Graduate Course Descriptions

AMS 501 Differential Equations and Boundary Value Problems I
Examples of initial and boundary value problems in which differential equations arise. Existence and uniqueness of solutions, systems of linear differential equations, and the fundamental solution matrix. Power series solutions, Sturm-Louisville theory, eigenfunction expansion, Green's functions.
Spring, 3 credits, ABCF grading
AMS 501 Webpage

AMS 502 Differential Equations and Boundary Value Problems II
Analytic solution techniques for, and properties of solutions of, partial differential equations, with concentration on second order PDEs. Techniques covered include: method of characteristics, separation of variables, eigenfunction expansions, spherical means, GreenÕs functions and fundamental solutions, and Fourier transforms. Solution properties include: energy conservation, dispersion, dissipation, existence and uniqueness, maximum and mean value principles.
Prerequisite: AMS 501
Fall, 3 credits, ABCF grading
AMS 502 webpage

AMS 503 Applications of Complex Analysis II
A study of those concepts and techniques in complex function theory that are of interest for their applications. Pertinent material is selected from the following topics: harmonic functions, calculus of residues, conformal mapping, and the argument principle. Application is made to problems in heat conduction, potential theory, fluid dynamics, and feedback systems.
Spring, 3 credits, ABCF grading
AMS 503 webpage

AMS 504 Foundations of Applied Mathematics II
An introductory course for the purpose of developing certain concepts and techniques that are fundamental in modern approaches to the solution of applied problems. An appropriate selection of topics is based on the concepts of metric spaces, compactness, sequences and convergence, continuity, differentiation and integration, function sequences, contraction mapping theorem. Strong emphasis on proofs.
Fall, 3 credits, ABCF grading
AMS 504 webpage

AMS 505 Applied Linear Algebra II
Review of matrix operations. Elementary matrices and reduction of general matrices by elementary operations, canonical forms, and inverses. Applications to physical problems. Coscheduled as AMS 505 or HPH 695.
Fall, 3 credits, ABCF grading
AMS 505 webpage

AMS 506 Finite Structures II
Problem solving in combinatorial analysis and graph theory using generating functions, recurrence relations, PolyaÕs enumeration formula, graph coloring, and network flows.
3 credits, ABCF grading
AMS 506 webpage

AMS 507 Introduction to Probability II
The topics include sample spaces, axioms of probability, conditional probability and independence, discrete and continuous random variables, jointly distributed random variables, characteristics of random variables, law of large numbers and central limit theorem, Markov chains. Note: Crosslisted with HPH 696.
Fall, 3 credits, ABCF grading
AMS 507 webpage

AMS 510 Analytical Methods for Applied Mathematics and Statistics II
Review of techniques of multivariate calculus, convergence and limits, matrix analysis, vector space basics, and Lagrange multipliers.
Fall, 3 credits, ABCF grading
AMS 510 webpage

AMS 511, Foundation of Quantitative Finance
Introduction to capital markets, securities pricing, and modern portfolio theory, including the organization and operation of securities market, the Efficient Market Hypothesis and its implications, the Capital Asset Pricing Model, the Arbitrage Pricing Theory, and more general factor models. Common stocks and their valuation, statistical analysis, and portfolio selection in a single-period, mean-variance context will be explored along with its solution as a quadratic program. Fixed income securities and their valuation, statistical analysis, and portfolio selection. Discussion of the development and use of financial derivatives. Introduction to risk neutral pricing, stochastic calculus, and the Black-Scholes Formula. Whenever practical, examples will use real market data. Numerical exercises and projects in a high-level programming environment will also be assigned.
Prerequisites: AMS505 or 510
3 credits, ABCF grading
AMS 511 webpage

