sign in sign up
<<
Home , Numerical linear algebra, Social choice theory

Institute of Mathematical Sciences 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.

MATH 231 MATH 242 Principles of Real Analysis I. Countable sets, least upper . Curves and surfaces, Gaussian bounds, and metric space topology including compactness, curvature, isometries, analysis, covariant differentiation completeness, connectivity, and uniform . Related with applications to physics and geometry (intended for topics as time permits. physicists and mathematicians). Prerequisite: Math 231 MATH 232 recommended. Principles of Real Analysis II. A rigorous study of calculus MATH 243 in Euclidean Spaces including multiple Riemann Integrals, Topics in Geometry. Selected topics in Riemannian geometry, derivatives of transformations, and the inverse function low dimensional manifold , elementary Lie groups and theorem. Prerequisite: Math 231. Lie algebra, and contemporary applications in MATH 235 and physics. Prerequisite: permission of instructor. Complex Analysis. Topics include but not limited to: Algebraic MATH 244 properties of complex numbers. Topological properties of the Algebraic Topology. An introduction to algebraic topology. complex plane. Differentiation and holomorphic functions. Basics of category theory, simplicial homology and Complex series, power series. Local and global Cauchy’s cohomology, relative homology, exact sequences, Poincare theorem. Mobius transformation. Cauchy’s integral theorem. duality, CW complexes, DeRahm cohomology, applications to Classification of singularities, Calculus of residues, contour knot theory. integration. Conformal Mapping. If time permits, the Laplace MATH 245 and the Fourier transforms. Prerequisite: ; Math Topics in Geometry and Topology. Topic varies from year to 231 recommended. year and will be chosen from: Differential Topology, Euclidean MATH 236 and Non-Euclidean Geometries, Knot Theory, Algebraic Complex Variables and Applications. Complex differentiation, Topology, and Projective Geometry. Cauchy-Riemann , Cauchy integral formula, Taylor MATH 247 and Laurent expansions, residue theory, contour integration Topology. Topological spaces, product spaces, quotient including branch point contours, uses of Jordan’s lemma, spaces, Hausdorff spaces, compactness, connectedness, path Fourier and Laplace transform integrals, conformal mapping. connectedness, fundamental groups, homotopy of maps, MATH 239 and covering spaces. Corequisite: Math 231 or permission of . Fourier analysis begins with the examination instructor. of the difficulties involved in attempting to reconstruct arbitrary MATH 248 functions as infinite combinations of elementary trigonometric Knot Theory. An introduction to the theory of knots and links functions. Topics in this course will include Fourier series, from combinatorial, algebraic, and geometric perspectives. summability, types and questions of convergence, and the Topics will include knot diagrams,p-colorings, Alexander, Fourier transform (with, if time permits, applications to PDEs, Jones, and HOMFLY polynomials, Seifert surfaces, genus, medical imaging, linguistics, and number theory). Seifert matrices, the fundamental group, representations of MATH 240 knot groups, covering spaces, surgery on knots, and important Modern Geometry. Geometry from a modern viewpoint. families of knots. Prerequisite: Topology (Math 247), or Algebra Euclidean geometry, , hyperbolic geometry, (Math 271), or permission of instructor. elliptical geometry, projective geometry, and fractal geometry. MATH 249 Additional topics may include algebraic varieties, differential Discrete Geometry. The goal of this course is to introduce forms, or Lie groups. students to the basics of discrete and convex geometry. Topics MATH 241 covered will include convex bodies, lattices, quadratic forms, Hyperbolic Geometry. An introduction to hyperbolic and interactions between them, such as the fundamentals geometry in dimensions two and three. Topics will include: of Minkowski’s theory, shortest vector problem, reduction Poincaré disk model, upper half-space model, hyperbolic , LLL, and connections to computational complexity isometries, linear fractional transformations, hyperbolic and theoretical science. Additional topics may trigonometry, cross-ratio, hyperbolic manifolds, and hyperbolic include an introduction to optimization questions, such as knots. packing, and convering problems.

