Graduate Math Courses

The following list is indicative of our course offerings.  Not all courses are offered every year.  Unless otherwise indicated, all of these are four-unit courses.

NOTE: Effective Spring 2014, certain Math courses are being renumbered at CGU.  This list shows the new numbers and also notes, where applicable, the former numbers.

MATH 231 Principles of Real Analysis I. Countable sets, least upper bounds, and metric space topology including compactness, completeness, connectivity, and uniform convergence. Related topics as time permits.
MATH 232 Principles of Real Analysis II. A rigorous study of calculus in Euclidean Spaces including multiple Riemann Integrals, derivatives of transformations, and the inverse function theorem. Prerequisite: Math 231.
MATH 235 Functions of a Complex Variable. Complex numbers and functions. Complex differentiation. Cauchy-Riemann equations. Holomorphic and harmonic functions. Complex power series, elementary functions. Cauchy's theorem and the deformation theorem. Consequences of Cauchy's theorem: Cauchy's integral formula, Liouville's theorem, fundamental theorem of algebra, Cauchy's formula for derivatives and Morera's theorem. Taylor series, uniqueness and maximum principle. Laurent series, singularities. Residue theorem, calculation of residues. Estimation of integrals. Prerequisite: Math 231 or permission of the instructor.
MATH 236 Complex Variables and Applications. Complex differentiation, Cauchy-Riemann equations, Cauchy integral formula, Taylor and Laurent expansions, residue theory, contour integration including branch point contours, uses of Jordan's lemma, Fourier and Laplace transform integrals, conformal mapping.
MATH 239 Fourier Analysis. Fourier analysis begins with the examination of the difficulties involved in attempting to reconstruct arbitrary functions as infinite combinations of elementary trigonometric functions. Topics in this course will include Fourier series, summability, types and questions of convergence, and the Fourier transform (with, if time permits, applications to PDEs, medical imaging, linguistics, and number theory).
MATH 241 Hyperbolic Geometry. An introduction to hyperbolic geometry in dimensions two and three. Topics will include: Poincaré disk model, upper half-space model, hyperbolic isometries, linear fractional transformations, hyperbolic trigonometry, cross-ratio, hyperbolic manifolds, and hyperbolic knots.
MATH 242 Differential Geometry. Curves and surfaces, Gaussian curvature, isometries, tensor analysis, covariant differentiation with applications to physics and geometry (intended for physicists and mathematicians). Prerequisite: Math 231 recommended.
MATH 243 Topics in Geometry. Selected topics in Riemannian geometry, low dimensional manifold theory, elementary Lie groups and Lie algebra, and contemporary applications in mathematics and physics. Prerequisite: permission of instructor.
MATH 244 Algebraic Topology. An introduction to algebraic topology. Basics of category theory, simplicial homology and cohomology, relative homology, exact sequences, Poincare duality, CW complexes, DeRahm cohomology, applications to knot theory.
MATH 245 Topics in Geometry and Topology. Topic varies from year to year and will be chosen from: Differential Topology, Euclidean and Non-Euclidean Geometries, Knot Theory, Algebraic Topology, and Projective Geometry.
MATH 247 Topology. Topological spaces, product spaces, quotient spaces, Hausdorff spaces, compactness, connectedness, path connectedness, fundamental groups, homotopy of maps, and covering spaces. Corequisite: Math 231 or permission of instructor.
MATH 248 Knot Theory. An introduction to the theory of knots and links from combinatorial, algebraic, and geometric perspectives. Topics will include knot diagrams,p-colorings, Alexander, Jones, and HOMFLY polynomials, Seifert surfaces, genus, Seifert matrices, the fundamental group, representations of knot groups, covering spaces, surgery on knots, and important families of knots. Prerequisite: Topology (Math 247), or Algebra (Math 271), or permission of instructor.
MATH 249 Discrete Geometry. The goal of this course is to introduce students to the basics of discrete and convex geometry. Topics covered will include convex bodies, lattices, quadratic forms, and interactions between them, such as the fundamentals of Minkowski’s theory, shortest vector problem, reduction algorithms, LLL, and connections to computational complexity and theoretical computer science. Additional topics may include an introduction to optimization questions, such as packing, and convering problems.
