


Graduate Courses in Applied Mathematics (642)


16:642:516
Applied Partial Differential Equations (3)
Theory and applications of partial differential equations. Firstorder equations: linear and quasilinear. Hyperbolic systems: shocks. Classification of secondorder linear equations. Hyperbolic: characteristics, wave equation. Elliptic: maximum principles, Laplace's and reducedwave equations. Parabolic: heat equation. Fourier and Laplace transforms.
Prerequisites: Advanced calculus, differential equations.

16:642:527,528
Methods of Applied Mathematics I,II (3,3)
Appropriate topics from linear algebra, linear operators in Hilbert space, linear integral equations, boundaryvalue problems, calculus of variations, numerical solution of ordinary and partial differential equations.
Prerequisite: Permission of instructor.

16:642:550
Linear Algebra and Applications (3)
Vector spaces, bases, and dimension. Linear operators, quadratic forms, and their matrix representations. Eigenvalues, eigenvectors, diagonalizability, Jordan, and other canonical forms. Applications to systems of linear differential equations.

16:642:551
Applied Algebra (3)
Basic algebraic structures, including groups and their representations, finite fields, and Boolean algebra. Applications to physics, counting arguments, switching circuits, and coding theory. Automata theory.
Prerequisite: 16:642:550.

16:642:561562
Introduction to Mathematical Physics I,II (3,3)
Study of models of classical and/or quantum mechanical physical systems, with emphasis on the use of rigorous mathematical techniques.
Prerequisites: Linear algebra, advanced calculus.

16:642:563
(F) Rigorous Results in Statistical Mechanics I: Equilibrium (3)
Treats the subject ab initio. Deals with general questions such as the existence of the thermodynamic limit, covergence of lowdensity expansions, correlation inequalities, and coexistence of phases.
Prerequisite: Permission of instructor. Students should have either a general mathematical background equivalent to that of a secondyear graduate student in mathematics or knowledge of statistical mechanics obtained from physics, chemistry, or engineering courses in the subject.

16:642:564
(S) Rigorous Results in Statistical Mechanics II: Nonequilibrium (3)
Ergodic theory, time evolution of infinite systems, heat flow in random systems, stationary nonequilibrium systems, exactly soluble models systems, and stochastic processes.
Prerequisite: Permission of instructor. Students should have either a general mathematical background equivalent to that of a secondyear graduate student in mathematics or knowledge of statistical mechanics obtained from physics, chemistry, or engineering courses in the subject.

16:642:573,574
Numerical Analysis I,II (3,3)
A general survey of basic topics in numerical analysis, typically including the approximation of functions, numerical integration, numerical solution of linear and nonlinear systems of equations, numerical techniques for unconstrained function minimization, approximation of matrix eigenvalues and eigenvectors, the numerical solution of initial and boundary value problems for ordinary differential equations, and the approximation of some simple model problems in partial differential equations.
Prerequisites: Advanced calculus, linear algebra, and differential equations.

16:642:575
Numerical Solutions of Partial Differential Equations (3)
The study of numerical methods (e.g., finite difference, finite element, finite volume methods) for the approximation of elliptic, parabolic, and hyperbolic partial differential equations, concentrating on the key ideas underlying the derivation of numerical schemes, their stability and accuracy, and the fast solution of the discrete equations resulting from application of the approximation schemes.
Prerequisite: Partial differential equations.

16:642:577,578
Selected Mathematical Topics in System Theory (3,3)
Selection of topics from mathematical system theory (e.g., realization, control, stability, optimization, and feedback), with emphasis on qualitative aspects. Algebraic techniques in linear system theory; geometrical and functional analytic techniques in the study of nonlinear control systems.
Prerequisites: Linear algebra, differential equations.

16:642:581
(S) Graph Theory (3)
Advanced introduction to graph theory. Topics include matching theory, connectivity, graph coloring, planarity, extremal graph theory, and the main techniques (elementary, probabilistic, algebraic, and polyhedral) for analyzing the structure and properties of graphs.
Prerequisites: 01:640:350 and 411, or permission of instructor; 01:640:477 is recommended.

16:642:582,583
Combinatorics I,II (3,3)
Advanced introduction to combinatorial theory and applications. Typical topics include hypergraphs, probabilistic methods, algebraic methods, matching theory, packing and covering, Ramsey theory, partially ordered sets and lattices, block designs, errorcorrecting codes, and matroids. Topics and emphasis vary depending on instructor.
Prerequisites: 01:640:350 and 411, or permission of instructor; 01:640:477 and 16:640:551 are recommended.

16:642:585
Mathematical Models of Social and Policy Problems (3)
Mathematical models of problems in social sciences and the public and private policy areas, emphasizing discrete models. Transportation and communication networks. Energy modeling. Pollution models. Models from economics, psychology, sociology, and political science, dealing with such issues as currency movement, land development, learning, smallgroup behavior, and power in legislatures. Development of requisite mathematical tools about graphs, signed graphs, Markov chains, and nperson games.
Prerequisites: Linear algebra, elementary probability.

16:642:586
(S) Theory of Measurement (3)
Foundations of measurement from a mathematical point of view. Homomorphisms or relational systems; scale type; uniqueness theory; ordinal, extensive, difference, and conjoint measurements; utility and expected utility; subjective probability; and applications to social and physical sciences.
Prerequisite: Undergraduate modern algebra or permission of instructor.