Institute of Mathematical Sciences | Graduate Math Courses Updated 2020-10-23 | 1 MATH 250 MATH 257 Statistical Methods for Clinical Trials Data. A second course Intermediate Probability (2-UNIT COURSE). Continuous in biostatistics. Emphasis on the most commonly used random variables; functions; joint density functions; statistical methods in pharmaceutical and other medical marginal and conditional distributions; functions of random research. Topics such as design of clinical trials, power and variables; conditional expectation; covariance and correlation; sample size determination, contingency table analysis, odds moment generating functions; law of large numbers; ratio and relative risk, survival analysis. Chebyshev’s theorem and central limit theorem. Students may MATH 251 not receive credit for both Math 251 and Math 257. Probability. The main elements of at an MATH 258 intermediate level. Topics include combinatorial analysis, Statistical Linear Models. An introduction to analysis of vari- conditional probabilities, discrete and continuous random ance (including one-way and two-way fixed effects ANOVA) variables, probability distributions, central limit theorem, and and (including simple linear regression, multi- numerous applications. Students may not receive credit for ple regression, variable selection, stepwise regression and anal- both Math 251 and Math 257. ysis of residual plots). Emphasis will be on both methods and MATH 252 applications to data. Statistical software will be used to analyze Statistical Theory. This course will cover in depth the data. Prerequisite: Math 252 or permission of instructor. mathematics behind most of the frequently used statistical MATH 259 tools such as point and interval estimation, hypothesis testing, Topics in . Topics vary from year to year and may goodness of fit, ANOVA, linear regression. This is a theoretical include: analysis of genetic data, experimental design, time se- course, but we will also be using statistical package to gain ries, computational methods, Bayesian analysis or other topics. some hands on experience with data. MATH 260 MATH 253 Monte Carlo Methods. This course introduces concepts Bayesian Statistics. An introduction to principles of data and statistical techniques that are critical to constructing analysis and advanced statistical modeling using Bayesian and analyzing effective simulations, and discusses certain inference. Topics include a combination of Bayesian applications for simulation and Monte Carlo methods. Topics principles and advanced methods; general, conjugate and include random number generation, simulation-based noninformative priors, posteriors, credible intervals, Markov optimization, model building, bias-variance trade-off, input Chain Monte Carlo methods, and hierarchical models. The selection using experimental design, Markov chain Monte Carlo emphasis throughout is on the application of Bayesian thinking (MCMC), and numerical integration. Prerequisite: Math 251. to problems in data analysis. Statistical software will be used MATH 264 as a tool to implement many of the techniques. Scientific Computing. Computational techniques applied to MATH 254 problems in the sciences and . Modeling of physical Computational Statistics. An introduction to computationally problems, computer implementation, analysis of results; use intensive statistical techniques. Topics may include: random of mathematical software; numerical methods chosen from: variable generation, Markov Chain Monte Carlo, tree based solutions of linear and nonlinear algebraic equations, solutions methods (CART, random forests), based techniques of ordinary and partial differential equations, finite elements, (support vector machines), optimization, other classification, linear programming, optimization algorithms, and fast-Fourier clustering & network analysis, the bootstrap, dimension transforms. reduction techniques, LASSO and the analysis of large data MATH 265 sets. Theory and applications are both highlighted. Algorithms . An introduction to the theory and will be implemented using statistical software. methods for numerical solution of mathematical problems. MATH 255 Core topics include: analysis of error and efficiency of Time Series. An introduction to the theory of statistical time methods; solutions of linear systems by series. Topics include decomposition of time series, seasonal and iterative methods; calculation of eigenvalue and models, forecasting models including causal models, trend eigenvectors; interpolation and approximation; numerical models, and smoothing models, autoregressive (AR), moving integration; solution of ordinary differential equations. average (MA), and integrated (ARIMA) forecasting models. MATH 267 Time permitting we will also discuss state space models, which Complexity Theory. Specific topics include finite include Markov processes and hidden Markov processes, and automata, pushdown automata, Turing machines, and their derive the famous Kalman filter, which is a recursive corresponding languages and grammars; undecidability; and to compute predictions. Statistical software will be used as a complexity classes, reductions, and hierarchies. tool to aid calculations required for many of the techniques. MATH 268 MATH 256 Algorithms. Algorithm design, computer implementation, Stochastic Processes. Continuation of Math 251. Properties and analysis of efficiency. Discrete structures, sorting and of independent and dependent random variables, conditional searching, time and space complexity, and topics selected expectation. Topics chosen from Markov processes, second from algorithms for arithmetic circuits, sorting networks, order processes, stationary processes, ergodic theory, parallel algorithms, computational geometry, parsing, and Martingales, and renewal theory. Prerequisite: Math 251 or pattern-matching. permission of instructor.

