MATH 400. Mathematics Teaching Practicum (1)

Practicum for teaching college mathematics. Includes preparation of syllabi, exams, lectures. Grading, alternative teaching styles, use of technology, interpersonal relations and motivation. Handling common problems and conflicts.

 

MATH 401. Abstract Algebra I (3)

Basic properties of groups, rings, modules and fields. Isomorphism theorems for groups; Sylow theorem; nil potency and solvability of groups; Jordan-Holder theorem; Gauss lemma and Eisenstein’s criterion; finitely generated modules over principal ideal domains with applications to abelian groups and canonical forms for matrices; categories and functors; tensor product of modules, bilinear and quadratic forms; field extensions; fundamental theorem of Galois theory, solving equations by radicals. Prereq: MATH 308.

 

MATH 402. Abstract Algebra II (3)

A continuation of MATH 401. Prereq: MATH 401.

 

MATH 405. Advanced Matrix Analysis. (3)

An advanced course in linear algebra and matrix theory. Topics include variational characterizations of eigenvalues of Hermitian matrices, matrix and vector norms, characterizations of positive definite matrices, singular value decomposition and applications, perturbation of eigenvalues. This course is more theoretical than MATH 431, which emphasizes computational aspects of linear algebra Prereq: MATH 307.

 

MATH 406. Mathematical Logic and Model Theory (3)

A study of formal logical systems and their models. Propositional logic and quantification. First order theories; consistency, compactness, and the Lowenheim Skolem theorem. Cross-listed as PHIL 406.

 

MATH 408. Introduction to Cryptology (3)

Introduction to the mathematical theory of secure communication. Topics include: classical cryptographic systems; one-way and trapdoor functions; RSA, DSA, and other public key systems; Primality and Factorization algorithms; birthday problem and other attack methods; elliptic curve cryptosystems; introduction to complexity theory; other topics as time permits. Prereq: MATH 303.

 

MATH 413. Graph Theory (3)

Building blocks of a graph, trees, connectivity, matchings, coverings, planarity, NP-complete problems, random graphs, and expander graphs; various applications and algorithms. Prereq: MATH 201 or MATH 307.

 

MATH 419. Applied Probability and Stochastic Processes for Biology (3)

Applications of probability and stochastic processes to biological systems. Mathematical topics will include: introduction to discrete and continuous probability spaces (including numerical generation of pseudo random samples from specified probability distributions), Markov processes in discrete and continuous time with discrete and continuous sample spaces, point processes including homogeneous and inhomogeneous Poisson processes and Markov chains on graphs, and diffusion processes including Brownian motion and the Ornstein-Uhlenbeck process.  Biological topics will be determined by the interests of the students and the instructor.  Likely topics include: stochastic ion channels, molecular motors and stochastic ratchets, actin and tubulin polymerization, random walk models for neural spike trains, bacterial chemotaxis, signaling and genetic regulatory networks, and stochastic predator-prey dynamics.  The emphasis will be on practical simulation and analysis of stochastic phenomena in biological systems.  Numerical methods will be developed using a combination of MATLAB, the R statistical package, MCell, and/or URDME, at the discretion of the instructor.  Student projects will comprise a major part of the course. Offered as BIOL 319, EECS 319, MATH 319, SYBB 319, BIOL 419, EBME 419, MATH 419, PHOL 419, and SYBB 419.

 

MATH 421. Fundamentals of Analysis I (3)

Abstract mathematical reasoning in the context of analysis in Euclidean space. Introduction to formal reasoning, sets and functions, and the number systems. Sequences and series; Cauchy sequences and convergence. Required for all mathematics majors. Additional work required for graduate students. (May not be taken for graduate credit by graduate students in the Department of Mathematics.) Offered as MATH 321 and MATH 421.

 

MATH 422. Fundamentals of Analysis II (3)

Continuation of MATH 321. Point-set topology in metric spaces with attention to ndimensional space; completeness, compactness, connectedness, and continuity of functions. Topics in sequences, series of functions, uniform convergence, Fourier series and polynomial approximation. Theoretical development of differentiation and Riemann integration. Required for all mathematics majors. Additional work required for graduate students. (May not be taken for graduate credit by graduate students in the Department of Mathematics.) Offered as MATH 322 and MATH 422. Prereq: MATH 321 or MATH 421.