MATH 250 Statistical Methods for Clinical Trials Data. A second course in biostatistics. Emphasis on the most commonly used statistical methods in pharmaceutical and other medical research. Topics such as design of clinical trials, power and sample size determination, contingency table analysis, odds ratio and relative risk, survival analysis.
MATH 251 Probability. The main elements of probability theory at an intermediate level. Topics include combinatorial analysis, conditional probabilities, discrete and continuous random variables, probability distributions, central limit theorem, and numerous applications. Students may not receive credit for both Math 251 and Math 257.
MATH 252 Statistical Theory. This course will cover in depth the mathematics behind most of the frequently used statistical tools such as point and interval estimation, hypothesis testing, goodness of fit, ANOVA, linear regression. This is a theoretical course, but we will also be using R statistical package to gain some hands on experience with data.
MATH 253 Bayesian Statistics. An introduction to principles of data analysis and advanced statistical modeling using Bayesian inference. Topics include a combination of Bayesian principles and advanced methods; general, conjugate and noninformative priors, posteriors, credible intervals, Markov Chain Monte Carlo methods, and hierarchical models. The emphasis throughout is on the application of Bayesian thinking to problems in data analysis. Statistical software will be used as a tool to implement many of the techniques.
MATH 255 Time Series. An introduction to the theory of statistical time series. Topics include decomposition of time series, seasonal models, forecasting models including causal models, trend models, and smoothing models, autoregressive (AR), moving average (MA), and integrated (ARIMA) forecasting models. Time permitting we will also discuss state space models, which include Markov processes and hidden Markov processes, and derive the famous Kalman filter, which is a recursive algorithm to compute predictions. Statistical software will be used as a tool to aid calculations required for many of the techniques.
MATH 256 Stochastic Processes. Continuation of Math 251. Properties of independent and dependent random variables, conditional expectation. Topics chosen from Markov processes, second order processes, stationary processes, ergodic theory, Martingales, and renewal theory. Prerequisite: Math 251 or permission of instructor.
MATH 257 Intermediate Probability (2-UNIT COURSE). Continuous random variables; distribution functions; joint density functions; marginal and conditional distributions; functions of random variables; conditional expectation; covariance and correlation; moment generating functions; law of large numbers; Chebyshev's theorem and central limit theorem. Students may not receive credit for both Math 251 and Math 257.
MATH 258 Statistical Linear Models. An introduction to analysis of variance (including one-way and two-way fixed effects ANOVA) and linear regression (including simple linear regression, multiple regression, variable selection, stepwise regression and analysis of residual plots). Emphasis will be on both methods and applications to data. Statistical software will be used to analyze data. Prerequisite: Math 252 or permission of instructor.
MATH 259 Topics in Statistics. Topics vary from year to year and may include: analysis of genetic data, experimental design, time series, computational methods, Bayesian analysis or other topics.
MATH 260 Monte Carlo Methods. This course introduces concepts and statistical techniques that are critical to constructing and analyzing effective simulations, and discusses certain applications for simulation and Monte Carlo methods. Topics include random number generation, simulation-based optimization, model building, bias-variance trade-off, input selection using experimental design, Markov chain Monte Carlo (MCMC), and numerical integration. Prerequisite: Math 251.
MATH 261A Introduction to C++ Programming (2-UNIT COURSE). Formerly Math 301A. This course is designed for graduate students in mathematics and financial engineering program, who seek to acquire the fundamental C++ programming skills. Topics include basic fundamentals of C++ programming, data types, control statements, arrays, strings, pointers, functions, classes and objects, simple algorithms and procedural problem solving for common tasks in finance.
MATH 262 Fundamentals of Scientific Computing. This course will help students develop skills in scientific computing in a PC/workstation environment, preparing them for the mathematics clinic, work in industrial applied mathematics, and research in applied mathematics. Students will be introduced to the Matlab programming environment, database development in MySQL, and the scientific typesetting language LaTeX. Programming competence will be developed by working through examples in basic numerical analysis including iterative methods for solving nonlinear equations, approximate integration and differentiation, interpolation, and numerical linear algebra. Content will vary, depending on the interest of the students.