Institute of Mathematical Sciences | Graduate Math Courses Updated 2020-10-23 | 2 MATH 269 birational geometry, tangent spaces, nonsingularity and Representations of High Dimensional Data. In today’s intersection theory. Prerequisite: Math 271; recommended world, data is exploding at a faster rate than computer previous courses in Analysis, Galois Theory, Differential architectures can handle. For that reason, mathematical Geometry and Topology are helpful but not required; or techniques to analyze large-scale objects must be developed. permission of the instructor. One mathematical method that has gained a lot of recent MATH 277 attention is the use of sparsity. Sparsity captures the idea Topics in Algebra. Topics vary from year to year and will be that high dimensional signals often contain a very small chosen from: Representation Theory, Algebraic Geometry, amount of intrinsic information. In this course, we will explore Commutative Algebra, Algebraic Number Theory, Coding various mathematical notions used in high dimensional Theory, Algebraic , Algebraic , including wavelet theory, Fourier analysis, Matroid Theory. Prerequisite: Math 271 or permission of compressed sensing, optimization problems, and randomized instructor. linear algebra. Students will learn the mathematical theory, and perform lab activities working with these techniques. MATH 280 Introduction to Partial Differential Equations. Classifying MATH 271 PDEs, the method of characteristics, the heat , wave Abstract Algebra I. Groups, rings, fields and additional topics. equation, and Laplace’s equation, separation of variables, Topics in group theory include groups, subgroups, quotient Fourier series and other orthogonal expansions, convergence groups, Lagrange’s theorem, symmetry groups, and the iso- of orthogonal expansions, well-posed problems, existence morphism theorems. Topics in Ring theory include Euclidean and uniqueness of solutions, maximum principles and energy domains, PIDs, UFDs, fields, polynomial rings, ideal theory, and methods, Sturm-Liouville theory, Fourier transform methods the isomorphism theorems. In recent years, additional topics and Green’s functions, Bessel functions. Prerequisites: have included the Sylow theorems, group actions, modules, Differential equations course and Math 231, or permission representations, and introductory category theory. of instructor. MATH 272 MATH 281 Abstract Algebra II: Galois Theory. Topics covered will include Dynamical Systems. The theory of continuous dynamical polynomial rings, extensions, classical constructions, systems was developed largely in response to the reality splitting fields, algebraic closure, separability, Fundamental that most nonlinear differential equations lack exact analytic Theorem of Galois Theory, Galois groups of polynomials and solutions. In addition to being of in their own right, solvability. Prerequisite: Math 271. This course is independent such nonlinear equations arise naturally as mathematical of Math 274 (Abstract Algebra II: Representation Theory), and models from many disciplines including biology, chemistry, students may receive credit for both courses. physiology, ecology, physics, and engineering. This course is MATH 273 an introduction to and survey of characteristic behavior of Linear Algebra. Topics may include approximation in inner such dynamical systems. Applications will be an integral part product spaces, similarity, the spectral theorem, Jordan of the course with examples including mechanical vibrations, canonical form, the Cayley Hamilton Theorem, polar and biological rhythms, circuits, insect outbreaks, and chemical singular decomposition, Markov processes, behavior of oscillations. Prerequisite: Differential equations course and systems of equations. Math 231, or permission of instructor. MATH 274 MATH 283 Abstract Algebra II: Representation Theory. Topics covered Mathematical Modeling. Introduction to the construction and will include group rings, characters, relations, interpretation of deterministic and stochastic models in the induced representations, application of representation theory, biological, social, and physical sciences, including simulation and other select topics from module theory. Prerequisite: Math studies. Students are required to develop a model in an area of 271. This course is independent of Math 272 (Abstract Algebra their interest. II: Galois Theory), and students may receive credit for both MATH 285 courses. Wavelets and their Applications. An introduction to MATH 275 wavelet analysis with a focus on applications. Wavelets are Number Theory. Topics covered will include the fundamental an important tool in modern signal and image processing theorem of arithmetic, Euclid’s algorithm, congruences, as well as other areas of . The main Diophantine problems, quadratic reciprocity, arithmetic objective of this course is to develop the theory behind functions, and distribution of primes. If time allows, we may wavelets and similar constructions. Theoretical topics may also discuss some geometric methods, coming from lattice include Fourier analysis, the discrete and continuous wavelet point counting, such as Gauss’s circle problem and Dirichlet transform, Shannon’s Sampling Theorem, statistical studies divisor problem, as well as some applications of Number of wavelet signal extraction, and the Heisenberg uncertainty Theory to and . principle. Application based topics may include wavelet based MATH 276 compression, signal processing methods, communications, Algebraic Geometry. Topics include affine and projective and sensing mechanisms where wavelets play a crucial role. varieties, the Nullstellensatz, rational maps and morphisms, Students will gain hands-on experience with real-world imaging techniques.