 

MATH 423. Introduction to Real Analysis I (3)

General theory of measure and integration. Measures and outer measures. Lebesgue measure on n-space. Integration. Convergence theorems. Product measures and Fubini’s theorem. Signed measures. Hahn-Jordan decomposition, Radon-Nikodym theorem, and Lebesgue decomposition. SpaceP-integrable function. Lebesgue differentiation theorem in nspace. Prereq: MATH 322 or MATH 422.

 

MATH 424. Introduction to Real Analysis II (3)

Measures on locally compact spaces. Riesz representation theorem. Elements of functional analysis. Normed linear space. Hahn-Banach, Banach-Steinhaus, open mapping, closed graph theorems. Weak topologies. Banach-Alaoglu theorem. Function spaces. Stone-Weierstrass and Ascoli theorems. Basic Hilbert space theory. Application to Fourier series. Additional topics: Haar measure on locally compact groups. Prereq: MATH 423.

 

MATH 425. Complex Analysis I (3)

Analytic functions. Integration over paths in the complex plane. Index of a point with respect to a closed path; Cauchy’s theorem and Cauchy’s integral formula; power series representation; open mapping theorem; singularities; Laurent expansion; residue calculus; harmonic functions; Poisson’s formula; Riemann mapping theorem. More theoretical and at a higher level than MATH 324. Prereq: MATH 322.

 

MATH 426. Geometry and Complex Analysis. (3)

The theme of this course will be the interplay between geometry and complex analysis, algebra and other fields of mathematics. An effort will be made to highlight significant, unexpected connections between major fields, illustrating the unity of mathematics. The choice of text(s) and syllabus itself will be flexible, to be adapted to the range of interests and backgrounds of pre-enrolled students. Possible topics include: the Mobius group and its subgroups, hyperbolic geometry, elliptic functions, Riemann surfaces, applications of conformal mapping, and potential theory in classical physical models. Offered as MATH 326 and MATH 426.

 

MATH 427. Convexity and Optimization (3)

Introduction to the theory of convex sets and functions and to the extremes in problems in areas of mathematics where convexity plays a role. Among the topics discussed are basic properties of convex sets (extreme points, facial structure of polytopes), separation theorems, duality and polars, properties of convex functions, minima and maxima of convex functions over convex set, various optimization problems. Offered as MATH 327, MATH 427, and OPRE 427.

 

MATH 428. Fourier Analysis (3)

Introduction to the mathematical aspects of Fourier analysis and synthesis. Accessible to upper level undergraduates and graduate students in the sciences and engineering. Periodic functions. Fourier series. Convergence theorems. The classical sine and cosine series. General orthogonal systems. Multiple Fourier series. Applications. Fourier integrals and Fourier Transforms. L^1 and L^2 theory. Inversion theorems. Classical sine and cosine transforms. Multiple Fourier Transform. Spherical symmetry. Other important transforms. Applications. In addition to the prerequisite listed below, MATH 321 and MATH 201 are recommended. Prereq: MATH 224.

 

MATH 431. Introduction to Numerical Analysis I (3)

Numerical linear algebra for scientists and engineers. Matrix and vector norms, computer arithmetic, conditioning and stability, orthogonality. Least squares problems: QR factorization, normal equations and Singular Value Decomposition. Direct solution of linear system: Gaussian elimination and Cholesky factorization. Eigenvalues and eigenvectors: the QR algorithm, Rayleigh quotient, inverse iteration. Introduction to iterative methods. Students will be introduced to MATLAB. Prereq: MATH 201 or MATH 308.

 

MATH 432. Numerical Differential Equations (3)

Numerical solution of differential equations for scientists and engineers. Solution of ordinary differential equations by multistep and single step methods. Stability, consistency, and convergence. Stiff equations. Finite difference schemes. Introduction to the finite element method. Introduction to multigrid techniques. The diffusion equation: numerical schemes and stability analysis. Introduction to hyperbolic equations. MATLAB will be used in this course.

