**

Independent study and special topics seminars in various branches of applied mathematics, change from year to year. Recent topics include Mathematics of Speculation, Scientific Computation, Coding and Information Theory, Topics in Chaotic Dynamics, and Software for Mathematical Experiments.

**APMA 1930I - Random Matrix Theory**In the past few years, random matrices have become extremely important in a variety of fields such as computer science, physics and statistics. They are also of basic importance in various areas of mathematics. This class will serve as an introduction to this area. The focus is on the basic matrix ensembles and their limiting distributions, but several applications will be considered. Prerequisites: MATH 0200 or 0350; and MATH 0520 or 0540; and APMA 0350, 0360, 1650, and 1660. APMA 1170 and MATH 1010 are recommended, but not required.

**APMA 1930K. Stability of Differential Equations in Applications**Basic stability and instability analysis of differential equations will be covered. Various examples of physical and biological applications will be studied and shared with the class.

**APMA 1930M. Applied Asymptotic Analysis**

Many problems in applied mathematics and physics are nonlinear and are intractable to solve using elementary methods. In this course we will systematically develop techniques for obtaining quantitative information from nonlinear systems by exploiting small scale parameters. Topics will include: regular and singular perturbations, boundary layer theory, multiscale and averaging methods and asymptotic expansions of integrals. Along the way, we will discuss many applications including nonlinear waves, coupled oscillators, nonlinear optics, fluid dynamics and pattern formation.

**APMA 1930U. Introduction to Stochastic Differential Equations**This seminar course serves as an introduction to stochastic differential equations at the senior undergraduate level. Topics covered include Brownian motion and white noise, stochastic integrals, the Itô calculus, existence and uniqueness of solutions to Itô stochastic differential equations, and the Feynman-Kac formula. More advanced topics, such as fractional Brownian motion, Lévy processes, and stochastic control theory, may be addressed depending on the interests of the class and time restrictions.

**APMA 1940R - Linear and Nonlinear Waves**

From sound and light waves to water waves and traffic jams, wave phenomena are everywhere around us. In this seminar, we will discuss linear and nonlinear waves as well as the propagation of wave packets. Among the tools we shall use and learn about are numerical simulations in Matlab and analytical techniques from ordinary and partial differential equations. We will also explore applications in nonlinear optics and to traffic flow problems. Prerequisites: MATH 0180 and either APMA 0330-0340 or APMA 0350-0360. No background in partial differential equations is required.

**APMA 1940S. Topics in Applied Differential Equations**

The course will cover several topics of ordinary differential equations arising from other disciplines such as physics, chemistry, biology, and engineering, with an emphasis on the modeling of various underlining equations. The course will also be supplemented with a use of computer algebra systems like MATHEMATICA.

**APMA 1940U. Filtering of Prediction of Hidden Markov Models**

This course is built around the problem of estimating noisily observed dynamics from a sequence of data. This is a fascinating engineering problem with strong ties to probability theory and with numerous scientific and technological applications. Topics covered will include conditioning and optimal estimates, filtering and interpolation of Markov chains, elements of stochastic calculus and Wiener processes, and the continuous-time Kalman filter. Applications of hidden Markov models to problems arising in finance, genetics, and epidemiological modeling will also be considered.

**APMA 1940V. Randomized algorithms for counting, integration and optimization**We consider the construction and analysis of random methods for approximating sums and integrals, and related questions. As a prototype example, consider the problem of counting the number of vectors of a fixed dimension with integer components that satisfy a collection of linear equality and inequality constraints. Depending on the number of constraints, this could be like a problem of counting the number of needles in a haystack, and straightforward enumeration is impossible. There are now a variety of randomized methods that can (with varying degrees of success) attack this problem and other problems with similar difficult features. We will survey some of the methods and the problem classes to which they can be applied.