Institute of Mathematical Sciences | Graduate Math Courses Updated 2020-10-23 | 3 MATH 286 MATH 306 Stochastic Methods in . Queuing theory, Optimization. The course emphasizes nonlinear programming. , discrete event simulation, inventory modeling, It covers numerical methods for finite-dimensional and Markov decision processes. optimization problems with fairly smooth functions. Both non- MATH 287 constrained and constrained optimizations will be discussed. Deterministic Methods in Operations Research. Linear, Certain degree of emphasis will be given to the convergence integer, nonlinear, and dynamic programming. Applications analysis of the numerical methods. Prerequisite: multivariable to transportation problems, inventory analysis, classical calculus and . optimization problems, and network analysis, including project MATH 334 planning and control. Applied Analysis. The course integrates concepts from real MATH 288 and with applications to differential equa- Social Choice and Decision-Making. This course focuses tions. We will introduce theoretical concepts such as metric, on the modeling of individual and group decisions using Banach and Hilbert spaces as well as abstract methods such as techniques from . Topics will include: basic the contraction mapping theorem, with a focus on applications concepts of game theory and , rather than exhaustivity. We will apply these concepts to the representations of games, Nash equilibria, theory, study of integral equations, ordinary differential equations and non-cooperative games, cooperative games, games, partial differential equations. The main methods will be Fourier paradoxes, impossibility theorems, Shapley value, power series and the Sturm-Liouville theory. indices, problems, and applications. MATH 335 MATH 289 Integral Transforms and Applications. Transforms covered will Special Topics in Mathematics. Topics vary from year to year include: Fourier, Laplace, Hilbert, Hankel, Mellin, Radon, and Z. and may include: algebraic geometry, algebraic topology, fluid The course will be relevant to mathematicians and enginners dynamics, partial differential equations, games and gambling, working in communications, signal and image processing, Bayesian analysis, geometric group theory or other topics. continuous and digital filters, wave propagation in fluids and solids, etc. MATH 293 Mathematics Clinic. The Mathematics Clinic provides applied, MATH 337 real-world research experience. A team of 3-5 students will Real and Functional Analysis I. Abstract measures, Lebesgue work on an open-ended research problem from an industrial measure on Rn, and Lebesgue-Stieljes measures on R. The partner, under the guidance of a faculty advisor. Problems Lebesgue integral and limit theorems. Product measures and involve a wide array of techniques from mathematical the Fubini theorem. Additional related topics as time permits. modeling as well as from engineering and computer science. Prerequisite: Math 231 and Math 232. Clinic projects generally address problems of sufficient MATH 338 magnitude and complexity that their analysis, solution Real and Functional Analysis II. Continuation of Math 337. and exposition require a significant team effort. Students Some of the topics covered will be: Banach and Hilbert spaces; are normally expected to enroll in Math 393 (Advanced Lp-spaces; complex measures and the Radon-Nikodym Mathematics Clinic) in the subsequent semester. Prerequisite: theorem. Prerequisite: Math 337. permission of instructor. MATH 351 MATH 294 Time Series Data Analysis. Analysis of time series data Methods of Applied Mathematics. Applications of linear by means of particular models such as ARIMA. Spectral algebra and differential equations in modeling. Concepts such analysis. Associated methods of inference and applications. as vector spaces, solvability of linear systems, eigenvalues and Prerequisite: permission of instructor. eigenvectors, decomposition, norms, MATH 352 matrix exponentials, solution of ODEs, nonlinear dynamical Nonparametric Statistics. Treatment of statistical questions systems, bifurcations, phase-plane analysis, stability of fixed which do not depend on specific parametric models. Examples points, and Floquet theory are reviewed. They are illustrated are testing for symmetry of a distribution and testing for through examples such as Leslie population models, Google’s equality of two distributors. Elementary combinatorial PageRank algorithm, Markov chains, image compression, methods will play a major role in the course. Prerequisite: Math population dynamics, linear and nonlinear oscillators, predator- 252 or permission of instructor. prey models, spread of epidemics, enzyme kinetics, gene/ protein regulatory networks and the like. Familiarity with MATH 353 undergraduate level differential equations and linear algebra Asymptotic Methods in Statistics with Applications. Modes will be helpful. 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 quan- tiles, statistics, and extreme order statistics; asymptoti- cally efficient estimation and hypothesis testing. Prerequisite: Math 251 and 252; linear algebra; analysis (Math 231 and 232 or equivalent).