16:642:587
Selected Topics in Discrete Mathematics (3)
Choice of topics depends on year and instructor.
Prerequisite: Permission of instructor.

16:642:588
(S) Arithmetic Combinatorics (3)
Structure of sumsets, inverse problems, arithmetic progressions in dense sets, regularity lemmas, pseudorandomness, Fourier analytics techniques, geometry of numbers.
Prerequisite: 16:642:582 or permission of instructor.

16:642:591,592
Topics in Probability and Ergodic Theory I,II (3,3)
Basic probability theory and its applications. Topics include stochastic independence, distributions and densities, Markov processes, stationary processes, the law of large numbers, and the central limit theorem. Also includes a broad range of applications to communications engineering, economics, biology, and physics.
Corequisites: 16:640:501,502.

16:642:593
(F) Mathematical Foundations for Industrial and Systems Engineering (3)
Underlying mathematical principles of system modeling. Foundations of the real number system and calculus of functions of one variable, with emphasis on logical principles and methods of proof.
Prerequisite: Permission of instructor.

16:642:611,612
Selected Topics in Applied Mathematics (3,3)
Topics of current interest.
Prerequisite: Permission of instructor.

16:642:613
Mathematical Foundations of Systems Biology (3)
Basic ODE models including the drug infusion, epidemics, and HodginHuxley models. Signal transduction networks: the mathematics of ultrasensitivity and bistability; monotone systems. Chemotaxis, robustness, and adaptation. Reverse engineering of gene and protein networks. PDE analysis of diffusion and transport. (Topics will vary.)
Prerequisite: Permission of instructor.

16:642:614
Selected Topics in Systems Biology (3)
Selection of mathematical topics (e.g., dynamical systems,
control theory, statistical mechanics, combinatorics, numerical
methods, probability, and statistics) applied to questions from systems
biology (gene regulation/signal transduction networks, population
biology and ecology, evolution, neuroscience, physiology, biomechanics).
Mischaikow, Sontag. Prerequisite: Permission of instructor.

16:642:621,622
Mathematical Finance I,II (3,3)
Introduction to stochastic processes and stochastic calculus, and their application to continuoustime finance and the mathematical theory of derivative security pricing.
Prerequisites: Probability Theory (Math 01:640:477), Advanced Calculus for Engineering (Math 01:640:421), or Elementary Partial Differential Equations (Math 01:640:423), and Linear Algebra (Math 01:640:250).

16:642:623
Computational Finance (3)
Implementation of models for pricing and hedging derivative securities, using C++ programming projects, with emphasis on Monte Carlo simulation, finite difference solution of
partial differential equations, binomial and trinomial trees, and the
fast Fourier transform (FFT).
Prerequisites: 16:332:503, 16:642:573, 621.
Corequisites: 16:642:574 or 575, 622.

16:642:624
Credit Risk Modeling (3)
Singlename credit derivatives; structural, reduced form, or intensity models; credit default swaps; multiname credit derivatives; topdown and bottomup models; collateralized debt obligations; tranche options; risk management.
Prerequisites: 16:642:573, 621. Corequisite: 16:642:622.

16:642:625
Portfolio Theory and Applications (3)
Introduction to modern quantitative portfolio management,
from theoretical foundations to the advanced applications, emphasizing models
and mathematical, numerical, and statistical techniques.
Prerequisites: 16:642:621, 16:220:507. Corequisite: 16:220:508.

16:642:626
FixedIncome Securities and Derivative Modeling (3)
Fixedincome securities products; interest rate and yield curve models; models for fixedincome pricing and risk management; exotic fixedincome derivative modeling; mortgagebacked securities; stochastic volatility models.
Prerequisites: 16:642:573, 621. Corequisite: 16:642:622.

16:642:627
HighFrequency Finance and Stochastic Control (3)
An introduction to stochastic control and estimation with
emphasis on the HamiltonJacobiBellman equation, Kalman filtering, and
applications to highfrequency finance and trading.
Prerequisites: 16:332:503, 16:642:621, 16:220:507, 16:960:563.
Corequisite: 16:220:508.

16:642:628
Topics in Mathematical Finance (3)
Topics rotate from semester to semester and year to year.
Recent topics have included (a) Energy Risk, Commodities, and Derivative Modeling,
(b) FixedIncome Products and Derivative Modeling, and (c) Quantitative Risk
Management.
Prerequisite: Permission of mathematical finance program director.

16:642:629
Special Research Projects in Mathematical Finance (1)
Research project performed in connection with the master's essay for the mathematical finance option of the master of science degree in mathematics, often as part of an industry internship in quantitative finance.
Prerequisite: Permission of mathematical finance program director.

16:642:630
Seminar in Mathematical Finance (0.5)
Seminar in mathematical finance theory, industry practice,
and career preparation for students pursuing the mathematical finance option of
the master of science degree in mathematics.
Prerequisite: Permission of mathematical finance program director.

16:642:661,662
Selected Topics in Mathematical Physics (3,3)
Topics of current interest in areas such as classical mechanics, statistical mechanics, ergodic theory, nonrelativistic quantum mechanics, and quantum field theory.
Prerequisite: Permission of instructor.






