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Current MSCS Graduate Courses

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Fall 2008

MATH 506 Model Theory I (D. Marker)
PREREQUISITES:
Math 502 or Phil 562.
DESCRIPTION:
Introduction to stability theory; categoricity, stability, forking, finite equivalence relation theorem, indiscernibles, orthogonality.
MATH 515 Number Theory II (A. Cojocaru)
PREREQUISITES:
Math 514
DESCRIPTION:
Introduction to classical, algebraic, and analytic number theory. Algebraic number fields, units, ideals, and P-adic theory. Riemann Zeta-function, Dirichlet's theorem, prime number theorem.
MATH 516 Abstract Algebra I (D. Radford)
PREREQUISITES:
MATH 330 and 425.
DESCRIPTION:
Structure of groups, Sylow theorems, solvable groups, structure of rings, polynomial rings, projective and injective modules, finitely generated modules over a PID.
MATH 531 Advanced Topics in Algebra (M. Popa)
PREREQUISITES:
Math 516 and Math 517, or consent of instructor.
DESCRIPTION:
An introduction to commutative and homological algebra. Radical ideals and primary decomposition; Hilbert's Nullstellensatz; integral closure and normalization; flatness; regular sequences and depth; Cohen-Macaulay and regular rings; Exts and Tors; projective dimension; Serre's syzygy theorem.
MATH 533 Real Analysis I (A. Furman)
PREREQUISITES:
MATH 411 or 414 or the equivalent.
DESCRIPTION:
Introduction to real analysis. Lebesgue measure and integration, differentiation, L-p classes, abstract integration.
MATH 539 Functional Analysis I (S. Hurder)
PREREQUISITES:
Math 533.
DESCRIPTION:
Topological vector spaces, Hilbert spaces, Hahn-Banach theorem, open mapping, uniform boundedness principle, linear operators in a Banach space, compact operators.
MATH 549 Differentiable Manifolds I (M. Culler)
PREREQUISITES:
Math 445.
DESCRIPTION:
Smooth manifolds and maps, tangent and normal bundles, Sard's theorem and transversality, embedding, differential forms, Stokes' theorem, degree theory, vector fields.
MATH 552 Algebraic Geometry I (I. Coskun)
PREREQUISITES:
 
DESCRIPTION:
Basic commutative algebra, affine and projective varieties, regular and rational maps, function fields, dimension and smoothness, projective curves, schemes, sheaves, and cohomology, positive characteristic.
MATH 568 Topics in Algebraic Topology (B. Shipley)
PREREQUISITES:
Math 548 or consent of instructor.
DESCRIPTION:
Homotopy Theory: vector bundles, characteristic classes, fiber bundles, fibrations, stable homotopy groups, generalized cohomology theories, spectral sequences.
MATH 575 Integral Equations and Applications (C. Knessl)
PREREQUISITES:
Math 411 and Math 417 and Math 481; or consent of instructor.
DESCRIPTION:
Fredholm and Volterra equations, Fredholm determinants, separable and symmetric kernels, Neumann series, transform methods, Wiener-Hopf method, Cauchy kernels, nonlinear equations, perturbation methods.
MATH 580 Mathematics of Fluid Mechanics (A. Cheskidov)
PREREQUISITES:
Grade of C or better in Math 410, Math 417, and 481.
DESCRIPTION:
Development of concepts and techniques used in mathematical models of fluid motions, Euler and Navier Stokes equations, Vorticity and vortex motion, Waves and instabilities, Viscous fluids and boundary layers, asymptotic methods.
MATH 589 Teaching and Presentation of Mathematics (S. Hurder)
PREREQUISITES:
None.
DESCRIPTION:
No graduation credit awarded for students enrolled in the Master of Science in the Teaching of Mathematics degree program. Required for teaching assistants in MSCS. Strategies and techniques for effective teaching in college and for mathematical consulting. Observation and evaluation, classroom management, presenting mathematics in multidisciplinary research teams.
MATH 590 Advanced Topics in Applied Mathematics (S. Yau)
PREREQUISITES:
This course is important for students interested in financial mathematics. Prerequisites are Advanced Calculus and Elementary Probability.
DESCRIPTION:
This course covers stochastic process and filtering. Topics include Probability and random variables; Stochastic processes; Stochastic differential equations; Introduction to filtering theory; Nonlinear filtering theory; Linear filtering theory; Application to linear filtering theory; and Approximate nonlinear filters.