Institute of Mathematical Sciences | Graduate Math Courses Updated 2020-10-23 | 4 MATH 354 MATH 361A Reliability Theory. Structural properties and reliability of Numerical Methods for Finance (2-UNIT COURSE). This complex systems; classes of life distributions based on aging; course focuses on pricing derivatives, but some topics of risk maintenance and replacement models; availability, reliability, management are also covered. Whereas the Mathematical and mean time between failures for complex systems; Markov Finance course (Math 358) shows the student how to in- models for systems; elementary renewal theory. Prerequisite: struments using closed-form (analytical) formulae, this course Math 251. Math 256 would be helpful but not essential. focuses on the instruments that can be best analyzed with MATH 355 numerical methods: structured loans, mortgage-backed secu- Linear Statistical Models. A discussion of linear statistical rities, etc. Topics include binomial and trinomial tree (lattice), models in both the full and less-than-full rank cases, the finite differences, Monte Carlo simulation, and an introduction Gauss-Markov theorem, and applications to regression to copulae. Prerequisite: Math 256. Corequisite: Math 358. analysis, analysis of variance, and analysis of covariance. MATH 362 Topics in design of experiments and multivariate analysis. Numerical Methods for Differential Equations. This course Prerequisite: Linear algebra and a year course in probability is devoted to numerical integration of ordinary and partial and statistics. differential equations. Methods such as Runge-Kutta and MATH 357 Adams formulas, predictor-corrector methods, stiff equation Deterministic and Stochastic Control. The course consists solvers and shooting method for BVPs are described. of two parts. The first part is lecture-based, and will cover Other techniques based on interpolation via Lagrange both deterministic and stochastic control. In deterministic and Chebyshev polynomials and cubic splines, numerical control, we will cover the , Pontryagin’s differentiation using finite differences, spectral and pseudo- maximum principle, the linear regulator, and controllability. spectral methods are introduced. Numerical methods for In stochastic control, we will review some stochastic analysis, solving elliptic, parabolic and hyperbolic partial differential and then will cover dynamic programming, viscosity solutions, equations are presented including discussion of truncation the stochastic version of Pontryagin’s maximum principle, error, consistency, stability, accuracy and convergence. Topics and backward stochastic differential equations. The second such as explicit vs. implicit schemes; implementation of part will be a seminar on applications of optimal control to Dirichlet, Neumann and Robin boundary conditions; operator engineering and to problems. The splitting; Godunov methods for hyperbolic systems; direct and course will cover a new method based on to iterative methods for elliptic systems; Fourier and Chebyshev solve backward stochastic differential equations analytically, based spectral and pseudo-spectral methods are discussed. with potential new applications. Prerequisite: Math 256 The Level-Set Method and its related PDEs may also be (Stochastic Processes). Knowledge of Ito calculus is helpful but considered. A background in differential equations and some not required, and will be reviewed in the course. computational skills (e.g. MATLAB, Python or ) are assumed. MATH 358 MATH 365 Mathematical Finance. This course will cover the theory of Statistical Methods In Molecular Biology. Topics include option pricing, emphasizing the Black-Scholes model and statistical analysis of microarray data (the biological problem, models. Implementation of the theory and model modern microarray technology, high-dimensional small calibration are covered in the companion course, Numerical sample size data problem, key elements in a good experiment Methods for Finance, Math 361A. We will see the binomial design, identifying sources of variation and data preparation, no-arbitrage pricing model, state , Brownian motion, normalization techniques, statistical methods and algorithms stochastic integration, and Ito’s lemma, the Black-Scholes for error detection and correction, statistical analysis equation, risk-neutral pricing and Girsanov theorem, change of workflow, software tools including R and bioconductor, and numeraire and two term structure models: Vasicek and LIBOR. computational implementation of workflow methods and Prerequisite: Mature understanding of advanced calculus algorithms in R) and microarray data interpretation (use and probability (at the level of Math 251) and permission of of linear models for analysis and assessment of differential instructor. Math 256 would be helpful. expression, gene selection (Empirical Bayes, volcano plots) and gene classification (clustering, heatmaps)). Prerequisite: MATH 359 competency in scientific computing, linear algebra, an upper Computational