AMS 512 Capital Markets and Portfolio Theory
Development of capital markets and portfolio theory in both continuous time and multi-period settings. Utility theory and its application to the determination of optimal consumption and investment policies. Asymptotic growth under conditions of uncertainty. Applications to problems in strategic asset allocation over finite horizons and to problems in public finance. Whenever practical, examples will use real market data. Numerical exercises and projects in a high-level programming environment will also be assigned.
Prerequisite: AMS 511
3 credits, ABCF grading
AMS 512 webpage

AMS 513 Financial Derivatives and Stochastic Calculus II
Further development of derivative pricing theory including the use of equivalent martingale measures, the Girsanov Theorem, the Radon-Nikodym Derivative, and a deeper, more general understanding of the Arbitrage Theorem. Numerical approaches to solving stochastic PDEÕs will be further developed. Applications involving interest rate sensitive securities and more complex options will be introduced. Whenever practical, examples will use real market data. Numerical exercises and projects in a high-level programming environment will also be assigned.
Prerequisite: AMS 511
3 credits, ABCF grading
AMS 513 webpage

AMS 514 Computational Finance II
Review of foundations: stochastic calculus, martingales, pricing, and arbitrage. Basic principles of Monte Carlo and the efficiency and effectiveness of simulation estimators. Generation of pseudo- and quasi-random numbers with sampling methods and distributions. Variance reduction techniques such as control variates, antithetic variates, stratified and Latin hypercube sampling, and importance sampling. Discretization methods including first and second order methods, trees, jumps, and barrier crossings. Applications in pricing American options, interest rate sensitive derivatives, mortgage-backed securities and risk management. Whenever practical, examples will use real market data. Extensive numerical exercises and projects in a general programming environment will also be assigned.
Prerequisite: AMS 512 and AMS 513
3 credits, ABCF grading
AMS 514 webpage

AMS 515 Case Studies in Computational Finance II
Actual applications of Quantitative Finance to problems of risk assessment, product design, portfolio management, and securities pricing will be covered. Particular attention will be paid to data collection and analysis, the design and implementation of software, and, most importantly, to differences that occur between Òtheory and practiceÓ in model application, and to the development of practical strategies for handling cases in which Òmodel failureÓ makes the naive use of quantitative techniques dangerous. Extensive use of guest lecturers drawn from the industry will be made.
Prerequisite: AMS 512 and AMS 513
3 credits, ABCF grading
AMS 515 webpage

AMS 516, Statistical Methods in Finance
The course introduces statistical methodologies in quantitative finance. Financial applications and statistical methodologies are intertwined in all lectures. The course will cover regression analysis and applications to the Capital Asset Pricing Model and multifactor pricing models, principal components and multivariate analysis, statistical methods for financial time series; value at risk, smoothing techniques and estimation of yield curves, and estimation and modeling of volatilities.
3 credits, ABCF grading
AMS 516 webpage

AMS 517, Topics in Statistical Methods for Finance: Credit Risk Modeling
The course will cover structural and reduced-form approach to pricing credit default, Markovian models (or rating-based) pricing methods, statistical inference of relative risks, counting process, correlated (or dependent) default times, copula methods and pricing models for CDOs.
3 credits, ABCF grading
AMS 517 webpage

AMS 526 Numerical Analysis I
Direct and indirect methods for solving simultaneous linear equations and matrix inversion, conditioning, and round-off errors. Computation of eigenvalues and eigenvectors.
Co-requisite: AMS 505
Fall, 3 credits, ABCF grading
AMS 526 webpage

AMS 527 Numerical Analysis II
Numerical methods based upon functional approximation: polynomial interpolation and approximation; and numerical differentiation and integration. Solution methods for ordinary differential equations. AMS 527 may be taken whether or not the student has completed AMS 526.
Spring, 3 credits, ABCF grading
AMS 527 webpage

AMS 528 Numerical Analysis III
An introduction to scientific computation, this course considers the basic numerical techniques designed to solve problems of physical and engineering interest. Finite difference methods are covered for the three major classes of partial differential equations: parabolic, elliptic, and hyperbolic. Practical implementation will be discussed. The student is also introduced to the important packages of scientific software algorithms. AMS 528 may be taken whether or not the student has completed AMS 526 or AMS 527.
Spring, 3 credits, ABCF grading
AMS 528 webpage

