Courses

Undergraduate Courses

APMA 0090. Introduction to Modeling
Topics of Applied Mathematics, introduced in the context of practical applications where defining the problems and understanding what kinds of solutions they can have is the central issue. Computations are performed in MATLAB; instruction is provided.

APMA 0160. Introduction to Computing Sciences
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 0330, 0340. Methods of Applied Mathematics I,II
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
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
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
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 (BIOL 1490)
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
Traditional and Bayesian statistical inferences on biopolymer data including; sequence alignment; structure prediction; regulatory signals; significances of searches; phylogeny; and functional genomics. Emphasis is on discrete high dimensional objects common in field. Statistical topics: parameter estimation; hypothesis testing and false discovery rates; statistical decision theory; and Bayesian posterior inference. Prerequisites: APMA 1650 and BIOL 1470 or BIOL 1500, and programming experience minimally Matlab.

APMA 1170. Introduction to Computational Linear Algebra
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
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
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)
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
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 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
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
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 1670. Statistical Analysis of Time Series
Time series analysis is an important branch of mathematical statistics with many applications to signal processing, econometrics, geology, etc. The course emphasizes methods for analysis in the frequency domain, in particular, estimation of the spectrum of a time-series, but time domain methods are also covered. Prerequisite: elementary probability and statistics on the level of APMA 1650-1660. Offered in alternate years.

APMA 1680. Nonparametric Statistics
A systematic treatment of the distribution-free alternatives to classical statistical tests. These non-parametric tests make minimum assumptions about distributions governing the generation of observations, yet are of nearly equal power to the classical alternatives. Prerequisite: APMA 1650 or equivalent.

APMA 1690. Computational Probability and Statistics
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 1700. The Mathematics of Insurance
The course consists of two parts. The first treats life contingencies, i.e., the construction of models for individual life insurance contracts. The second treats the Collegective Theory of Risk, which constructs mathematical models for the insurance company and its portfolio of policies as a whole. Suitable also for students proceeding to the Institute of Actuaries examinations. Prerequisites: Probability to the level of APMA 1650 or MATH 1610. Offered in alternate years.

APMA 1710. Information Theory (CSCI 1850, ENGN 1510)
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 1930. Senior Seminar
Independent study and special topics seminars in various branches of applied mathematics, changing 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 1940. Cryptography (MATH 1580)
(Fall SO1) Topics include symmetric ciphers, public key ciphers, complexity, digital signatures, applications and protocols. MATH 1530 is not required for this course. What is needed from abstract algebra and elementary number theory will be covered. Prerequisite: MATH 0520 or MATH 0540.

APMA 1940. Mathematical Models in Computational Biology (Section I) (Spring S01)
This course is designed to introduce students to the use of mathematical models in biology as well as some more recent topics in cojputatinoal bilogy. Mathematical techniques will involve difference equations and dynamical systems theory, ordinary differential equations and some partial differential equations. These techniques will be applied in the study of many biologial applications such as: (i) Difference equations: population dynamics, red blood cell production, population genetics; (ii) Ordinary differential equations: predator-prey models, Lotka-Volterra model, modeling the evolution of the genome, heart beat model/cycle, transmission dynamics of HIV and gonorrhea; (iii) Partial differential equations: tumor growth, modeling evolution of the genome, pattern formation. Prerequisites: APMA 0330 and APMA 0340

APMA 1940. The History of Mathematics (Section II)
The course will not be a systematic survey but will focus on specific topics in the history of mathematics such as Archimedes and integration, Oresme and graphing, Newton and infinitesimals, simple harmonic motion, the discovery of 'Fourier' series, the Monte Carlo method, reading and analyzing the original texts. A basic knowledge of calculus will be assumed.

APMA 1940. Introduction to Mathematical Models in Computational Biology (Section III)
This course is an introduction to the use of mathematical models in biology. Mathematical techniques will involve difference equations and dynamical systems theory, ordinary differential equations and some partial differential equations. These techniques will be applied in the study of many biological applications such as: population dynamics, transmission dynamics of HIV and tumor growth. Prerequisites: APMA 0330 and APMA 0340.

APMA 1950, APMA1960. Independent Study

Graduate Level Courses

APMA 2050, APMA 2060. Mathematical Methods of Applied Science Introduces science and engineering graduate students to a variety of fundamental mathematical methods. Topics include linear algebra, complex variables, Fourier-series, Fourier and Laplace transforms and their applications, ordinary differential equations, tensors, curvilinear coordinates, integral equations, partial differential equations, and calculus of variations.

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 2130. Methods of Applied Mathematics: Partial Differential Equations Solution methods and basic theory for first and second order partial differential equations. Geometric interpretation and solution of linear and nonlinear first order equations by characteristics; formation of caustics and propagation of discontinuities. Classification of second order equations and issues of well-posed problems. Green's functions and maximum principles for elliptic systems. Characteristic methods and discontinuous solutions for hyperbolic systems.