Statistics. This course will cover standard division course in statistics, familiarity with basic biology, methods typically discussed in a computational statistics access to a computer on which the Elluminate software can be course. Areas include optimization methods: Newton, quasi- run (requires latest JAVA runtime environment). Newton. Fisher scoring, iteratively reweighted least squares, EM algorithm. These methods presented for both settings: MATH 366 univariate and multivariate. Combinatorial optimization: . Data mining is the process for discovering Simulated Annealing and genetic algorithm. Simulation and patterns in large data sets using techniques from mathematics, Monte Carlo Integration methods: exact and approximation. computer science and statistics, with applications ranging MCMC methods: Metropolis-Hasting algorithm and Gibbs from biology and neuroscience to history and . Sampling. R/R-studio will be used for all computational needs. Students will learn advanced data mining techniques that are Prerequisite: Math 251 or equivalent. commonly used in practice, including linear classifiers, support vector machines, clustering, dimension reduction, transductive learning and topic modeling.

Institute of Mathematical Sciences | Graduate Math Courses Updated 2020-10-23 | 5 MATH 368 sign of matrix, derivatives), matrix convexity; matrix means Numerical Methods for Matrix Computations. This course is and inequalities, properties of positive and nonnegative an in-depth study of numerical linear algebra and the matrix matrices, primitive matrices, stochastic and doubly stochastic computations that arise in solving linear systems, least- matrices; mean transformation; singular values. squares problems, and eigenvalue problems for both dense MATH 384 and sparse matrices. It will cover many of the fundamental Advanced Partial Differential Equations. Advanced topics in algorithms such as LU decomposition, Jacobi, Gauss-Seidel the study of linear and nonlinear partial differential equations. and SOR iterations, methods (e.g., Conjugate Topics may include the theory of distributions; Hilbert spaces; Gradient and GMRES), QR and SVD factorization of matrices, conservation laws, characteristics and entropy methods; fixed eigenvalue problems via power, inverse, and Arnoldi iterations. point theory; critical point theory; the calculus of variations It will also introduce condition numbers and error analysis, and numerical methods. Applications to fluid mechanics, forward and backward stability, constrained and unconstrained , mathematical biology and related fields. optimization problems. The course is designed for those who Prerequisite: Math 280 or 282; Math 2323 recommended. wish to use matrix computations in their own research. A background in linear algebra and some computational skills MATH 385 (e.g., MATLAB, Python or C) are assumed. Mathematical Modeling in Biology. With examples selected from a wide range of topics in biology and physiology, MATH 381 this course introduces both discrete and continuous bio- Mathematical Models in . This course mathematical modeling, including deterministic and stochastic introduces the students to mathematical modeling of fluid approaches. Methods for simplifying and analyzing the flow and transport phenomena. We start with vectors, , resulting models are also described. These include non- notation, and derivation of the Navier-Stokes equations dimensionalization and scaling, perturbation methods, analysis that govern fluid flow. We then use these to model a wide of stability and bifurcations, and numerical simulations. range of problems. Depending on the students’ , The selection of topics will vary from year to year but may topics may be chosen from: hydrostatics, minimal surfaces, include: enzyme kinetics, transport across cell membranes, soap films, and static shapes of sessile and hanging drops; the Hodgkin-Huxley model of excitable cells, gene regulatory shape oscillations of liquid drops in microgravity; pulsatile networks and systems biology, cardio-vascular, respiratory and flow in compliant tubes as a model for blood flow in the renal systems, population dynamics, predator-prey systems, cardiovascular system; slender-body theory and swimming of population genetics, epidemics and spread of infectious bacteria with flagella; linear acoustics and sound propagation; diseases, cell cycle modeling, circadian rhythms, glucose- oscillations of gas bubbles, cavitation, and acoustics of insulin kinetics, bio-molecular switches, pattern formation, bubbly liquids; dynamics of polymer molecules in shear flows morphogenesis, etc. Prerequisite: advanced calculus, linear as models of non-Newtonian fluids; dynamics of magnetic algebra, differential equations, probability theory, and basic suspensions (ferrofluids); lubrication theory, thin liquid films numerical methods. and coating flows; thermo-capillary migration of drops and bubbles; flow in porous media, ground-water flow and MATH 386 Darcy’s law; flow of rivers and estuaries; shallow water