**APMA 1940W. Randomized algorithms for counting, integration and optimization**We consider the construction and analysis of random methods for approximating sums and integrals, and related questions. As an example, consider the problem of counting the number of vectors with integer components that satisfy a collection of linear equality and inequality constraints. Depending on the number of constraints, this could be a problem of counting the number of needles in a haystack, and straightforward enumeration is impossible. There are now a variety of randomized methods that can attack this problem and other problems with similar difficult features. We survey some of the methods and the problems to which they apply. The prerequisite for the course is APMA 1630 or APMA 1655.

** APMA 1970. Independent Study **

### GRADUATE LEVEL COURSES

**APMA 2110. Real Analysis (MATH 2210) **

Provides the basis of real analysis which is fundamental to many of the other courses in the program: metric spaces, measure theory, and the theory of integration and differentiation.

**APMA 2120. Hilbert Spaces and Their Applications (MATH 2220)**

A continuation of APMA 2110: Metric spaces, Banach spaces, Hilbert spaces, the spectrum of bounded operators on Banach and Hilbert spaces, compact operators, applications to integral and differential equations.

**APMA 2190, APMA 2200. Nonlinear Dynamical Systems: Theory and Applications**

Basic theory of ordinary differential equations, flows, and maps. Two-dimensional systems. Linear systems. Hamiltonian and integrable systems. Lyapunov functions and stability. Invariant manifolds, including stable, unstable, and center manifolds. Bifurcation theory and normal forms. Nonlinear oscillations and the method of averaging. Chaotic motion, including horseshoe maps and the Melnikov method. Applications in the physical and biological sciences.

**APMA 2210. Topics in Differential Equations**

A variety of topics in nonlinear dynamics, based in part on the interests of the students, will be covered. Among the possible topics are: bifurcation theory, degree theory, infinite-dimensional systems, delay-differential equations, exponential dichotomies, skew-product flows, and monotone dynamical systems. The prerequisite for the course is a solid (rigorous) grounding in nonlinear dynamics, typically APMA 2190-2200 or equivalent.

**APMA 2230, APMA 2240. Partial Differential Equations (MATH 2370, 2380)**

The theory of the classical partial differential equations; the method of characteristics and general first order theory. The Fourier transform, the theory of distributions, Sobolev spaces, and techniques of harmonic and functional analysis. More general linear and nonlinear elliptic, hyperbolic, and parabolic equations and properties of their solutions, with examples drawn from physics, differential geometry, and the applied sciences. Semester II concentrates on special topics chosen by the instructor.

**APMA 2410. Fluid Dynamics I (ENGN 2810)**

Formulation of the basic conservation laws for a viscous, heat conducting, compressible fluid. Molecular basis for thermodynamic and transport properties. Kinematics of vorticity and its transport and diffusion. Introduction to potential flow theory. Viscous flow theory; the application of dimensional analysis and scaling to obtain low and high Reynolds number limits.

**APMA 2420. Fluid Dynamics II (ENGN 2820)**

A continuation of APMA 2410. Topics include: Low Reynolds number flows, boundary layer theory, wave motion, stability and transition, acoustics, and compressible flows.

**APMA 2550. Numerical Solution of Partial Differential Equations I**

Finite difference methods for solving time-depend initial value problems of partial differential equations. Fundamental concepts of consistency, accuracy, stability and convergence of finite difference methods will be covered. Associated well-posedness theory for linear time-dependent PDEs will also be covered. Some knowledge of computer programming expected.

**APMA 2560. Numerical Solution of Partial Differential Equations II**

An introduction to weighted residual methods, specifically spectral, finite element and spectral element methods. Topics include a review of variational calculus, the Rayleigh-Ritz method, approximation properties of spectral end finite element methods, and solution techniques. Homework will include both theoretical and computational problems.

**APMA 2570. Numerical Solution of Partial Differential Equations III**

We will cover advanced topics rotating between spectral methods, theory of finite element methods, and discontinuous Galerkin methods for partial differential equations. Algorithm formulation, analysis, and efficient implementation issues will be addressed. See Banner for current course description. APMA 2550/APMA 2560 or equivalent knowledge in numerical methods will be a prerequisite.

