16:642:516 Applied Partial Differential Equations (3) Theory and applications of partial differential equations. First-order equations: linear and quasi-linear. Hyperbolic systems: shocks. Classification of second-order linear equations. Hyperbolic: characteristics, wave equation. Elliptic: maximum principles, Laplace's and reduced-wave 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, boundary-value 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.

16:642:561-562 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 low-density expansions, correlation inequalities, and coexistence of phases. Prerequisite: Permission of instructor. Students should have either a general mathematical background equivalent to that of a second-year 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 second-year 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, error-correcting 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, small-group behavior, and power in legislatures. Development of requisite mathematical tools about graphs, signed graphs, Markov chains, and n-person 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 sum-sets, 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 Hodgin-Huxley 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 continuous-time 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) Single-name credit derivatives; structural, reduced form, or intensity models; credit default swaps; multiname credit derivatives; top-down and bottom-up 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 Fixed-Income Securities and Derivative Modeling (3) Fixed-income securities products; interest rate and yield curve models; models for fixed-income pricing and risk management; exotic fixed-income derivative modeling; mortgage-backed securities; stochastic volatility models. Prerequisites: 16:642:573, 621. Corequisite: 16:642:622.

16:642:627 High-Frequency Finance and Stochastic Control (3) An introduction to stochastic control and estimation with emphasis on the Hamilton-Jacobi-Bellman equation, Kalman filtering, and applications to high-frequency 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) Fixed-Income 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.