56:645:503-504Theory of Functions of a Complex Variable I,II (3,3) Analytic functions and the Cauchy Integral Theorem. Conformal mappings. Laplace transforms.
56:645:505-506Analysis I,II (3,3) Infinite series, especially Fourier series. Epsilon-delta proofs of continuity and differentiability. Convergence tests. Measure theory and integration.
56:645:507Contemporary Issues--Teaching beyond Regurgitation (3) Discusses traditional as well as contemporary approaches to teaching mathematics. Comparisons within these contexts are investigated. The intricate connections between geometry and algebra serve as a segue to a deeper analysis of calculus and linear and abstract algebra. Selected readings from NCTM publications are a course requirement.
56:645:508Mathematical Reasoning (3)This course develops two fundamental components of writing mathematics:
reasoning (thinking about the proof) and writing (formulating and
writing the ideas precisely using logical statements). The course begins with
illustrative examples and general guidelines.
56:645:511Computer Science (3) A survey of computer science, both theoretical and practical, for the pure mathematician. Topics could include time-complexity of algorithms, NP-completeness, Turing machines, factoring and primality testing, Strassen's matrix reduction algorithm, and the fast Fourier transform.
56:645:527-528Methods of Applied Mathematics I,II (3,3) Derivation of the heat and wave equations. Existence theorems for ordinary differential equations, series solutions. Bessel and Legendre equations. Sturm-Liouville Theorem. Pre- or corequisite: 56:645:549.
56:645:530Manifolds (3) Topological and differential manifolds. Surfaces. Fundamental groups and coverings. Differential forms and de Rham cohomology.
56:645:531Geometry (3) Review and reevaluation of Euclid's geometry. Axiomatic development of Euclidean and hyperbolic geometries. The parallel postulate. The impossibility of trisecting an angle or duplicating a cube.
56:645:532Differential Geometry (3) Curves and surfaces in Euclidean space. Riemannian manifolds, connections, and curvature.
56:645:533-534Introduction to the Theory of Computation I,II (3,3) 645:533: Introduction to formal languages, automata, and computability: regular languages and finite state automata; context-free grammars and languages; pushdown automata; the Church-Turing thesis; Turing machines; decidability and undecidability; Rice's theorem. 645:534: Second course in the sequence; addresses key topics in computability and complexity theory, such as recursive and recursively enumerable sets; the Recursion Theorem; Turing reductions and completeness; Kolmogorov complexity; space and time complexity; NP-completeness; hierarchy theorems; probabilistic complexity classes, and interactive proof systems.
56:645:535-536Algebra for Computer Scientists I,II (3,3) Linear and abstract algebra, including group theory, with applications to image processing, data compression, error correcting codes, and encryption.
56:645:537Computer Algorithms (3) Algorithm design techniques: divide-and-conquer, greedy method, dynamic programming, backtracking, and branch-and-bound. Advanced data structures, graph algorithms, and algebraic algorithms. Complexity analysis, complexity classes, and NP-completeness. Introduction to approximation algorithms and parallel algorithms.
56:645:538Combinatorial Optimization (3) Algorithmic techniques for solving optimization problems over discrete structures, including integer and linear programming, branch-and-bound, greedy algorithms, divide-and-conquer, dynamic programming, local optimization, simulated annealing, genetic algorithms, and approximation algorithms.
56:645:540Computational Number Theory and Cryptography (3) Primes and prime number theorems and numerical applications; the Chinese remainder theorem and its applications to computers and Hashing functions; factoring numbers; cryptography; computation aspects of the topics emphasized. Students required to do some simple programming.
56:645:541Introduction to Computational Geometry (3) Algorithms and data structures for geometric problems that arise in various applications, such as computer graphics, CAD/CAM, robotics, and geographical information systems (GIS). Topics include point location, range searching, intersection, decomposition of polygons, convex hulls, and Voronoi diagrams.
56:645:542Parallel Supercomputing (3) Fundamental issues in the design and development of programs for parallel supercomputers; programming models and performance optimization techniques; application examples and programming exercises on a contemporary parallel machine; cost models and performance analysis and evaluation.
56:645:545Topology (3) Point set topology, fundamental group and coverings. Singular homology and cohomology, the Brouwer degree and fixed-point theorems, the sphere retraction theorem, invariance of domains.
56:645:549-550Linear Algebra and Applications (3,3) Finite dimensional vector spaces, matrices, and linear operators. Eigenvalues, eigenvectors, diagonalizability, and Jordan canonical form. Applications.
56:645:551-552Abstract Algebra (3,3) Introductory topics in rings, modules, groups, fields, and Galois theory. Pre- or corequisite: 56:645:549.
56:645:554Applied Functional Analysis (3) Infinite dimensional vector spaces, especially Banach and Hilbert vector spaces. Orthogonal projections and the spectral decomposition theorem. Applications to differential equations and approximation methods.
