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Course Criteria
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3.00 Credits
Staff. Prerequisite(s): Math 570/571. Discusses advanced topics in logic.
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3.00 Credits
Staff. Prerequisite(s): Math 312 or Math 412, or with the permission of the instructor. Advanced geometric methods used in geometric modeling,computer graphics, computer vision, and robotics.
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3.00 Credits
Staff. Generating functions, enumeration methods, Polya's theorem, combinatorial designs, discrete probability, extremal graphs, graph algorithms and spectral graph theory, combinatorial and computational geometry. Applied Linear Analysis. (M) Staff. Prerequisite(s): Math 241 and one semester of: Math 360/361 or Math 508/509. Application of techniques from linear algebra to real problems in economics, engineering, physics, etc. and the difficulties involved in their implementation. Particular emphasis is placed on solving equations, the eigenvalue problem for symmetric matrices and the metric geometry of spaces of matrices. Applications to problems such as options pricing, image reconstruction, airplane and ship design, oil prospecting, etc. (these topics will vary from year to year). Analysis of the numerical algorithms available to solve such problems, rates of convergence, accuracy and stability. Applied Linear Analysis. (M) Staff. Prerequisite(s): Math 680 or with the permission of the instructor. Continuation of Math 680.
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3.00 Credits
Staff. This course will discuss Mathematical Foundations of Computer Security in the Fall and will be followed in Spring by Math 691, Advanced Topics in Mathematical Foundations of Computer Security. In Fall we will study basic topics in cryptography and network security protocols. "What is to distinguish a digital dollar when it is as easily reproducible as the spoken word How do we converse privately when every syllable is bounced off a satellite and smeared over an entire continent How should a bank know that it really is Bill Gates requesting from his laptop in Fiji a transfer of $100,000,.....,000 to another bank Fortunately, the mathematics of cryptography can help. Cryptography provides techniques for keeping information secret, for determining that information has not been tampered with, and for determing who authored pieces of information." (From the Foreword by R. Rivest to the "Handbook of Applied Cryptography" by Menezes, van Oorschot, and Vanstone.) Textbook: Douglas R. Stinson. "Cryptography: Theory and Practice". Publisher: Chapman & Hall/CRC; 3 edition (November 1, 2005) ISBN: 1584885084.
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3.00 Credits
Staff. The course discusses advanced topics in mathematical foundations of semantics of programming languages and programming structures. Numerical Analysis. (M) Staff. Prerequisite(s): Math 320/321. A study of numerical methods for matrix problems, ordinary and partial differential equations, quadrature and the solution of algebraic or transcendental equations. Emphasis will be on the analysis of those methods which are particularly suited to automatic high-speed computation. Numerical Analysis. (M) Staff. Prerequisite(s): Math 692 or with the permission of the instructor. Continuation of Math 692. (PHYS654) Mathematical Foundations of Theoretical Physics. (M) Staff. Selected topics in mathematical physics, such as mathematical methods of classical mechanics, electrodynamics, relativity, quantum mechanics and quantum field theory.
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3.00 Credits
Staff. Selected topics in mathematical physics, such as mathematical methods of classical mechanics, electrodynamics, relativity, quantum mechanics and quantum field theory.
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3.00 Credits
Staff. Prerequisite(s): Math 694 or permission of the instructor. This interdisciplinary course discusses advanced topics in mathematical physics. Topics may include elliptic operators, heat kernels, complexes and the Atiyah-Singer index theorem, Feynman graphs and anomalies, computing Abelian and non-Abelian anomalies, and the relation of anomalies to the index theorem.
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3.00 Credits
Staff. Prerequisite(s): Math 696 or permission of the instructor. Continuation of Math 696. Topics may include the family index theorem, equivariant cohomology and loop spaces, the homological algebra of BRST invariance and the Wess-Zumino consistency condition, the descent equations, and worldsheet anomalies in string theory. Topics in Algebra. (M) Staff. Topics from the literature. The specific subjects will vary from year to year. Topics in Algebra. (M) Staff. Topics from the literature. The specific subjects will vary from year to year. Advanced Number Theory. (M) Staff. Prerequisite(s): Math 620/621. Ramification theory, adeles and ideles, Tate's thesis, group cohomology and Galois cohomology, class field theory in terms of ideles and cohomology, Lubin-Tate formal groups, Artin and Swan conductors, central simple algebras over local and global fields, general Hasse principles. Other topics may include the following: zero-dimensional Arakelov theory, Tate duality, introduction to arithmetic of elliptic curves, local and global epsilon factors in functional equations, p-adic L-functions and Iwasawa theory, modular forms and functions and modular curves. Advanced Number Theory. (M) Staff. Prerequisite(s): Math 720 or with the permission of the instructor. Continuation of Math 720. (MATH694) Topics in Algebraic Geometry. (M) Staff. Prerequisite(s): Either Math 622/623 or Math 624/625. Topics from the literature. The specific subjects will vary from year to year. Topics in Algebraic Geometry. (M) Staff. Prerequisite(s): Either Math 622/623 or Math 624/625. Topics from the literature. The specific subject will vary from year to year. (MATH540) Topics in Algebraic and Differential Topology. (M) Staff. Prerequisite(s): Math 618/619. Topics from the literature. The specific subjects will vary from year to year. Topics in Algebraic and Differential Topology. (M) Staff. Prerequisite(s): Math 618/619. Topics from the literature. The specific subjects will vary from year to year. Topics in Classical Analysis. (M) Staff. Prerequisite(s): Math 608 and Math 609 and permission from the instructor. Harmonic analysis in Euclidean space, Riemann surfaces, Discontinuous groups and harmonic analysis in hyperbolic space, Pseudodifferential operators and index theorems, Variational methods in non-linear PDE, Hyperbolic equations and conservation laws, Probability and stochastic processes, Geometric measure theory, Applications of analysis to problems in differential geometry. The specific subjects will vary from year to year. Topics in Classical Analysis. (M) Staff. Prerequisite(s): Math 748 or with the permission of the instructor. Continuation of Math 748. Topics in Functional Analysis. (M) Staff. Topics from the literature. The specific subjects will vary from year to year. Topics in Functional Analysis. (M) Staff. Topics from the literature. The specific subjects will vary from year to year. Topics in Operator theory. (M) Staff. Topics from the literature. The specific subjects will vary from year to year. Topics in Operator Theory. (M) Staff. Topics from the literature. The specific subjects will vary from year to year. Topics in Differential Geometry. (M) Staff. Prerequisite(s): Math 660/661. Topics from the literature. The specific subjects will vary from year to year. Topics in Differential Geometry. (M) Staff. Prerequisite(s): Math 660/661. Topics from the literature. The specific subjects will vary from year to year.
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3.00 Credits
Staff. Prerequisite(s): Math 694. Corequisite(s): Math 695. This course is designed to bring mathematicians with no physics background up to speed on the basic theories of physics: mechanics,relativity,quantum mechanics,classical fields,quantum filed theory, the standard model,strings,superstrings,and M-theory. Graduate Seminars
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3.00 Credits
Staff. Seminar on current and recent literature in algebra.
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