APMA 2140. Methods of Applied Mathematics: Integral Equations
Integral equations. Fredholm and Volterra theory, expansions in orthogonal functions, theory of Hilbert-Schmidt. Singular integral equations, method of Wiener-Hopf. Calculus of variations and direct methods.

APMA 2160. Methods of Applied Mathematics: Asymptotics
Calculus of asymptotic expansions, evaluation of integrals. Solution of linear ordinary differential equations in the complex plane, WKB method, special functions. May be taken concurrently with APMA 2140.

APMA 2170. Functional Analysis and Applications
Topics vary according to interest of instructor and class.

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
The main topic is universality in chaos, in particular a rigorous and self-contained treatment of Feigenbaum's notion of quantitative universality and renormalization in one-dimensional maps.

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 2260. Introduction to Stochastic Control Theory
This course serves as an introduction to the theory of stochastic control and dynamic programming technique. Optimal stopping, total expected (discounted) cost problems, and long-run average cost problems will be discussed in discrete time setting. The last part of the course deals with continuous time deterministic control and game problems. The course requires some familiarity with probability theory.

APMA 2270, APMA 2280. Topics in the Control of Systems
Topics will vary from year to year, but will include optimal control theory, computational methods for control, and algebraic methods in systems science.

APMA 2330. Foundations of Continuum Mechanics (ENGN 2210)
An introduction to the mathematical foundations of continuum mechanics. Vectors and tensors, properties and basic operations. Kinematics of deformation. Conservation laws, thermodynamics. Stress. Constitutive equations. Elastic, viscous, and viscoelastic response. Linearization. Simple problems in finite and linear elasticity, and in Navier-Stokes flows. Creep and relaxation in linear viscoelasticity.

APMA 2340. Linear Elasticity (ENGN 2240)
General theorems in linear elasticity. Basic singular solutions. Boussinesq-Papkovich and Galerkin representations. Curvilinear coordinates. Cavity, inclusion, crack and contact problems. States of plane strain, plane stress, and axial symmetry. Complex variable techniques. Torsion. Thermoelasticity.

APMA 2350. Advanced Elasticity (ENGN 2270)
Large elastic deformations. Controllable deformations of incompressible materials. Initial stress problems. Elastic stability. Additional topics may include membrane theory, fiber-reinforced materials, second-order elasticity.

APMA 2360. Topics in Continuum Mechanics (ENGN 2280)
Advanced topics such as finite elasticity theory, initial stress problems, elastic stability, quasiconvexification in membrane theory, and mechanics of fiber-reinforced materials.

APMA 2370. Plasticity (ENGN 2290)
Theory of the inelastic behavior of materials with negligible time effects. Experimental background for metals and fundamental postulates for plastic stress-strain relations. Variational principles for incremental elastic-plastic problem, uniqueness. Upper and lower bound theorems of limit analysis and shakedown. Slip line theory. Representative problems in strucutral analysis, metal forming, indentation, strain and stress concentrations at notches, and ductile failure.

APMA 2380. Stress Waves in Solids (ENGN 2260)
Interested students should register for ENGN 2260.

APMA 2390. Viscoelasticity (ENGN 2250)
Interested students should register for ENGN 2250.

APMA 2410. Fluid Dynamics I (ENGN 2810)
An introduction to fundamental concepts of the mechanics and thermodynamics of fluid flow. Major topics include compressible and incompressible flows, ciscous and inviscid flows, and vorticity dynamics.

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 2470, APMA 2480. Topics in Fluid Dynamics
Initial review of topics selected from flow stability, turbulence, turbulent mixing, surface tension effects, and thermal convection. Followed by focused attention on the dynamics of dispersed two-phase flow and complex fluids.

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
Examines the development and analysis of spectral methods for the solution of time-dependent partial differential equations. Topics include key elements of approximation and stability theory for Fourier and polynomial spectral methods as well as attention to temporal integration and numerical aspects. Some knowledge of computer programming expected.

APMA 2570. Numerical Solution of Partial Differential Equations III
We will cover finite element methods for ordinary differential equations and for elliptic, parabolic and hyperbolic partial differential equations. Algorithm development, analysis, and computer implementation issues will be addressed. In particular, we will discuss in depth the discontinuous Galerkin finite element method.

APMA 2580. Computational Fluid Dynamics
An introduction to computational fluid dynamics with emphasis on incompressible flows. Reviews the basic discretization methods (finite differences and finite volumes) following a pedagogical approach from basic operators to the Navier-Stokes equations. Suitable for first-year graduate students, more advanced students, and senior undergraduates. Requirements include three to four computer projects. Material from APMA 1170 and APMA 1180 is appropriate as prerequisite, but no prior knowledge of fluid dynamics is necessary.

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 likely 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. There will be no exams.