MATH 264 Scientific Computing. Computational techniques applied to problems in the sciences and engineering. Modeling of physical problems, computer implementation, analysis of results; use of mathematical software; numerical methods chosen from: solutions of linear and nonlinear algebraic equations, solutions of ordinary and partial differential equations, finite elements, linear programming, optimization algorithms, and fast-Fourier transforms.
MATH 265 Numerical Analysis. An introduction to the theory and methods for numerical solution of mathematical problems. Core topics include: analysis of error and efficiency of methods; solutions of linear systems by Gaussian elimination and iterative methods; calculation of eigenvalue and eigenvectors; interpolation and approximation; numerical integration; solution of ordinary differential equations.
MATH 267 Complexity Theory. Specific topics include finite automata, pushdown automata, Turing machines, and their corresponding languages and grammars; undecidability; and complexity classes, reductions, and hierarchies.
MATH 268 Algorithms. Algorithm design, computer implementation, and analysis of efficiency. Discrete structures, sorting and searching, time and space complexity, and topics selected from algorithms for arithmetic circuits, sorting networks, parallel algorithms, computational geometry, parsing, and pattern-matching.
MATH 271 Abstract Algebra I. Groups, rings, fields and additional topics. Topics in group theory include groups, subgroups, quotient groups, Lagrange's theorem, symmetry groups, and the isomorphism theorems. Topics in Ring theory include Euclidean domains, PIDs, UFDs, fields, polynomial rings, ideal theory, and the isomorphism theorems. In recent years, additional topics have included the Sylow theorems, group actions, modules, representations, and introductory category theory.
MATH 272 Abstract Algebra II: Galois Theory. Topics covered will include polynomial rings, field extensions, classical constructions, splitting fields, algebraic closure, separability, Fundamental Theorem of Galois Theory, Galois groups of polynomials and solvability. Prerequisite: Math 271. This course is independent of Math 274 (Abstract Algebra II: Representation Theory), and students may receive credit for both courses.
MATH 273 Linear Algebra. Topics will be chosen from among: Similarity of matrices and the Jordan form; the Cayley Hamilton Theory, limits of sequences and series of matrices: iterative solutions of large systems of linear algebraic equations; the Perron-Frobenius theory of nonnegative matrices; estimating eigenvalues of matrices. Prerequisite: Math 231.
MATH 274 Abstract Algebra II: Representation Theory. Topics covered will include group rings, characters, orthogonality relations, induced representations, application of representation theory, and other select topics from module theory. Prerequisite: Math 271. This course is independent of Math 272 (Abstract Algebra II: Galois Theory), and students may receive credit for both courses.
MATH 275 Number Theory. Topics covered will include the fundamental theorem of arithmetic, Euclid's algorithm, congruences, Diophantine problems, quadratic reciprocity, arithmetic functions, and distribution of primes. If time allows, we may also discuss some geometric methods, coming from lattice point counting, such as Gauss's circle problem and Dirichlet divisor problem, as well as some applications of Number Theory to coding theory and cryptography.
MATH 276 Algebraic Geometry. Topics include affine and projective varieties, the Nullstellensatz, rational maps and morphisms, birational geometry, tangent spaces, nonsingularity and intersection theory. Prerequisite: Math 271; recommended previous courses in Analysis, Galois Theory, Differential Geometry and Topology are helpful but not required; or permission of the instructor.
MATH 277 Topics in Algebra. Topics vary from year to year and will be chosen from: Representation Theory, Algebraic Geometry, Commutative Algebra, Algebraic Number Theory, Coding Theory, Algebraic Combinatorics, Algebraic Graph Theory, Matroid Theory. Prerequisite: Math 271 or permission of instructor.
MATH 280 Introduction to Partial Differential Equations. Classifying PDEs, the method of characteristics, the heat equation, wave equation, and Laplace's equation, separation of variables, Fourier series and other orthogonal expansions, convergence of orthogonal expansions, well-posed problems, existence and uniqueness of solutions, maximum principles and energy methods, Sturm-Liouville theory, Fourier transform methods and Green's functions, Bessel functions. Students may not receive credit for both Math 280 and Math 282.