**APMA 2580. Computational Fluid Dynamics**

We rotate topics between incompressible and compressible fluid dynamics from one year to the next. See Banner for the most current course description. Prerequisite: APMA 2550/APMA 2560 or equivalent knowledge in numerical methods.

**APMA 2610. Recent Applications of Probability and Statistics**

This is a topics course, covering a selection of modern applications of probability and statistics in the computational, cognitive, engineering, and neural sciences. The course will be rigorous, but the emphasis will be on application. Topics will include: Markov chains and their applications to MCMC computing and hidden Markov models; Dependency graphs and Bayesian networks; parameter estimation and the EM algorithm; Kalman and particle filtering; Nonparametric statistics ("learning theory"), including consistency, bias/variance tradeoff, and regularization; the Bayesian approach to nonparametrics, including the Dirichlet and other conjugate priors; principle and independent component analysis; Gibbs distributions, maximum entropy, and their connections to large deviations. Each topic will be introduced with several lectures on the mathematical underpinnings, and concluded with a computer project, carried out by each student individually, demonstrating the mathematics and the utility of the approach.

**APMA 2630, APMA 2640. Theory of Probability (MATH 2630, MATH 2640)**

A two-semester course in probability theory. Semester I includes an introduction to probability spaces and random variables, the theory of countable state Markov chains and renewable processes, laws of large numbers and the central limit theorems. Measure theory is first used near the end of the first semester (APMA 2110 may be taken concurrently). Semester II provides a rigorous mathematical foundation to probability theory and covers conditional probabilities and expectations, limit theorems for sums of random variables. martingales, ergodic theory, Brownian motion and an introduction to stochastic process theory.

**APMA 2660. Stochastic Processes**

Review of the theory of stochastic differential equations and reflected SDEs, and of the ergodic and stability theory of these processes. Introduction to the theory of weak convergence of probability measures and processes. Concentrates on applications to the probabilistic modeling, control, and approximation of modern communications and queuing networks; emphasizes the basic methods, which are fundamental tools throughout applications of probability.

**APMA 2670. Mathematical Statistics I**

Advanced Statistical Inference. Emphasis on the theoretical aspects of the subject. Frequentist and Bayesian approaches, and their interplay. Topics include: general theory of inference, point and set estimation, hypothesis testing, and modern computational methods (E-M Algorithm, Markov Chain Monte Carlo, Bootstrap). Students should have prior knowledge of probability theory, at the level of APMA 2630 or higher.

**APMA 2680. Mathematical Statistics II**

This course provides a solid presentation of modern nonparametric statistical methods. Topics include: density estimation, adaptive smoothing, cross-validation, bootstrap, classification and regression trees and their connection to the Huffman code, projection pursuit, the ACE algorithm for time series prediction, support vector machines, and learning theory. The course will provide the mathematical underpinnings, but it will also touch upon some applications in computer vision/speech recognition, and biological, neural, and cognitive sciences. Prerequisite: APMA 2760

**APMA 2810, 2820. Seminars in Applied Mathematics Topics Courses**Graduate level seminars in various branches of applied mathematics change from year to year. APMA 2810 Topics Courses are offered during the Fall semester, and APMA 2820 Topics Courses are offered during the Spring semester. The following courses have been offered in past semesters, however for current listings, please see BANNER.

**APMA 2812C. Interacting Particle Systems**The course will provide an introduction to both static and dynamic interacting particle systems. Topics covered will include basic constructions of interacting particle systems and their limits, including mean-field approximations, analysis of their ergodic properties, including Gibbs measures, Markov random fields and phase transitions. A broad range of applications, ranging from queueing networks to biology and statistical physics, will be used to illustrate the theory. Prerequisite: A good knowledge of measure-theoretic probability theory (at the level of APMA 2630 and 2640)

**APMA 2970. Preliminary Examination Preparation **

**APMA 2980. Research in Applied Mathematics **

**APMA 2990. Thesis Preparation**