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 2690, APMA 2700. Topics in Statistics and its Applications
Advanced topics varying from year to year, including: non-parametric methods for density estimation, regression and prediction in time-series; cross-validation and adaptive smoothing techniques; bootstrap; recursive partitioning, projection-pursuit, ACE algorithm; non-parametric classification and clustering; stochastic Metropolis-type simulation and global optimization algorithms; Markov random fields and statistical mechanics; applications to image processing, speech recognition and neural networks.

APMA 2720. Information Theory
Information theory and its relationship with probability, statistics, and data compression. Entropy. The Shannon-McMillan-Breiman theorem. Shannon's source coding theorems. Statistical inference; hypothesis testing; model selection; the minimum description length principle. Information-theoretic proofs of limit theorems in probability: Law of large numbers, central limit theorem, large deviations, Markov chain convergence, Poisson approximation, Hewitt-Savage 0-1 law. Prerequisites: APMA 2630; APMA 1710.

APMA 2810. Seminars in Applied Mathematics (FalI)

APMA 2810R. Computational Biology Methods for Gene/Protein Networks and Structural Proteomics (BIOL 2930)
The course presents computational and statistical methods for gene and protein networks and structural proteomics. It emphasizes: (1) Probabilistic models for gene regulatory networks via microarray, chromatic immunoprecipitation, and cisregulatory data; (2) Signals transduction pathways via tandem mass spectrometry data; and (3) Molecular modeling forligand-receptor coupling and docking. The course is recommended for graduate students.

APMA 2810S. Topics in Control
The course deals with advanced topics in control. Current topics include direct methods (including stochastic finite element method for Levy processes) and subsystems and embeddings.

APMA 2810T. Nonlinear Partial Differential Equations
This course introduces techniques useful for solving many nonlinear partial differential equations, with emphasis on elliptic problems. PDEs from a variety of applications will be discussed. Contact the instructor about prerequisites.

APMA 2810U. Topics in Differential Equations
This course is a sequel to APMA 2170 and concentrates on similar material.

APMA 2810V. Topics in Partial Differential Equations
The course will cover an introduction of the L_p theory of second order elliptic and parabolic equations, finite difference approximations of elliptic and parabolic equations, and some recent developments in the Navier-Stokes equations and quasi-geostrophic equations. Some knowledge of real analysis will be expected.

APMA 2810W. Advanced Topics in High Order Numerical Methods for Convection Dominated Problems
This is an advanced seminar course. We will cover several topics in high order numberical methods for convection dominated problems, including methods for solving Boltzmann type equations, methods for solving unsteady and steady Hamilton-Jacobi equations, and methods for solving moment models in semi-conductor device simulations. Prerequisite: APMA 2550 or equivalent knowledge of numerical analysis.

APMA 2820. Seminar in Applied Mathematics (Spring)

APMA 2820J. Numerical Linear Algebra
Solving large systems of linear equations: The course will use the text of Treften and Bao that includes all the modern concepts of solving linear questions.

APMA 2820P. Foundations in Statistical Inference in Molecular Biology
In molecular biology, inferences in high dimensions with missing data are common. A conceptual framework for Bayesian and frequentist inferences in this setting include: sequence alignment, RNA secondary structure prediction, database search, and tiled arrays. Statistical topics: parameter estimation, hypothesis testing, recursions, and characterization of posterior spaces. This is a core course in proposed Ph.D. program in computational molecular biology.

APMA 2820R. Structure Theory of Control Systems
The course deals with the following problems: Given a family of control systems S and a family of control systems S', when does there exist an appropriate embedding of S into S'? Most of the course with deal with families of linear control systems. Knowledge of control theory and mathematical sophistication are required.

APMA 2820U. Structure Theory of Control Systems
The course deals with the following problems: given a family of control systems S and a family of control systems S', when does there exist an appropriate embedding of S into S'? Most of the course will deal with the families of linear control systems. Knowledge of control theory and mathematical sophistication are required.

APMA 2820V. Progress in the Theory of Shock Waves
The course will begin with a self-contained introduction to the theory of "hyperbolic conservation laws," that is quasilinear first order systems of partial differential equations whose solutions spontaneously develop singularities that propage as shock waves. Then a number of recent developments will be discussed. The aim is to familiarize the students with the current status of the theory as well as with the ever expanding areas of applications of the subject.

APMA 2820W. An introduction to the Theory of Large Deviations
The theory of large deviations attempts to estimate the probability of rare events and identify the most likely way they happen. The course will begin with a review of the general framework, standard techniques (change-of-measure, subadditivity, etc.), and elementary examples (e.g., Sanov's and Cramer's Theorems). We then will cover large deviations for diffusion processes and the Wentsel-Freidlin theory. The last part of the course will be one or two related topics, possibly drawn from (but not limited to) risk-sensitive control; weak convergence methods; Hamilton-Jacobi-Bellman equations; Monte Carlo methods. Prerequisites: APMA 2630 and APMA 2640.

APMA 2820X. Boundary Conditions for Hyperbolic Systems: Numerical and Far Field
Description to follow.

Updated March 21, 2007