MATH 281 Dynamical Systems. The theory of continuous dynamical systems was developed largely in response to the reality that most nonlinear differential equations lack exact analytic solutions. In addition to being of interest in their own right, such nonlinear equations arise naturally as mathematical models from many disciplines including biology, chemistry, physiology, ecology, physics, and engineering. This course is an introduction to and survey of characteristic behavior of such dynamical systems. Applications will be an integral part of the course with examples including mechanical vibrations, biological rhythms, circuits, insect outbreaks, and chemical oscillations. Prerequisite: Math 280 or 282.
MATH 282 Partial Differential Equations. Second-order PDEs and the modeling situations in which they occur. Separation of variables, Fourier analysis, Green's function and variational methods. Prerequisite: Math 231 or permission of the instructor. Students may not receive credit for both Math 280 and Math 282.
MATH 283 Mathematical Modeling. Introduction to the construction and interpretation of deterministic and stochastic models in the biological, social, and physical sciences, including simulation studies. Students are required to develop a model in an area of their interest.
MATH 284 Applied Analysis. Formerly Math 280. The course integrates concepts from real and functional analysis with applications to differential equations. We will introduce theoretical concepts such as metric, Banach and Hilbert spaces as well as abstract methods such as the contraction mapping theorem, with a focus on applications rather than exhaustivity. We will apply these concepts to the study of integral equations, ordinary differential equations and partial differential equations. The main methods will be Fourier series and the Sturm-Liouville theory.
MATH 285 Wavelets and their Applications. Formerly Math 268. An introduction to wavelet analysis with a focus on applications. Wavelets are an important tool in modern signal and image processing as well as other areas of applied mathematics. The main objective of this course is to develop the theory behind wavelets and similar constructions. Theoretical topics may include Fourier analysis, the discrete and continuous wavelet transform, Shannon's Sampling Theorem, statistical studies of wavelet signal extraction, and the Heisenberg uncertainty principle. Application based topics may include wavelet based compression, signal processing methods, communications, and sensing mechanisms where wavelets play a crucial role. Students will gain hands-on experience with real-world imaging techniques.
MATH 286 Stochastic Methods in Operations Research. Queuing theory, decision theory, discrete event simulation, inventory modeling, and Markov decision processes.
MATH 287 Deterministic Methods in Operations Research. Linear, integer, nonlinear, and dynamic programming. Applications to transportation problems, inventory analysis, classical optimization problems, and network analysis, including project planning and control.
MATH 288 Social Choice and Decision-Making. This course focuses on the modeling of individual and group decisions using techniques from game theory. Topics will include: basic concepts of game theory and social choice theory, representations of games, Nash equilibria, utility theory, non-cooperative games, cooperative games, voting games, paradoxes, impossibility theorems, Shapley value, power indices, fair division problems, and applications.
MATH 289 Special Topics in Mathematics. Topics vary from year to year and may include: algebraic geometry, algebraic topology, fluid dynamics, partial differential equations, games and gambling, Bayesian analysis, geometric group theory or other topics.
MATH 293 Mathematics Clinic. Formerly Math 392. The Mathematics Clinic provides applied, real-world research experience. A team of 3-5 students will work on an open-ended research problem from an industrial partner, under the guidance of a faculty advisor. Problems involve a wide array of techniques from mathematical modeling as well as from engineering and computer science. Clinic projects generally address problems of sufficient magnitude and complexity that their analysis, solution and exposition require a significant team effort. Students are normally expected to enroll in Math 393 (Advanced Mathematics Clinic) in the subsequent semester. Prerequisite: permission of instructor.
MATH 294 Methods of Applied Mathematics. Derivation of classical equations of applied math including quasilinear hyperbolic equations, Laplace, Poisson and Helmholtz equations, advection-diffusion equation, wave equation, etc., in various orthogonal curvilinear coordinate systems. The gradient, divergence, curl and Laplacian operators in Cartesian, cylindrical and spherical coordinates, acting on scalar or vector fields. Methods of solution on bounded domains using eigenfunction expansions and the associated Sturm-Liouville eigenvalue problems, including Bessel functions and Legendre polynomials. Method of characteristics for systems of hyperbolic equations, including nonlinear waves and shocks. Solution on unbounded domains using Laplace and Fourier transforms. Calculation of the transforms and their inverses using integration in the complex plane, including residue theory and integrals involving branch cuts. Green's functions. Fredholm's solvability criteria for linear systems. Calculus of variations and the Euler-Lagrange equations. Analysis of simple nonlinear dynamical systems in the phrase plane. Applications of linear algebra to the study of differential equations. Matrix exponentials. Prerequisite: Advanced Calculus, Differential Equations, Linear Algebra, Complex Variables.