56:645:555Glimpses of Mathematics (3) The intuitive beginnings and modern applications of key ideas of mathematics, such as polyhedra and the fundamental theorem of algebra. Extensive use of computer-generated films to help visualize the methods and results.
56:645:556Visualizing Mathematics by Computer (3) Introduction to symbolic computational packages and scientific visualization through examples from calculus and geometry. Covers 2-D, 3-D, and animated computer graphics using Maple, Mathematica, and Geomview. No programming knowledge required.
56:645:557Signal Processing (3) Signal modeling: periodic, stationary, and Gaussian signals. System representation: Volterra representation, state space representation, simulation. Themes in system design: least square estimation, system identification, adaptive signal processing. Representation of discrete causal signals: role of Fourier analysis, convolutions, fast Fourier transforms. Realization of linear recurrent structures: controllability, observability and minimal realization, frequency domain analysis of signals, and the role Laplace transforms. Stability analysis: Lyapunov and linearization methods. Prediction, filtering, and identification: linear prediction, the LQR problem, Kalman filter.
56:645:558Theory and Computation in Probability and Queuing Theory (3) Basic probability structures, probability distributions, random number generations and simulations, queuing models, analysis of single queue, queuing networks, applications of queuing theory.
56:645:560Industrial Mathematics (3) Monte Carlo methods, wavelets, data acquisition and manipulation, filters, frequency domain methods, fast Fourier transform, discrete Fourier transform.
56:645:561Optimization Theory (3) Linear programming: optimization, simplex algorithm, nonlinear programming, game theory.
56:645:562Mathematical Modeling (3) Perturbation methods, asymptotic analysis, conservation laws, dynamical system and chaos, oscillations, stability theory. Applications may include traffic flow, population dynamics, and combustion.
56:645:563Statistical Reasoning (3) Random variables, uniform, Gaussian, binomial, Poisson distributions, probability theory, stationary processes, central limit theorem, Markov chains, Taguchi quality control.
56:645:570Special Topics in Pure Mathematics (3) Topics vary from semester to semester. Prerequisite: Permission of instructor. Course may be taken more than once.
56:645:571-572Computational Mathematics I,II (3,3) Newton's method, curve and surface fitting. Numerical solutions of differential equations and linear systems, eigenvalues and eigenvectors. Fast Fourier transform.
56:645:574Control Theory and Optimization (3) Controllability, observability, and stabilization for linear and nonlinear systems. Kalman and Nyquist criteria. Frequency domain methods, Liapunov functions.
56:645:575Qualitative Theory of Ordinary Differential Equations (3) Cauchy-Picard existence and uniqueness theorem. Stability of linear and nonlinear systems. Applications to equations arising in biology and engineering.
56:645:578Mathematical Methods in Systems Biology I (3)
Introduction to
computational and system biology, focusing on advanced mathematical tools. In
particular, ordinary and partial differential equations, control theory and
discrete mathematics (networks) will be used to address a wide set of
biological and biomedical applications. The latter will range from classical
prey-predator populations examples to cancer immuno and drug therapies, from
evolutionary math to gene networks.
Prerequisite: 56:645:508.
56:645:580Special Topics in Applied Mathematics (3) Topics vary from semester to semester. Prerequisite: Permission of instructor. Course may be taken more than once.
56:645:698Independent Study in Pure Mathematics (3) Study of a particular subject independently but with frequent consultations with a faculty member.
56:645:699Independent Study in Applied Mathematics (3) Study of a particular subject independently but with frequent consultations with a faculty member.
56:645:700Thesis in Pure Mathematics (3) Expository paper written under the close guidance of a faculty member.
56:645:701Thesis in Applied Mathematics (3) Expository paper written under the close guidance of a faculty member.
56:645:800Matriculation Continued (0) Continuous registration may be accomplished by enrolling for at least 3 credits in standard course offerings, including research courses, or by enrolling in this course for 0 credits. Students actively engaged in study toward their degree who are using university facilities and faculty time are expected to enroll for the appropriate credits.
56:645:877Teaching Assistantship (E6)
56:645:581Mathematical Methods in Systems Biology II (3)The course will focus on mathematical methods, which are of particular relevance for biological systems. Building up on Mathematical Methods in System Biology I, the course will first develop the theory of ordinary differential equations, dealing with equilibrium analysis, phase portraits, and stability. Then links between properties of networks and systems of ODEs will be explored. Finally, network dynamics will be addressed.Prerequisite: 56:645:508.
56:645:579Celestial MechanicsIn this course we examine the orbits of planets, satellites, asteroids, and spacecraft.
The first part of the course deals with classical astrodynamics, as practiced by space
agencies before 1990. The second part of the
course will explore features of three-body systems that cannot be approximated by
two-body conic sections. Topics include Lagrange points, hyperbolic maps, and stable and
unstable manifolds. We will see how these concepts have in recent years been used to find spacecraft orbits which require less fuel than the classical piecewise conic section orbits.