Spring 2009

MATH 512 Advanced Topics in Logic (A. Medvedev)
PREREQUISITES:
Abstract algebra is required. In model theory, a good understanding of Math 503 or Math 506 will be sufficient. In algebraic geometry, taking Math 552 in the fall will be a good idea. Dedication and brilliance may substitute for one of the last two.
DESCRIPTION:
Model Theory of Fields. We will try to start from scratch and get within sight of Hrushovski's Model-theoretic proof of the Mordell-Lang Conjecture in algebraic geometry. Along the way, we'll view algebraic geometry as model theory of algebraically closed fields, and we'll also develop the model theory of differentially closed fields, which are just like algebraically closed fields, but with a derivation. We'll need to build up some advanced model theory so that we can see the geometry through the logic. More details will appear on my web page during the summer.
MATH 517 Abstract Algebra II (A. Libgober)
PREREQUISITES:
MATH 516.
DESCRIPTION:
Rings and algebras, polynomials in several variables, power series rings, tensor products, field extensions, Galois theory, Wedderburn theorems.
MATH 518 Representation Theory (R. Takloo-Bighash)
PREREQUISITES:
MATH 517.
DESCRIPTION:
Major areas of representation theory, including structure of group algebras, Wedderburn theorems, characters and orthogonality relations, idempotents and blocks.
MATH 535 Complex Analysis I (L. DeMarco)
PREREQUISITES:
MATH 411 or 427.
DESCRIPTION:
Analytic functions as mappings. Cauchy theory. Power series. Partial fractions. Infinite products.
MATH 537 Introduction to Harmonic Analysis I (R. Shvydkoy)
PREREQUISITES:
Math 533, plus Math 417 or Math 535.
DESCRIPTION:
Text: J. Duoandikoetxea "Fourier Analysis", AMS, Grad Stud in Math, Vol 29. Excellent concise text on harmonic analysis with a good selection of exercises. Topics will include Fourier Transform, Hardy-Littlewood maximal function, singular integral operators and Hilbert transform, Littlewood-Paley theory, Sobolev and Besov spaces, BMO, and Hardy spaces, Bernstein inequalities, Carleson measures, applications to PDE. Evaluation of the course will consist of take-home problem sets every two weeks, written final or in-class presentation.
MATH 546 Advanced Topics in Analysis (A. Furman)
PREREQUISITES:
Approval of department.
DESCRIPTION:
Dynamics.
MATH 553 Algebraic Geometry II (H. Gillet)
PREREQUISITES:
Math 552.
DESCRIPTION:
Divisors and linear systems, differentials, Riemann-Roch theorem for curves, elliptic curves, geometry of curves and surfaces.
MATH 568 Topics in Algebraic Topology (L. Kauffman)
PREREQUISITES:
Math 548 or consent of instructor. Mathematical maturity, basic abstract algebra, some familiarity with topological ideas and ideas from graph theory. (We do not require algebraic topology, but this is useful background, and the course will be of interest to persons who are studying algebraic topology.)
DESCRIPTION:
State sums in Low Dimensional Topology. This course will study state sum invariants of knots and manifolds in dimensions three and four. A state sum is the analog of a partition function in statistical mechanics, transposed to a topological setting. This includes the Jones polynomial, its categorification by Khovanov and the recent work of Oszwath and Szabo on categorification of the Alexander polynomial.
MATH 569 Advanced Topics in Geometric and Differential Topology (P. Shalen)
PREREQUISITES:
Approval of the department.
DESCRIPTION:
The main topic will be the topology of 3-dimensional manifolds. Time permitting, the course will also cover some applications to hyperbolic geometry.
MATH 579 Singular Perturbations (J. Bona)
PREREQUISITES:
MATH 481 or consent of the instructor
DESCRIPTION:
Algebraic and transcendental equations, regular perturbation expansions of differential equations, matched asymptotic expansions, boundary layer theory, Poincare-Lindstedt, multiple scales, bifurcation theory, homogenization.
MATH 581 Special Topics in Fluid Mechanics (R. Abramov)
PREREQUISITES:
Grade of C or better in MATH 580
DESCRIPTION:
Geophysical flows for the atmosphere and ocean, barotropic flow, conserved quantities, chaotic dynamics, empirical statistical theories for geophysical flows, equilibrium statistical mechanics, mean response to small fluctuations.
MATH 586 Computational Finance (D. Nicholls)
PREREQUISITES:
Grade of C or better in MATH 220 and Stat 381; or consent of instructor.
DESCRIPTION:
Introduction to the mathematics of financial derivatives; options, asset price random walks, Black-Scholes model; partial differential techniques for option valuation, binomial models, numerical methods; exotic options, interest-rate derivatives.