MATH 306 Optimization. The course emphasizes nonlinear programming. It covers numerical methods for finite-dimensional optimization problems with fairly smooth functions. Both non-constrained and constrained optimizations will be discussed. Certain degree of emphasis will be given to the convergence analysis of the numerical methods. Prerequisite: multivariable calculus and numerical linear algebra.
MATH 335 Integral Transforms and Applications. Transforms covered will include: Fourier, Laplace, Hilbert, Hankel, Mellin, Radon, and Z. The course will be relevant to mathematicians and enginners working in communications, signal and image processing, continuous and digital filters, wave propagation in fluids and solids, etc.
MATH 337 Real and Functional Analysis I. Formerly Math 331. Abstract measures, Lebesgue measure on Rn, and Lebesgue-Stieljes measures on R. The Lebesgue integral and limit theorems. Product measures and the Fubini theorem. Additional related topics as time permits. Prerequisite: Math 131 and Math 132.
MATH 338 Real and Functional Analysis II. Formerly Math 332. Continuation of Math 337. Some of the topics covered will be: Banach and Hilbert spaces; Lp-spaces; complex measures and the Radon-Nikodym theorem. Prerequisite: Math 337.
MATH 351 Time Series Analysis. Analysis of time series data by means of particular models such as ARIMA. Spectral analysis. Associated methods of inference and applications. Prerequisite: permission of instructor.
MATH 352 Nonparametric and Computational Statistics. Treatment of statistical questions which do not depend on specific parametric models. Examples are testing for symmetry of a distribution and testing for equality of two distributors. Elementary combinatorial methods will play a major role in the course. Prerequisite: Math 252 or permission of instructor.
MATH 353 Asymptotic Methods in Statistics with Applications. Modes of convergence for random variables and their distributions; central limit theorems; laws of large numbers; statistical large smaple theory of functions of sample moments, sample quantiles, rank statistics, and extreme order statistics; asymptotically efficient estimation and hypothesis testing. Prerequisite: Math 251 and 252; linear algebra; analysis (Math 231 and 232 or equivalent).
MATH 354 Reliability Theory. Structural properties and reliability of complex systems; classes of life distributions based on aging; maintenance and replacement models; availability, reliability, and mean time between failures for complex systems; Markov models for systems; elementary renewal theory. Prerequisite: Math 251. Math 256 would be helpful but not essential.
MATH 355 Linear Statistical Models. A discussion of linear statistical models in both the full and less-than-full rank cases, the Gauss-Markov theorem, and applications to regression analysis, analysis of variance, and analysis of covariance. Topics in design of experiments and multivariate analysis. Prerequisite: Linear algebra and a year course in probability and statistics.
MATH 357 Deterministic and Stochastic Control. The course consists of two parts. The first part is lecture-based, and will cover both deterministic and stochastic control. In deterministic control, we will cover the calculus of variations, Pontryagin’s maximum principle, the linear regulator, and controllability. In stochastic control, we will review some stochastic analysis, and then will cover dynamic programming, viscosity solutions, the stochastic version of Pontryagin’s maximum principle, and backward stochastic differential equations. The second part will be a seminar on applications of optimal control to engineering and to mathematical finance problems. The course will cover a new method based on Malliavin calculus to solve backward stochastic differential equations analytically, with potential new applications. Prerequisite: Math 256 (Stochastic Processes). Knowledge of Ito calculus is helpful but not required, and will be reviewed in the course.
MATH 358 Mathematical Finance. This course will cover the theory of option pricing, emphasizing the Black-Scholes model and interest rate models. Implementation of the theory and model calibration are covered in the companion course, Numerical Methods for Finance, Math 361A. We will see the binomial no-arbitrage pricing model, state prices, Brownian motion, stochastic integration, and Ito’s lemma, the Black-Scholes equation, risk-neutral pricing and Girsanov theorem, change of numeraire and two term structure models: Vasicek and LIBOR. Prerequisite: Mature understanding of advanced calculus and probability (at the level of Math 251) and permission of instructor. Math 256 would be helpful.