waves; Image Processing. This will be a course on mathematical geophysical fluid dynamics and climate modeling. models for image processing and analysis. It will explore Prerequisites: , Differential Equations, Linear fundamental concepts such as image formation, image Algebra. representation, image quantization, point operations, change of contrast, image enhancement, noise, blur, image MATH 382 degradation, edge and contour detection (such as the Perturbation and Asymptotic Analysis. Non- Canny edge detector), corner detection, filtering, denoising, dimensionalization and scaling, regular and singular morphology, image transforms, Fourier shape descriptors, perturbation problems, asymptotic expansions; asymptotic image restoration, image segmentation, and applications. Time evaluation of integrals with Laplace’s approximation, Watson’s permitting, we will explore one or more concepts in computer lemma, steepest descents and stationary phase; perturbation vision, e.g. motion analysis, 3D shape reconstruction, or methods in ordinary and partial differential equations; feature detection and tracking. All theoretical concepts will be boundary layers and matched asymptotic expansions; method accompanied by computer exercises, to be performed either of multiple time scales; homogenization; WKB method, rays in Matlab or in Python. All students will be required to work in and geometrical optics. Prerequisite: differential equations. small teams on a final project of their choice. Math 383 MATH 387 Advanced Matrix Analysis. Topics covered will include: vector Discrete Mathematical Modeling. and matrix norms; Hilbert space; spectrum and eigenvalues; deals with countable quantities. The techniques used for unitary, normal and symmetric matrices; Jordan canonical discrete models often differ significantly from those used for form; the minimal polynomial and the companion matrix; the continuous models. This course explores some of the main real Jordan and Weyr canonical forms; matrix factorizations; techniques and problems that arise in discrete mathematical trace and ; positivity and absolute value; tensor modeling. Topics include combinatorial analysis, Markov product, block matrices and partial ordering, kernel functions, chains, graph theory, optimization, algorithmic behavior projections and positivity preserving mappings; matrix and phase transitions in random combinatorics. The goal calculus (matrix exponent, matrix square root, log of matrix,

Institute of Mathematical Sciences | Graduate Math Courses Updated 2020-10-23 | 6 is for students to acquire sufficient skills to solve real- MATH 453 world problems requiring discrete mathematical models. Financial Time Series. This course will cover standard Prerequisite: Probability and linear algebra. A previous course methods in handling time series data. These include univariate in discrete mathematics would be helpful. arima and garch models. Some multivariate models will also be MATH 388 discussed. Prerequisite: Math 251 and Math 252. Continuous Mathematical Modeling. A course aimed at MATH 454 the construction, simplification, analysis and interpretation Statistical Learning. This course is targeted at statisticians and of mathematical models, primarily in the form of partial financial engineering practitioners who wish to use cutting- differential equations, arising in the physical and biomedical edge statistical learning techniques to analyze their data. The sciences. Derivation and methods of solution: method of main goal is to provide a toolset to deal with vast and complex characteristics, separation of variables, Fourier and Laplace data that have emerged in fields ranging from biology to transforms. Examples such as traffic flow, steady and transient finance to marketing to astrophysics in the past twenty years. heat conduction, potential flow, advection-diffusion processes, The course presents some of the most important modeling wave propagation and acoustics. Dimensional analysis and and prediction techniques, along with relevant applications. scaling, , and bifurcation analysis. Students Topics include principal component analysis, linear regression, will normally work on a modeling project as part of the classification, resampling methods, shrinkage approaches, course. Familiarity with vector calculus, complex variables and tree-based methods, clustering, and Bayesian MCMC modeling. differential equations will be helpful. Prerequisite: permission MATH 458A of instructor; Math 294 recommended. Quantitative Risk Management (2-UNIT COURSE). This MATH 389 course will focus on the calculation of Value-at-Risk, risk Advanced Topics in Mathematics. Topics will vary from year theory, and extreme value theory. We will also study coherent to year. measures of risk, the Basle accords, and, if time allows, the role MATH 393 of BIS. There will be a practical assignment with data coming Advanced Mathematics Clinic. Normally a continuation of from Riskmetrics. We will discuss practical issues following Math 293. The Mathematics Clinic