AMS 530 Principles in Parallel Computing
This course is designed for both academic and industrial scientists interested in parallel computing and its applications to large-scale scientific and engineering problems. It focuses on the three main issues in parallel computing: analysis of parallel hardware and software systems, design and implementation of parallel algorithms, and applications of parallel computing to selected problems in physical science and engineering. The course emphasizes hands-on practice and understanding of algorithmic concepts of parallel computing.
Prerequisite: A course in basic computer science such as operating systems or architectures or some programming experience
Spring, 3 credits, ABCF grading
AMS 530 webpage

AMS 532, Laboratory Rotations and Journal Club
This is a two semester course in which students spend at least 8 weeks in each of three different laboratories actively participating in the research of participating Computational Biology faculty. In addition, students will attend and actively participate in research discussions at weekly Journal Club meetings on topics from the current literature using the skills and knowledge acquired during the rotations.
0 credits, ABCF grading
AMS 532 webpage

AMS 533: Numerical Methods and Algorithms in Computational Biology
This class will survey many of the key techniques used in diverse aspects of computational biology. We will focus on how to successfully formulate a statement of the problem to be solved, and how that formulation can guide in selecting the most suitable computational approach. A set of problems from a diverse range of problems in biology will be used as examples. Note: Informatic methods for genomic analysis (such as data mining and analysis of nucleic acid and protein sequences) will not be covered. These topics are covered thoroughly in CSE 549.
3 credits, ABCF grading
AMS 533 webpage

AMS 535 Introduction to Computational Structural Biology and Drug Design
This course will provide an introduction to Computational Structural Biology with application to Drug Design. Methods and applications that use computation to model biological systems involved in human disease will be emphasized. The course aims to foster collaborative learning and will consist of presentations by the instructor, guest lecturers, and by course participants with the goal of summarizing key methods, topics, and papers relevant to Computational Structural Biology.
0-3 credits, ABCF grading May be repeated for credit
AMS 535 webpage

AMS 536 Molecular Modeling of Biological Molecules
This course is designed for students who wish to gain hands-on experience modeling biological molecules at the atomic level. In conjunction with the individual interests, Molecular Mechanics, Molecular Dynamics, Monte Carlo, Docking (virtual screening), or Quantum Mechanics software packages can be used to study relevant biological systems(s). Projects will include setup, execution, and analysis. Course participants will give literature presentations relevant to the simulations being performed and a final project report will be required. Familiarity with UNIX (Linux) is desirable.
Prerequisite: AMS 535 or permission of instructor
0-3 credits, ABCF grading May be repeated for credit
AMS 536 webpage

AMS 540 Linear Programming
Formulation of linear programming problems and solutions by simplex method. Duality, sensitivity analysis, dual simplex algorithm, decomposition. Applications to the transportation problem, two-person games, assignment problem, and introduction to integer and nonlinear programming. This course is offered as both MBA 540 and AMS 540.
Prerequisite: A course in linear algebra
3 credits, ABCF grading
AMS 540 webpage

AMS 542 Analysis of Algorithms
Techniques for designing efficient algorithms, including choice of data structures, recursion, branch and bound, divide and conquer, and dynamic programming. Complexity analysis of searching, sorting, matrix multiplication, and graph algorithms. Standard NP-complete problems and polynomial transformation techniques. This course is offered as both AMS 542 and CSE 548.
Spring, 3 credits, ABCF grading
AMS 542 webpage

AMS 544 Discrete and Nonlinear Optimization
Theoretical and computational properties of discrete and nonlinear optimization problems: integer programming, including cutting plane and branch and bound algorithms, necessary and sufficient conditions for optimality of nonlinear programs, and performance of selected nonlinear programming algorithms. This course is offered as both MBA 544 and AMS 544.
Prerequisite: AMS 540 or MBA 540
3 credits, ABCF grading
AMS 544 webpage