MATH 359 Simulation. This course will introduce the students to the general concepts and tools of simulation analysis using pseudo random numbers generated on a computer. Starting with a background in calculus-based probability theory, the students will learn how to combine the mathematics of probability with the utility of the computer to find approximate solutions to a variety of mathematical problems arising in analysis, probability and statistics, stochastic processes, optimization, and general modeling. In undertaking this study, students will discover that many otherwise intractable problems can often be attacked using simulation techniques that are relatively easy to implement, thus adding to their general problem solving capabilities. Prerequisite: Math 251 or equivalent.
MATH 361A Numerical Methods for Finance (2-UNIT COURSE). This course focuses on pricing derivatives, but some topics of risk management are also covered. Whereas the Mathematical Finance course (Math 358) shows the student how to price instruments using color-form (analytical) formulae, this course focuses on the instruments that can be best analyzed with numerical methods: structured loans, mortgage-backed securities, etc. Topics include binomial and trinomial tree (lattice), finite differences, Monte Carlo simulation, and an introduction to copulae. Prerequisite: Math 256. Corequisite: Math 358.
MATH 361B Credit Risk (2-UNIT COURSE). After a survey of statistical and actuarial methods of credit risk management, we will study the modern option pricing methods of credit risk. This includes both the reduced-from and the structural (Merton) approach. Prerequisite: Math 256. Corequisite: Math 358.
MATH 362 Numerical Methods for Partial Differential Equations. Finite difference, finite element, and spectral methods for elliptic, parabolic and hyperbolic partial differential equations; discussion of discretization schemes, truncation error, consistency, stability, accuracy and convergence; explicit vs. implicit schemes; implementation of Dirichlet, Neumann and Robin boundary conditions; operator splitting; Godunov methods for hyperbolic systems; direct and iterative methods for elliptic systems; Gauss-Seidel, SOR and multigrid methods; Fourier and Chebyshev based spectral and pseudo-spectral methods. Prerequisite: partial differential equations and numerical analysis.
MATH 365 Computational Methods for Molecular Biology. Topics include statistical analysis of microarray data (the biological problem, modern microarray technology, high-dimensional small sample size data problem, key elements in a good experiment design, identifying sources of variation and data preparation, normalization techniques, statistical methods and algorithms for error detection and correction, statistical analysis workflow, software tools including R and bioconductor, and computational implementation of workflow methods and algorithms in R) and microarray data interpretation (use of linear models for analysis and assessment of differential expression, gene selection (Empirical Bayes, volcano plots) and gene classification (clustering, heatmaps)). Prerequisite: competency in scientific computing, linear algebra, an upper division course in statistics, familiarity with basic biology, access to a computer on which the Elluminate software can be run (requires latest JAVA runtime environment).
MATH 368 Advanced Numerical Analysis. Numerical linear algebra including LU decomposition, Jacobi, Gauss-Seidel and SOR iterations, Krylov subspace methods (Conjugate Gradient, GMRES), QR and SVD factorization of matrices, eigenvalue problems via power, inverse, QR and Arnoldi iterations, error analysis, forward and backward stability; numerical integration of ODEs including Runge-Kutta and Adams formulas, predictor-corrector methods, stiff equation solvers and shooting method for BVPs; other numerical methods including interpolation via Lagrange and Chebyshev polynomials and cubic splines, integration and quadrature with trapezoidal and Simpson rules, Newton-Cotes formulae, Gaussian quadrature, and singular integrals, root-finding via one-point iteration, bisection, Newton and secant methods, numerical differentiation using finite differences, spectral and pseudo-spectral methods. Prerequisite: advanced calculus and elementary numerical analysis.
MATH 381 Fluid Dynamics. This is an advanced course in fluid dynamics which introduces the students to mathematical and computational modeling of flow and transport phenomena. Topics include: review of vectors and tensors; derivation of the governing conservation equations (Navier-Stokes); constitutive equations; exact solutions; Stokes/Rayleigh problems; low Reynolds number (Stokes) flows; flow in Hele-Shaw cells; lubrication theory; thin films; boundary layer theory; hydrodynamic stability; Kelvin-Helmholtz and Rayleigh instabilities; stability of parallel shear flows; interfacial phenomena; surface tension; contact angles and contact lines; finite-difference computational methods; vorticity-streamline formulations; staggered grids; boundary integral and volume-of-fluid methods for flows with interfaces.