provides applied, real-world the 2008-2009 financial crisis. Prerequisite: Math 256 & research experience. A team of 3-5 students will work on an knowledge of derivatives. open-ended research problem from an industrial partner, MATH 458B under the guidance of a faculty advisor. Problems involve a Optimal Portfolio Theory (2-UNIT COURSE). This course will wide array of techniques from mathematical modeling as well touch briefly on the (one-period) CAPM. Then we will move as from engineering and computer science. Clinic projects to the dynamic CAPM (Merton’s model), first in discrete time, generally address problems of sufficient magnitude and then in continuous time. We will also cover related approaches complexity that their analysis, solution and exposition require a favored by practitioners, such as Black-Litterman. We will also significant team effort. Prerequisite: permission of instructor. cover estimation problems, and, if time allows, Roll’s critique of MATH 398 the CAPM. Along the way, we will study dynamic programming Independent Study (Master’s Students). Directed research or in discrete and continuous time. Prerequisite: Math 256. reading with individual faculty. Corequisite: Math 358. MATH 451 MATH 461 Statistical Mechanics and Lattice Models. An intermediate- Level-Set Methods. This course provides an introduction level graduate statistical mechanics course emphasizing to level-set methods and dynamic implicit surfaces for fundamental techniques in mathematical physics. Topics describing moving fronts and interfaces in a variety of include: random walks, lattice models, asymptotic/ settings. Mathematical topics include: construction of signed thermodynamic limit, critical phenomena, transfer matrix, distance functions; the level-set equation; Hamilton-Jacobi duality, polymer model, mean field, variational method, equations; motion of a surface normal to itself; re-initialization; , finite-size scaling. Prerequisite: Math extrapolation in the normal direction; and the particle level- 251 or equivalent. The course will assume a basic familiarity set method. Applications will include image processing and with thermodynamics, as well as undergraduate-level real computer vision, image restoration, de-noising and de-blurring, analysis and linear algebra. image segmentation, surface reconstruction from unorganized data, one- and two-phase fluid dynamics (both compressible MATH 452 and incompressible), solid/fluid structure interaction, computer Large-Scale Inference. This is a course for graduate students graphics simulation of fluids (i.e. smoke, water), heat flow, in statistics, mathematics and other fields that require an and Stefan problems. Appropriate for students in applied advanced background and deeper understanding of the theory and computational mathematics, computer graphics, science, and methods of high-dimensional data analysis. Bayesian or engineering. Prerequisite: advanced calculus, numerical hierarchical models and Empirical Bayesian approaches will methods, computer programming. 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.

Institute of Mathematical Sciences | Graduate Math Courses Updated 2020-10-23 | 7 MATH 462 Mathematics of Machine Learning. Machine learning is a rapidly growing field that is concerned with finding patterns in data. It is responsible for tremendous advances in technology, from personalized product recommendations to speech recognition in cell phones. This course material covers theoretical foundations, algorithms, and methodologies for machine learning, emphasizing the role of probability and optimization and exploring a variety of real-world applications. Through lecture examples and programming projects, students will learn how to apply powerful machine learning techniques to new problems. Students are expected to have a solid foundation in calculus and linear algebra as well as exposure to the basic tools of logic and probability, and should be familiar with at least one modern, high-level programming language. MATH 466 Advanced Big Data Analysis. This graduate level course is designed to give students a snapshot of recent techniques used to analyze, statistically and algorithmically, extremely large datasets. To accomplish this goal, the course will start with an applied and quick introduction to necessary optimization background. From there we will introduce students to topics such as spectral graph clustering, fast kernel methods, compressed sensing, among others. We will highlight applications of these methods to diverse areas such as genomics and recommender systems, but the bedrock of the course will be theory. To that end, students are expected to have a solid foundation in probability and analysis, as well as comfort with algorithmic thinking. MATH 498 Independent Research (PhD students). Directed research or reading with individual faculty. MATH 499 Doctoral Study.

710 N. College Avenue | Claremont, CA 91711 | cgu.edu/ims

Institute of Mathematical Sciences | Graduate Math Courses Updated 2020-10-23 | 8