AMS 545 Computational Geometry
Study of the fundamental algorithmic problems associated with geometric computations, including convex hulls, Voronoi diagrams, triangulation, intersection, range queries, visibility, arrangements, and motion planning for robotics. Algorithmic methods include plane sweep, incremental insertion, randomization, divide-and-conquer, etc. This course is offered as both AMS 545 and CSE 555.
Spring, 3 credits, ABCF grading
AMS 545 webpage

AMS 546 Network Flows
Theory of flows in capacity-constrained networks. Topics include maximum flow, feasibility criteria, scheduling problems, matching and covering problems, minimum-length paths, minimum-cost flows, and associated combinatorial problems. This course is offered as both MBA 546 and AMS 546.
Spring, 3 credits, ABCF grading
AMS 546 webpage

AMS 547 Discrete Mathematics
This course introduces such mathematical tools as summations, number theory, binomial coefficients, generating functions, recurrence relations, discrete probability, asymptotics, combinatorics, and graph theory for use in algorithmic and combinatorial analysis. This course is offered as both CSE 547 and AMS 547.
Spring, 3 credits, ABCF grading
AMS 547 webpage

AMS 550 Operations Research: Stochastic Models
Includes Poisson processes, renewal theory, discrete-time and continuous-time Markov processes, Brownian motion, applications to queues, statistics, and other problems of engineering and social sciences. This course is offered as both MBA 550 and AMS 550.
Prerequisite: AMS 507 or equivalent
Spring, 3 credits, ABCF grading
AMS 550 webpage

AMS 552 Game Theory I
Elements of cooperative and noncooperative games. Matrix games, pure and mixed strategies, and equilibria. Solution concepts such as core, stable sets, and bargaining sets. Voting games, and the Shapley and Banzhaff power indices. This course is offered as both ECO 604 and AMS 552.
Prerequisite: Admission to graduate AMS program or permission of instructor
0-3 credits, ABCF grading
AMS 552 webpage

AMS 553 Simulation and Modeling
A comprehensive course in formulation, implementation, and application of simulation models. Topics include data structures, simulation languages, statistical analysis, pseudorandom number generation, and design of simulation experiments. Students apply simulation modeling methods to problems of their own design. This course is offered as CSE 529, AMS 553, and MBA 553.
Prerequisite: CSE 214 or equivalent; AMS 310 or 507 or equivalent; or permission of instructor
Spring, 3 credits, ABCF grading
AMS 553 webpage

AMS 554 Queuing Theory
Introduction to the mathematical aspects of congestion. Birth and death processes. Queues with service priorities and bulk-service queues. Analysis of transient- and steady-state behavior. Estimation of parameters. Applications to engineering, economic, and other systems. This course is offered as both MBA 554 and AMS 554.
3 credits, ABCF grading
AMS 554 webpage

AMS 555 Game Theory II
Refinements of strategic equilibrium, games with incomplete information, repeated games with and without complete information, and stochastic games. The Shapley value of games with many players, and NTU-values. This course is offered as both ECO 605 and AMS 555.
Spring, 0-3 credits, ABCF grading
AMS 555 webpage

AMS 556 Dynamic Programming
Stochastic and deterministic multistage optimization problems. Stochastic path problems. Principle of optimality. Recursive and functional equations. Method of successive approximations and policy iteration. Applications to finance, economics, inventory control, maintenance, inspection, and replacement problems. This course is offered as both MBA 556 and AMS 556.
Prerequisite: MBA/AMS 550 or MBA/AMS 558
3 credits, ABCF grading
AMS 556 webpage

AMS 565 Wave Propagation
Theory of propagation of vector and scalar waves in bounded and unbounded regions. Development of methods of geometrical optics. Propagation in homogeneous and anisotropic media.
Fall, 3 credits, ABCF grading
AMS 565 webpage