MATH 382 Perturbation and Asymptotic Analysis. Non-dimensionalization and scaling, regular and singular perturbation problems, asymptotic expansions; asymptotic evaluation of integrals with Laplace's approximation, Watson's lemma, steepest descents and stationary phase; perturbation methods in ordinary and partial differential equations; boundary layers and matched asymptotic expansions; method of multiple time scales; homogenization; WKB method, rays and geometrical optics. Prerequisite: differential equations.
MATH 384 Advanced Partial Differential Equations. Formerly Math 282. Advanced topics in the study of linear and nonlinear partial differential equations. Topics may include the theory of distributions; Hilbert spaces; conservation laws, characteristics and entropy methods; fixed point theory; critical point theory; the calculus of variations and numerical methods. Applications to fluid mechanics, mathematical physics, mathematical biology and related fields. Prerequisite: Math 280 or 282; recommended Math 232.
MATH 385 Mathematical Modeling in Biology. With examples selected from a wide range of topics in biology and physiology, this course introduces both discrete and continuous bio-mathematical modeling, including deterministic and stochastic approaches. Methods for simplifying and analyzing the resulting models are also described. These include non-dimensionalization and scaling, perturbation methods, analysis of stability and bifurcations, and numerical simulations. The selection of topics will vary from year to year but may include: enzyme kinetics, transport across cell membranes, the Hodgkin-Huxley model of excitable cells, gene regulatory networks and systems biology, cardio-vascular, respiratory and renal systems, population dynamics, predator-prey systems, population genetics, epidemics and spread of infectious diseases, cell cycle modeling, circadian rhythms, glucose-insulin kinetics, bio-molecular switches, pattern formation, morphogenesis, etc. Prerequisite: advanced calculus, linear algebra, differential equations, probability theory, and basic numerical methods.
MATH 386 Image Processing. Formerly Math 336. This seminar exposes the student to modern image processing theory and techniques. The course will be composed of two parts. The first part will consist of lectures on the basics of image modeling and image processing. Specific topics of study include the cartoon/texture decomposition, level set representation, wavelet processing, and the Scale Invariant Feature Transform (SIFT). Furthermore, several computational approaches will be discussed including variational methods, a-contrario techniques, graph based approaches, and Bayesian analysis. During the first part, the student will select a topic of interest and the second part will explore the student chosen topics in more depth through student presentations and group discussions. Prerequisite: Experience with MATLAB is strongly recommended. Undergraduate courses in probability, linear algebra, and differential equations would be helpful.
MATH 387 Discrete Mathematical Modeling. Formerly Math 389. Discrete mathematics deals with countable quantities. The techniques used for discrete models often differ significantly from those used for continuous models. This course explores some of the main techniques and problems that arise in discrete mathematical modeling. Topics include combinatorial analysis, Markov chains, graph theory, optimization, algorithmic behavior and phase transitions in random combinatorics. The goal is for students to acquire sufficient skills to solve real-world problems requiring discrete mathematical models. Prerequisite: Probability and linear algebra. A previous course in discrete mathematics would be helpful.
MATH 388 Continuous Mathematical Modeling. A course aimed at the construction, simplification, analysis, interpretation and evaluation of mathematical models that shed light on problems arising in the physical and social sciences. Derivation and methods for solution of model equations, heat conduction problems, simple random walk processes, simplification of model equations, dimensional analysis and scaling, perturbation theory, and a discussion of self-contained modular units that illustrate the principle modeling ideas. Students will normally be expected to develop a modeling project as part of the course requirements. Prerequisite: permission of instructor.
MATH 389 Advanced Topics in Mathematics. Topics will vary from year to year.
MATH 393 Advanced Mathematics Clinic. Normally a continuation of Math 293. The Mathematics Clinic provides applied, real-world research experience. A team of 3-5 students will work on an open-ended research problem from an industrial partner, under the guidance of a faculty advisor. Problems involve a wide array of techniques from mathematical modeling as well as from engineering and computer science. Clinic projects generally address problems of sufficient magnitude and complexity that their analysis, solution and exposition require a significant team effort. Prerequisite: permission of instructor.