 

MATH 433. Numerical Solutions of Nonlinear Systems and Optimization (3)

The course provides an introduction to numerical solution methods for systems of nonlinear equations and optimization problems. The course is suitable for upper-undergraduate and graduate students with some background in calculus and linear algebra. Knowledge of numerical linear algebra is helpful. Among the topics which will be covered in the course are Nonlinear systems in one variables; Newton’s method for nonlinear equations and unconstrained minimization; Quasi-Newton methods; Global convergence of Newton’s methods and line searches; Trust region approach; Secant methods; Nonlinear least squares. Prereq: MATH 223, MATH 201, MATH 431 or permission.

 

MATH 434. Optimization of Dynamic Systems (3)

Fundamentals of dynamic optimization with applications to control. Variational treatment of control problems and the Maximum Principle. Structures of optimal systems; regulators, terminal controllers, time-optimal controllers. Sufficient conditions for optimality. Singular controls. Computational aspects. Selected applications. Cross-listed as EECS 421.

 

MATH 435. Ordinary Differential Equations. (3)

A second course in ordinary differential equations. Existence, uniqueness, and continuation of solutions of ODE. Linear systems, fundamental matrix, qualitative methods (phase plane). Dependence on initial data and parameters (Gronwall’s inequality, nonlinear variation of parameters). Stability for linear and nonlinear equations, linearization, Poincare-Bendixson theory. Additional topics may include regular and singular perturbation methods, autonomous oscillations, entrainment of forced oscillators, and bifurcations. Prereq: MATH 224 and either MATH 201 or MATH 307.

 

MATH 439. Integrated Numerical and Statistical Computation (3)

This course will embed numerical methods into a Bayesian framework. The statistical framework will make it possible to integrate a priori information about the unknowns and the error in the data directly into the most efficient numerical methods. A lot of emphasis will be put on understanding the role of the priors, their encoding into fast numerical solvers, and how to translate qualitative or sample-based information – or lack thereof – into a numerical scheme. Confidence on computed results will also be discussed from Bayesian perspective, at the light of the given data and a priori information. The course should be of interest to anyone working on signal and image processing statistics, numerical analysis and modeling.

 

MATH 440. Computational Inverse Problems (3)

This course will introduce various computational methods for solving inverse problems under different conditions. First the classical regularization methods will be introduced, and the computational challenges, which they pose, will be addressed. Following this, the statistical methods for solving inverse problems will be studied and their computer implementation discussed. We will then see how to combine the two approaches to best exploit their potentials. Applications arising from various areas of science, engineering and medicine will be discussed throughout the course.

 

MATH 441. Mathematical Modeling. (3)

Mathematics is a powerful language for describing real world phenomena and providing predictions that otherwise are hard or impossible to obtain. The course gives the students pre-requisites for translating qualitative descriptions given in the professional non-mathematical language into the quantitative language for mathematics. While the variety in the subject matter is wide, some general principles and methodologies that a modeler can pursue are similar in many applications. The course focuses on these similarities. The course is based on representative case studies that are discussed and analyzed in the classroom, the emphasis being on general principles of developing and analyzing mathematical models. The examples will be taken from different fields of science and engineering, including life sciences, environmental sciences, biomedical engineering and physical sciences. Modeling relies increasingly on computation, so the students should have basic skills for using computers and programs like Matlab or Mathematica. Prereq: MATH 224 or MATH 228.

 

MATH 444. Mathematics of Data Mining and Pattern Recognition. (3)

This course will give an introduction to a class of mathematical and computational methods for the solution of data mining and pattern recognition problems. By understanding the mathematical concepts behind algorithms designed for mining data and identifying patterns, students will be able to modify to make them suitable for specific applications. Particular emphasis will be given to matrix factorization techniques. The course requirements will include the implementations of the methods in MATLAB and their application to practical problems. Prereq: MATH 201 or MATH 307.

 

MATH 445. Introduction to Partial Differential Equations (3)

Method of characteristics for linear and quasilinear equations. Second order equations of elliptic, parabolic, type; initial and boundary value problems. Method of separation of variables, eigenfunction expansions, Sturm-Liouville theory. Fourier, Laplace, Hankel transforms; Bessel functions, Legendre polynomials. Green’s functions. Examples include: heat diffusion, Laplace’s equation, wave equations, one dimensional gas dynamics and others. Appropriate for seniors and graduate students in science, engineering, and mathematics. Prereq: MATH 201 and MATH 224.

