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Course Criteria
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3.00 Credits
Staff. Prerequisite(s): Math 548 or with the permission of the instructor. Continuation of Math 548.
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3.00 Credits
Staff. Prerequisite(s): Math 371 or Math 503. Propositional logic: semantics, formal deductions, resolution method. First order logic: validity, models, formal deductions; Godel's completeness theorem, Lowenheim-Skolem theorem: cut-elimination, Herbrand's theorem, resolution method.Computability: finite automata, Turing machines, Godel's incompleteness theorems. Algorithmically unsolvable problems in mathematics.
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3.00 Credits
Staff. Prerequisite(s): Math 570 or with the permission of the instructor. Continuation of Math 570.
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3.00 Credits
Staff. Topics will include: the axioms, ordinal and cardinal arithmetic, formal construction of natural numbers and real numbers within set theory, formal treatment of definition by recursion. Mathematical Theory of Computation. (M) Staff. Prerequisite(s): Math 320/321. This course will discuss advanced topics in Mathematical Theory of Computation. Mathematical Theory of Computation. (M) Staff. Prerequisite(s): Math 574 or with the permission of the instructor. Continuation of Math 574. Combinatorial Analysis and Graph Theory. (M) Staff. Prerequisite(s): Permission of the instructor. Generating functions, enumeration methods, Polya's theorem, combinatorial designs, discrete probability, extremal graphs, graph algorithms and spectral graph theory, combinatorial and computational geometry. Combinatorial Analysis and Graph Theory. (M) Staff. Prerequisite(s): Math 580 or with the permission of the instructor. Continuation of Math 580. Applied Mathematics and Computation. (M) Staff. Prerequisite(s): Math 240-241. Math 312, Math 360. Knowledge of Math 412 and Math 508 is recommended. This course offers first-hand experience of coupling mathematics with computing and applications. Topics include: Random walks, randomized algorithms, information theory, coding theory, cryptography, combinatorial optimization, linear programming, permutation networks and parallel computing. Lectures will be supplemented by informal talks by guest speakers from industry about examples and their experience of using mathematics in the real world. Applied Mathematics and Computation. (M) Staff. Prerequisite(s): Math 582 or with the permission of the instructor. Continuation of Math 582.
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3.00 Credits
Staff. Prerequisite(s): Math 241, knowledge of linear algebra and basic physics. In the last 25 years there has been a revolution in image reconstruction techniques in fields from astrophysics to electron microscopy and most notably in medical imaging. In each of these fields one would like to have a precise picture of a 2 or 3 dimensional object which cannot be obtained directly.The data which is accesible is typically some collection of averages. The problem of image reconstruction is to build an object out of the averaged data and then estimate how close the reconstruction is to the actual object. In this course we introduce the mathematical techniques used to model measurements and reconstruct images. As a simple representative case we study transmission X-ray tomography (CT).In this context we cover the basic principles of mathematical analysis, the Fourier transform, interpolation and approximation of functions, sampling theory, digital filtering and noise analysis.
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3.00 Credits
Staff. Prerequisite(s): Math 584 or with the permission of the instructor. Continuation of Math 584. Advanced Applied Mathematics. (M) Staff. Prerequisite(s): Math 241. This course offers first-hand experience of coupling mathematics with applications. Topics will vary from year to year. Among them are: Random walks and Markov chains, permutation networks and routing, graph expanders and randomized algorithms, communication and computational complexity, applied number theory and cryptography. Advanced Applied Mathematics. (M) Staff. Prerequisite(s): Math 590 or with the permission of the instructor. Continuation of Math 590.
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3.00 Credits
Staff. Prerequisite(s): Math 241 or Permission of Instructor. Physics 151 would be helpful for undergraduates. Introduction to mathematics used in physics and engineering, with the goal of developing facility in classical techniques. Vector spaces, linear algebra, computation of eigenvalues and eigenvectors, boundary value problems, spectral theory of second order equations, asymptotic expansions, partial differential equations, differential operators and Green's functions, orthogonal functions, generating functions, contour integration, Fourier and Laplace transforms and an introduction to representation theory of SU(2) and SO(3). The course will draw on examples in continuum mechanics, electrostatics and transport problems.
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3.00 Credits
Topology and Geometric Analysis. (A) Staff. Prerequisite(s): Math 500/501 or with the permission of the instructor. Differentiable functions, inverse and implicit function theorems. Theory of manifolds: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields and differential forms: Frobenius' theorem, integration on manifolds, Stokes' theorem in n dimensions, de Rham cohomology. Introduction to Lie groups and Lie group actions. Topology and Geometric Analysis. (B) Staff. Prerequisite(s): Math 600 or with the permission of the instructor. Covering spaces and fundamental groups, van Kampen's theorem and classification of surfaces. Basics of homology and cohomology, singular and cellular; isomorphism with de Rham cohomology. Brouwer fixed point theorem, CW complexes, cup and cap products, Poincare duality, Kunneth and universal coefficient theorems, Alexander duality, Lefschetz fixed point theorem. Algebra. (A) Staff. Prerequisite(s): Math 370/371 or Math 502/503. Group theory: permutation groups, symmetry groups, linear algebraic groups, Jordan-Holder and Sylow theorems, finite abelian groups, solvable and nilpotent groups, p-groups, group extensions. Ring theory: Prime and maximal ideals, localization, Hilbert basis theorem, integral extensions, Dedekind domains, primary decomposition, rings associated to affine varieties, semisimple rings, Wedderburn's theorem, elementary representation theory. Linear algebra: Diagonalization and canonical form of matrices, elementary representation theory, bilinear forms, quotient spaces, dual spaces, tensor products, exact sequences, exterior and symmetric algebras. Module theory: Tensor products, flat and projective modules, introduction to homological algebra, Nakayama's Lemma. Field theory: separable and normal extensions, cyclic extensions, fundamental theorem of Galois theory, solvability of equations. Algebra. (B) Staff. Prerequisite(s): Math 602 or with the permission of the instructor. Continuation of Math 602. First Year Seminar in Mathematics. (A) Staff. Prerequisite(s): Open to first year Mathematics graduate students. Others need permission of the instructor. This is a seminar for first year Mathematics graduate student, supervised by faculty. Students give talks on topics from all areas of mathematics at a level appropriate for first year graduate students. Attendance and preparation will be expected by all participants, and learning how to present mathematics effectively is an important part of the seminar. First Year Seminar in Mathematics. (B) Staff. Prerequisite(s): Open to first year Mathematics graduate students. Ohters need permission of the instructor. Continuation of Math 604.
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3.00 Credits
Staff. Possible topics: harmonic analysis on locally compact abelian groups; almost periodic functions; direct integral decomposition theory, Types I, II and III: induced representations, representation theory of semisimple groups. Differential Geometry. (M) Staff. Prerequisite(s): Math 600/601, Math 602/603. Riemannian metrics and connections, geodesics, completeness, Hopf-Rinow theorem, sectional curvature, Ricci curvature, scalar curvature, Jacobi fields, second fundamental form and Gauss equations, manifolds of constant curvature, first and second variation formulas, Bonnet-Myers theorem, comparison theorems, Morse index theorem, Hadamard theorem, Preissmann theorem, and further topics such as sphere theorems, critical points of distance functions, the soul theorem, Gromov-Hausdorff convergence. Differential Geometry. (M) Staff. Prerequisite(s): Math 660 or with the permission of the instructor. Continuation of Math 660.
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3.00 Credits
Staff. Prerequisite(s): Math 570/571. Discusses advanced topics in logic.
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