MATH 451 Statistical Mechanics and Lattice Models. An intermediate-level graduate statistical mechanics course emphasizing fundamental techniques in mathematical physics. Topics include: random walks, lattice models, asymptotic/thermodynamic limit, critical phenomena, transfer matrix, duality, polymer model, mean field, variational method, renormalization group, finite-size scaling. Prerequisite: Math 251 or equivalent. The course will assume a basic familiarity with thermodynamics, as well as undergraduate-level real analysis and linear algebra.
MATH 453 Financial Time Series. Formerly Math 463. This course will cover standard methods in handling time series data. These include univariate arima and garch models. Some multivariate models will also be discussed. Prerequisite: Math 251 and Math 252.
MATH 458A Quantitative Risk Management (2-UNIT COURSE). This course will focus on the calculation of Value-at-Risk, risk theory, and extreme value theory. We will also study coherent measures of risk, the Basle accords, and, if time allows, the role of BIS. There will be a practical assignment with data coming from Riskmetrics. We will discuss practical issues following the 2008-2009 financial crisis. Prerequisite: Math 256 & knowledge of derivatives.
MATH 458B Optimal Portfolio Theory (2-UNIT COURSE). This course will touch briefly on the (one-period) CAPM. Then we will move to the dynamic CAPM (Merton's model), first in discrete time, then in continuous time. We will also cover related approaches favored by practitioners, such as Black-Litterman. We will also cover estimation problems, and, if time allows, Roll's critique of the CAPM. Along the way, we will study dynamic programming in discrete and continuous time. Prerequisite: Math 256. Corequisite: Math 358.
MATH 461 Level-Set Methods. This course provides an introduction to level-set methods and dynamic implicit surfaces for describing moving fronts and interfaces in a variety of settings. Mathematical topics include: construction of signed distance functions; the level-set equation; Hamilton-Jacobi equations; motion of a surface normal to itself; re-initialization; extrapolation in the normal direction; and the particle level-set method. Applications will include image processing and computer vision, image restoration, de-noising and de-blurring, image segmentation, surface reconstruction from unorganized data, one- and two-phase fluid dynamics (both compressible and incompressible), solid/fluid structure interaction, computer graphics simulation of fluids (i.e. smoke, water), heat flow, and Stefan problems. Appropriate for students in applied and computational mathematics, computer graphics, science, or engineering. Prerequisite: advanced calculus, numerical methods, computer programming.
MATH 462 Bayesian Inference and Machine Learning. This is a seminar course concentrating on the mathematical theory and techniques of machine learning. The course consists of two parts. The first part concentrates on variational modeling and techniques in machine learning. A rigorous approach to variational mathematics will be developed, and modern applications of variational modeling will be explored. Topics include support vector machines, L1 regularization, linear discriminate analysis, and PDE modeling. The second half of the course covers Bayesian analysis with special emphasis on non-parametric Bayesian modeling. The connection with probabilistic graphical structures, Poisson point processes, and Levy processes will be made. Topics in the area include conjugate priors, Markov-Chain Monte Carlo and Gibbs sampling methods and mean field theories. If time permits, and depending on the interests of students, we may also cover compressed sensing, operator splitting, or Bregman iterations.
MATH 466 Mathematical Foundations of Data-Intensive Algorithms. The computational science fields are becoming increasingly data- and information-oriented. As vast datasets are available in scientific, commercial, and social domains, new set of problems become of utmost interest. How does the Digital Sky Survey use millions of pictures to map the universe? How does Google know which pages you want to see? How does Amazon know which items are you likely to buy next? All of these are examples of high-dimensional data, or data available in the form of a graph, which upon clever analysis yields interesting and impressive insights. In this seminar, we will discuss the mathematical foundations behind these techniques, which bring together fields of computer science, mathematics, and probability and statistics. Topics covered will include properties of high-dimensional data and random graphs, random walks on graphs and the related PageRank algorithm, algorithms for massive data problems, including recommendation systems, and probabilistic reasoning in graphical models, such as Bayesian Networks and Factor Graphs. The seminar will be focused on the theory behind these problems and techniques, with occasional homework exercises that deal with actual data. Prerequisites: At least some proof-based math course, probability and linear algebra.