 

MATH 446. Numerical Methods for Partial Differential Equations (3)

This course is an introduction to numerical methods of PDEs, and in particular, to finite element methods (FEM), emphasizing the interconnection between the functional analytic viewpoint of PDEs and the practical and effective computation of the numerical approximations.  In particular, the emphasis is on showing that many of the useful and elegant ideas in finite dimensional linear algebra have a natural counterpart in the infinite dimensional setting of Hilbert spaces, and that the same techniques that guarantee the existence and uniqueness of the solutions in fact provide also stable computational methods to approximate the solutions. The topics covered in this course include Fourier analysis, weak derivatives, weak forms, generalized functions; Sobolev spaces, trace theorem, compact embedding theorems, Poincare inequalities; Riesz theory, Fredholm theory; Finite Element Method (FEM): Grid generation, existence, stability and convergence of solutions for elliptic problems; Semi-discretization of parabolic and hyperbolic equations; Stiffness; Numerical solution of linear systems by iterative methods. A quintessential part of this course comprises numerical implementation of the finite element method. Matlab is used as the programming tool both in demonstrations and examples in the class as well as in home assignments. Recommended Preparation: linear algebra, multivariate calculus, and ordinary differential equations.

 

MATH 449. Dynamical Models for Biology and Medicine. (3)

Introduction to discrete and continuous dynamical models with applications to biology and medicine. Topics include: population dynamics and ecology; models of infectious diseases; population genetics and evolution; biological motion (reaction-diffusion and chemotaxis); Molecular and cellular biology (biochemical kinetics, metabolic pathways, immunology). The course will introduce students to the basic mathematical concepts and techniques of dynamical systems theory (equilibria, stability, bifurcations, discrete and continuous dynamics, diffusion and wave propagation, elements of system theory and control). Mathematical exposition is supplemented with introduction to computer tools and techniques (Mathematica, Matlab). Prereq:MATH 224 or MATH 228, or BIOL/EBME 300, and MATH 201.

 

MATH 461. Introduction to Topology (3)

Metric spaces, topological spaces, and continuous functions. Compactness, connectedness, path connectedness. Topological manifolds; topological groups. Polyhedra, simplical complexes. Fundamental groups. Prereq: MATH 224.

 

MATH 462. Algebraic Topology (3)

The fundamental group and covering spaces; van Kampen’s theorem. Higher homotopy groups; long-exact sequence of a pair. Homology theory; chain complexes; short and long exact sequences; Mayer-Vietoris sequence. Homology of surfaces and complexes; applications. Prereq: MATH 461.

 

MATH 465. Differential Geometry (3)

Manifolds and differential geometry. Vector fields; Riemannian metrics; curvature; intrinsic and extrinsic geometry of surfaces and curves; structural equations of Riemannian geometry; the Gauss-Bonnet theorem. Prereq: MATH 321.

 

MATH 467. Differentiable Manifolds (3)

Differentiable manifolds and structures on manifolds. Tangent and cotangent bundle; vector fields; differential forms; tensor calculus; integration and Stokes’ theorem. May include Hamiltonian systems and their formulation on manifolds; symplectic structures; connections and curvature; foliations and integrability. Prereq: MATH 322.

 

MATH 471. Advanced Engineering Mathematics (3)

Vector analysis, Fourier series and integrals. Laplace transforms, separable partial differential equations, and boundary value problems. Bessel and Legendre functions. Emphasis on techniques and applications. Students may not take both MATH 345 and MATH 471 for credit. Prereq: MATH 224.

 

MATH 473. Introduction to fundamental mathematical techniques for image processing and computer vision (3)

Topics include but not limited to image denoising, image compression, image segmentation, image registration, and compressive sensing. Main tools are discrete Fourier/wavlet analysis, optimization and calculus of variation and partial differential equations. Accessible to upper level undergraduate and graduate students from mathematics, statistics, sciences, engineering and medicine. Prereq Math 330 or equivalent.

