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.
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.
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.
Complex Analysis. Topics include but not limited to: Algebraic properties of complex numbers. Topological properties of the complex plane. Differentiation and holomorphic functions. Complex series, power series. Local and global Cauchy’s theorem. Mobius transformation. Cauchy’s integral theorem. Classification of singularities, Calculus of residues, contour integration. Conformal Mapping. If time permits, the Laplace and the Fourier transforms. Prerequisite: linear algebra; Math 231 recommended.
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.
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).
Modern Geometry. Geometry from a modern viewpoint. Euclidean geometry, discrete geometry, hyperbolic geometry, elliptical geometry, projective geometry, and fractal geometry. Additional topics may include algebraic varieties, differential forms, or Lie groups.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Computational Statistics. An introduction to computationally intensive statistical techniques. Topics may include: random variable generation, Markov Chain Monte Carlo, tree based methods (CART, random forests), kernel based techniques (support vector machines), optimization, other classification, clustering & network analysis, the bootstrap, dimension reduction techniques, LASSO and the analysis of large data sets. Theory and applications are both highlighted. Algorithms will be implemented using statistical software.
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.
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.
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.
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.
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.
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.
Introduction to C++ Programming (2-UNIT COURSE). 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.
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.
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.
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.
Complexity Theory. Specific topics include finite automata, pushdown automata, Turing machines, and their corresponding languages and grammars; undecidability; and complexity classes, reductions, and hierarchies.
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.
Representations of High Dimensional Data. In today's world, data is exploding at a faster rate than computer architectures can handle. For that reason, mathematical techniques to analyze large-scale objects must be developed. One mathematical method that has gained a lot of recent attention is the use of sparsity. Sparsity captures the idea that high dimensional signals often contain a very small amount of intrinsic information. In this course, we will explore various mathematical notions used in high dimensional signal processing including wavelet theory, Fourier analysis, compressed sensing, optimization problems, and randomized linear algebra. Students will learn the mathematical theory, and perform lab activities working with these techniques.
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.
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.
Linear Algebra. Topics may include approximation in inner product spaces, similarity, the spectral theorem, Jordan canonical form, the Cayley Hamilton Theorem, polar and singular value decomposition, Markov processes, behavior of systems of equations.
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.
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.
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.
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.
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. Prerequisites: Differential equations course and Math 231, or permission of instructor.
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: Differential equations course and Math 231, or permission of instructor.
Partial Differential Equations.Theory and applications of quasi-linear and linear equations of first order, including systems. Theory of higher order linear equations, including classical methods of solutions for the wave, heat and potential equations.
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.
Applied Analysis. 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.
Wavelets and their Applications. 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.
Stochastic Methods in Operations Research. Queuing theory, decision theory, discrete event simulation, inventory modeling, and Markov decision processes.
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.
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.
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.
Mathematics Clinic. 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.
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.
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.
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.
Real and Functional Analysis I. 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 231 and Math 232.
Real and Functional Analysis II. 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.
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.
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.
Asymptotic Methods in Statistics with Applications. Modes of convergence for random variables and their distributions; central limit theorems; laws of large numbers; statistical large sample 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).
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.
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.
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.
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.
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.
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 closed-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.
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-form and the structural (Merton) approach. Prerequisite: Math 256. Corequisite: Math 358.
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 368 recommended.
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).
Data Mining. Data mining is the process for discovering patterns in large data sets using techniques from mathematics, computer science and statistics, with applications ranging from biology and neuroscience to history and economics. Students will learn advanced data mining techniques that are commonly used in practice, including linear classifiers, support vector machines, clustering, dimension reduction, transductive learning and topic modeling.
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: linear algebra, ODEs, and elementary numerical analysis.
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.
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.
Advanced Partial Differential Equations. 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; Math 2323 recommended.
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.
Image Processing. 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.
Discrete Mathematical Modeling. 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.
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 principal modeling ideas. Students will normally be expected to develop a modeling project as part of the course requirements. Prerequisite: permission of instructor; Math 294 recommended.
Advanced Topics in Mathematics. Topics will vary from year to year.
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.
Independent Study (Master's Students). Directed research or reading with individual faculty.
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.
Large-Scale Inference. This is a course for graduate students in statistics, mathematics and other fields that require an advanced background and deeper understanding of the theory and methods of high-dimensional data analysis. Bayesian hierarchical models and Empirical Bayesian approaches will be presented, as well as theory and application of multiple hypothesis testing such as false discovery rate. Applications will be presented on important and high-throughput medical problems particularly in genomic medicine.
Financial Time Series. 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.
Statistical Learning. This course is targeted at statisticians and financial engineering practitioners who wish to use cutting-edge statistical learning techniques to analyze their data. The main goal is to provide a toolset to deal with vast and complex data that have emerged in fields ranging from biology to finance to marketing to astrophysics in the past twenty years. The course presents some of the most important modeling and prediction techniques, along with relevant applications. Topics include principal component analysis, linear regression, classification, resampling methods, shrinkage approaches, tree-based methods, clustering, and Bayesian MCMC modeling.
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.
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.
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.
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.
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.
Independent Research (PhD students). Directed research or reading with individual faculty.