AMS 566 Compressible Fluid Dynamics
Physical, mathematical, and computational description in compressible fluid flows. Integral and differential forms of the conservation equations, one-dimensional flow, shocks and expansion waves in two and three dimensions, quasi-one-dimensional flow, transient flow, numerical methods for steady supersonic flow, numerical methods for transient flow.
Spring, 3 credits, ABCF grading
AMS 566 webpage

AMS 569 Probability Theory I
Probability spaces and sigma-algebras. Random variables as measurable mappings. Borel-Cantelli lemmas. Expectation using simple functions. Monotone and dominated convergence theorems. Inequalities. Stochastic convergence. Characteristic functions. Laws of large numbers and the central limit theorem. This course is offered as both AMS 569 and MBA 569.
Prerequisite: AMS 504 or equivalent
AMS 569 webpage
3 credits, ABCF grading

AMS 570 Introduction to Mathematical Statistics
Probability and distributions; multivariate distributions; distributions of functions of random variables; sampling distributions; limiting distributions; point estimation; confidence intervals; sufficient statistics; Bayesian estimation; maximum likelihood estimation; statistical tests.
Prerequisite: AMS 507
Spring, 3 credits, ABCF grading
AMS 570 webpage

AMS 571 Mathematical Statistics
Sampling distribution; convergence concepts; classes of statistical models; sufficient statistics; likelihood principle; point estimation; Bayes estimators; consistency; Neyman-Pearson Lemma; UMP tests; UMPU tests; Likelihood ratio tests; large sample theory.
Prerequisite: AMS 570 is preferred but not required
Fall, 3 credits, ABCF grading
AMS 571 webpage

AMS 572 Data Analysis I
Introduction to basic statistical procedures. Survey of elementary statistical procedures such as the t-test and chi-square test. Procedures to verify that assumptions are satisfied. Extensions of simple procedures to more complex situations and introduction to one-way analysis of variance. Basic exploratory data analysis procedures (stem and leaf plots, straightening regression lines, and techniques to establish equal variance). Coscheduled as AMS 572 or HPH 698.
Fall, 3 credits, ABCF grading
AMS 572 webpage

AMS 573 Design and Analysis of Categorical Data
Measuring the strength of association between pairs of categorical variables. Methods for evaluating classification procedures and inter-rater agreement. Analysis of the associations among three or more categorical variables using log linear models. Logistic regression.
Spring, 3 credits, ABCF grading
AMS 573 webpage

AMS 575 Internship in Statistical Consulting
Directed quantitative research problem in conjunction with currently existing research programs outside the department. Students specializing in a particular area work on a problem from that area; others work on problems related to their interests, if possible. Efficient and effective use of computers. Each student gives at least one informal lecture to his or her colleagues on a research problem and its statistical aspects.
Prerequisite: Permission of instructor
Fall and Spring, 3-4 credits, ABCF grading
AMS 575 webpage

AMS 577 Multivariate Analysis
The multivariate distribution. Estimation of the mean vector and covariance matrix of the multivariate normal. Discriminant analysis. Canonical correlation. Principal components. Factor analysis. Cluster analysis.
Prerequisites: AMS 572 and AMS 578
3 credits, ABCF grading
AMS 577 webpage

AMS 578 Regression Theory
Classical least-squares theory for regression including the Gauss-Markov theorem and classical normal statistical theory. An introduction to stepwise regression, procedures, and exploratory data analysis techniques. Analysis of variance problems as a subject of regression. Brief discussions of robustness of estimation and robustness of design.
Prerequisite: AMS 572 or equivalent
Spring, 3 credits, ABCF grading
AMS 578 webpage

AMS 582 Design of Experiments
Discussion of the accuracy of experiments, partitioning sums of squares, randomized designs, factorial experiments, Latin squares, confounding and fractional replication, response surface experiments, and incomplete block designs. Coscheduled as AMS 582 or HPH 699.