 

MATH 475. Mathematics of Imaging in Industry and Medicine (3)

The mathematics of image reconstruction; properties of radon transform, relation to Fourier transform; inversion methods, including convolution, backprojection, rho-filtered layergram, algebraic reconstruction technique (ART), and orthogonal polynomial expansions. Reconstruction from fan beam geometry limited angle techniques used in NMR; survey of applications. Prereq: PHYS 431 and MATH 345 or MATH 471.

 

MATH 478: Computational Neuroscience (3)

Computer simulations and mathematical analysis of neurons and neural circuits, and the computational properties of nervous systems. Students are taught a range of models for neurons and neural circuits, and are asked to implement and explore the computational and dynamic properties of these models. The course introduces students to dynamical systems theory for the analysis of neurons and neural circuits, as well as a cable theory, passive and active compartmental modeling, numerical integration methods, models of plasticity and learning, models of brain systems, and their relationship to artificial and neural networks. Term project required. Students enrolled in MATH 478 will make arrangements with the instructor to attend additional lectures and complete additional assignments addressing mathematical topics related to the course. Prereq: MATH 223 & MATH 224 or BIOL 300 & BIOL 306, or consent of department. Cross-listed as EBME 478, EECS 478, MATH 478, NEUR 478

 

MATH 482. High Dimensional Probability (3)

Behavior of random vectors, random matrices, and random projections in high dimensional spaces, with a view toward applications to data sciences.  Topics include tail inequalities for sums of independent random variables, norms of random matrices, concentration of measure, and bounds for random processes. Applications may include structure of random graphs, community detection, covariance estimation and clustering, randomized dimension reduction, empirical processes, statistical learning, and sparse recovery problems.  Additional work is required for graduate students. Offered as MATH 382, MATH 482, STAT 382 and STAT 482.

 

MATH 486. Quantum Computing, Information, and Devices (3)

An introduction to the math, physics, engineering, and computer science underlying the rapidly emerging fields of quantum computing, quantum information, and quantum devices. The course is taught by a group of faculty from physics, engineering, computer science, and math, and is geared towards students with diverse backgrounds and interests in these fields. Students will select a concentration in one of these four areas, and the coursework, while still covering all topics, will be adjusted to focus on the selected area in the most detail. Note that the listed prerequisites depend on choice of concentration. Topics will include: 1. (Mathematics) Introduction to linear algebra, convex geometry, fundamental theory of quantum information. 2. (Physics) Introduction to the quantum mechanics of two-level systems (qubits). Survey of physics and materials for qubit technologies. 3. (Computer Science) Basic quantum gates and circuits, introduction to the theory of algorithms, survey of quantum algorithms. 4. (Engineering) Quantum architectures, mapping algorithms onto circuits. The course consists of lectures, homework, and group projects. Group projects will aim to synthesize the diverse backgrounds of the students and instructors to capture the interdisciplinary nature of the field. Students taking the course for graduate credit will complete an additional literature research project and presentation, in addition to enhanced problem sets. Offered as CSDS 386, CSDS 486, ECSE 386, ECSE 486, MATH 386, MATH 486, PHYS 386, and PHYS 486. Prereq: (MATH 223 or MATH 227) and (MATH 224 or MATH 228) and (MATH 201 or MATH 307) and (PHYS 122 or PHYS 124) and PHYS 221.

 

MATH 491. Probability I (3)

Probabilistic concepts. Discrete probability, elementary distributions. Measure theoretic framework of probability theory. Probability spaces, sigma algebras, expectations, distributions. Independence. Classical results on almost sure convergence of sums of independent random variables. Kolmogorov’s law of large numbers. Recurrence of sums. Weak convergence of probability measures. Inversion, Levy’s continuity theorem. Central limit theorem. Introduction to the central limit problem. Prereq: MATH 423.

 

MATH 492. Probability II (3)

Conditional expectations. Discrete parameter martingales. Stopping times, optional stopping. Discrete parameter stationary processes and ergodic theory. Discrete time Markov processes. Introduction to continuous parameter stochastic processes. Kolmogorov’s consistency theorem. Gaussian processes. Brownian motion theory (sample path properties, strong Markov property, Martingales associated to Brownian motion, functional central limit theorem). Prereq: MATH 491.

 

MATH 494. Introduction to Information Theory (3)

This course is intended as an introduction to information and coding theory with emphasis on the mathematical aspects. It is suitable for advanced undergraduate and graduate students in mathematics, applied mathematics, statistics, physics, computer science and electrical engineering. Course content: Information measures-entropy, relative entropy, mutual information, and their properties. Typical sets and sequences, asymptotic equipartition property, data compression. Channel coding and capacity: channel coding theorem. Differential entropy, Gaussian channel, Shannon-Nyquist theorem. Information theory inequalities (400 level). Additional topics, which may include compressed sensing and elements of quantum information theory. Recommended preparation: MATH 201 or MATH 307. Offered as MATH 394, EECS 394, MATH 494 and EECS 494.

 

MATH 497.  Stochastic Models: Time Series and Markov Chains (3)

Introduction to stochastic modeling of data. Emphasis on models and statistical analysis of data with a significant temporal and/or spatial structure.  This course will analyze time and space dependent random phenomena from two perspectives:  Stationary Time Series: Spectral representation of deterministic signals, autocorrelation.  Power spectra.  Transmission of stationary signals through linear filters. Optimal filter design, signal-to-noise ratio.  Gaussian signals and correlation matrices. Spectral representation and computer simulation of stationary signals.  Discrete Markov Chains:  Transition matrices, recurrences and the first step analysis.  Steady rate.  Recurrence and ergodicity, empirical averages.  Long run behavior, convergence to steady state.  Time to absorption.  Eigenvalues and nonhomogeneous Markov chains.  Introduction to Gibbs fields and Markov Chain Monte Carlo (MCMC).  This course is related to STAT 538 but can be taken independently of it. Offered as: MATH 497 and STAT 437.

 

MATH 499. Special Topics (3)

Special topics in mathematics.

 

MATH 528. Analysis Seminar (1-3)

Continuing seminar on areas of current interest in analysis. Allows graduate and advanced undergraduate students to become involved in research. Topics will reflect interests and expertise of the faculty and may include functional analysis, convexity theory, and their applications. May be taken more than once for credit. Consent of department required.

 

MATH 535. Applied Mathematics Seminar (1-3)

Continuing seminar on areas of current interest in applied mathematics. Allows graduate and advanced undergraduate students to become involved in research. Topics will reflect interests and expertise of the faculty and may include topics in applied probability and stochastic processes, continuum mechanics, numerical analysis, mathematical physics or mathematical biology. May be taken more that once for credit.

 

MATH 549. Mathematical Life Sciences Seminar (1-3)

Continuing seminar on areas of current interest in the applications of mathematics to the life sciences. Allows graduate and advanced undergraduate students to become involved in research. Topics will reflect interests and expertise of the faculty and may include mathematical biology, computational neuroscience, mathematical modeling of biological systems, models of infectious diseases, computational cell biology, mathematical ecology and mathematical biomedicine broadly constructed. May be taken more than once for credit.

 

MATH 598. Stochastic Models: Diffusive Phenomena and Stochastic Differential Equations (3)

Introduction to stochastic modeling of data.  Emphasis on models and statistical analysis of data with significant temporal and/or spatial structure.  This course will analyze time and space dependent random phenomena from two perspectives: Brownian motion and diffusive processes: Classification of stochastic processes, finite dimensional distributions, random walks and their scaling limits, Brownian motion and its paths properties, general diffusive processes, Fokker-Planck-Kolmogorov equations, Poisson and point processes, heavy tail diffusions, Levy processes, tempered stable diffusions.  Stochastic calculus and stochastic differential equations: Wiener random integrals, mean-square theory, Brownian stochastic integrals and Ito formula, stochastic integrals for Levy processes, martingale property, basic theory and applications of stochastic differential equations.  This course is related to STAT 437 but can be taken independently of it. Offered as MATH 598 and STAT 538.

 

MATH 601. Reading and Research Problems (1-18)

Presentation of individual research, discussion, and investigation of research papers in a specialized field of mathematics.

 

MATH 651. Thesis (M.S.) (1-18)

 

MATH 701. Dissertation (Ph.D.) (1-18)

Prereq: Predoctoral research consent or advanced to Ph.D. candidacy milestone.