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Current MSCS Graduate Courses ( Spring 2012, Fall 2012, Spring 2013 )

Spring 2012

MATH 511 Descriptive Set Theory (C. Rosendal)
PREREQUISITES:
Department approval.
DESCRIPTION:
Advanced topics in logic.
MATH 517 Abstract Algebra II (I. Coskun)
PREREQUISITES:
MATH 516.
DESCRIPTION:
Rings and algebras, polynomials in several variables, power series rings, tensor products, field extensions, Galois theory, Wedderburn theorems.
MATH 531 Advanced Topics in Algebra (Whyte)
PREREQUISITES:
Department approval.
DESCRIPTION:
Lie Groups.
MATH 535 Complex Analysis I (M. Greenblatt)
PREREQUISITES:
MATH 411 or 427.
DESCRIPTION:
Analytic functions as mappings. Cauchy theory. Power series. Partial fractions. Infinite products.
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 553 Algebraic Geometry II ()
PREREQUISITES:
Math 552.
DESCRIPTION:
Divisors and linear systems, differentials, Riemann-Roch theorem for curves, elliptic curves, geometry of curves and surfaces.
MATH 569 Advanced Topics in Geometric and Differential Topology (D. Groves)
PREREQUISITES:
Math 547 or permission of the instructor.
DESCRIPTION:
Introduction to Geometric Group Theory. This course will consider some of the basic concepts of geometric group theory. Specifically, groups acting on metric spaces, and groups considered as metric spaces. The focus will be on coarse notions of (negative and nonpositive) curvature, particularly on groups and metric spaces which are hyperbolic in the sense of Gromov, though other topics will also be covered.
MATH 571 Topics in Algebraic Geometry (L. Ein)
PREREQUISITES:
Math 552 and 553 OR approval of the instructor.
DESCRIPTION:
We'll discuss various topics relating algebraic properties of the minimal resolution of the defining ideal of a projective subvariety of P^n and the geometric properties of the variety. In particular we will study the famous conjecture of Mark Green on the syzygies of a canonical curves and the theorem of Voisin on the syzygies of the generic canonical curves. We will also study recent results on syzygies by Eisenbud and Schreyer, and the recent theorem of Ein and Lazarsfeld on asymptotic syzygies.
MATH 577 Advanced Applied Partial Differential Equations (R. Shvydkoy)
PREREQUISITES:
Math 410 and 417 and 481.
DESCRIPTION:
Quasilinear and nonlinear first order PDE's, shock solutions, second order equations, cylinder and sphere problems, Wave, Laplace and diffusion equations, maximum principles, nonlinear wave motion.
MATH 582 Wave Propagation and Scattering I (J. Bona)
PREREQUISITES:
Math 417 and 481; or consent of the instructor.
DESCRIPTION:
Solutions of wave equations in multiple dimensions, vector, and dyadic waves; separable and nonseparable problems. Representations: Green's function integrals, complex integrals, spectral representations. Approximate solutions.
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.
MATH 590 Advanced Topics in Applied Mathematics (R. Abramov)
PREREQUISITES:
MATH 313, MATH 480 or approval of the department.
DESCRIPTION:
Introduction to ordinary differential equations, existence, uniqueness of solutions, dependence on parameters, autonomous and non-autonomous systems, linear systems, nonlinear systems, bifurcations, conservative systems.
MATH 591 Seminar on Mathematics Curricula (A. Castro Superfine)
PREREQUISITES:
Enrollment in the Doctor of Arts program in Mathematics or consent of the instructor.
DESCRIPTION:
Examination of research and reports on mathematics curricula. Analysis of research in teaching and learning mathematics. Developments in using technology in mathematics teaching.

Fall 2012

MATH 504 Set Theory I (D. Sinapova)
PREREQUISITES:
Math 430 or 502 or Phil 562.
DESCRIPTION:
Same as Phil 565. Naive and axiomatic set theory. Independence of the continuum hypothesis and the axiom of choice.
MATH 512 Advanced Topics in Logic (I. Goldbring)
PREREQUISITES:
DESCRIPTION:
MATH 514 Number Theory I (R. Takloo-Bighash)
PREREQUISITES:
None.
DESCRIPTION:
Introduction to classical, algebraic, and analytic number theory.
MATH 516 Abstract Algebra I (M. Popa)
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 520 Commutative and Homological Algebra (I. Coskun)
PREREQUISITES:
MATH 516 and MATH 517; or consent of the instructor.
DESCRIPTION:
Commutative rings; primary decomposition; integral closure; valuations; dimension theory; regular sequences; projective and injective dimension; chain complexes and homology; Ext and Tor; Koszul complex; homological study of regular rings.
MATH 533 Real Analysis I (S. Hurder)
PREREQUISITES:
MATH 411 or 414 or the equivalent.
DESCRIPTION:
Introduction to real analysis. Lebesgue measure and integration, differentiation, L-p classes, abstract integration.
MATH 547 Algebraic Topology I (M. Culler)
PREREQUISITES:
MATH 330 and 445.
DESCRIPTION:
The fundamental group and its applications,covering spaces, classification of compact surfaces, introduction to homology, development of singular homology theory, applications of homology.
MATH 550 Differentiable Manifolds II (A. Furman)
PREREQUISITES:
Math 549.
DESCRIPTION:
Vector bundles and classifying spaces, Lie groups and Lie algebras, tensors, Hodge theory, Poincare duality. Topics from elliptic operators, Morse theory, cobordism theory, de Rahm theory, characteristic classes.
MATH 552 Algebraic Geometry I (A. Prendergast-Smith)
PREREQUISITES:
None.
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 571 Techniques and Examples in Algebraic Geometry (I. Coskun)
PREREQUISITES:
Approval of the department. A year of algebraic geometry at the level of Hartshorne or Griffiths-Harris.
DESCRIPTION:
In this course, I will introduce techniques and examples that every working algebraic geometer should be familiar with. I will start with a unit on intersection theory with applications to homogeneous varieties. Then I will do some basics of the theory of curves and surfaces. Next, I will discuss some explicit birational geometry. Finally, I will end the course with an introduction to Bridgeland stability and the birational geometry of the Hilbert scheme of points on surfaces.
MATH 579 Singular Perturbations (C. Knessl)
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 583 Topics in Wave Propagation (C. Sparber)
PREREQUISITES:
Math 480 and Math 481; consent of instructor.
DESCRIPTION:
Rigorous, asymptotic, and numerical analysis of mathematical models for linear and nonlinear waves. Techniques include inverse scattering, asymptotic analysis, and finite-difference and spectral methods.
MATH 584 Applied Stochastic Models (I. Nenciu)
PREREQUISITES:
Stat 401 and MATH 417 and 481, or consent of the instructor.
DESCRIPTION:
Applications of stochastic models in chemistry, physics, biology, queueing, filtering, and stochastic control, diffusion approximations, Brownian motion, stochastic calculus, stochastically perturbed dynamical systems, first passage times.
MATH 589 Teaching and Presentation of Mathematics (B. Shipley)
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.

Spring 2013

MATH 512 Advanced Topics in Logic (D. Sinapova)
PREREQUISITES:
Approval of department.
DESCRIPTION:
Advanced topics in modern logic; e.g. large cardinals, infinitary logic, model theory of fields, o-minimality, Borel equivalence relations. Same as PHIL 569. May be repeated. Students may register in more than one section per term.
MATH 517 Abstract Algebra II (M. Popa)
PREREQUISITES:
MATH 516.
DESCRIPTION:
Rings and algebras, polynomials in several variables, power series rings, tensor products, field extensions, Galois theory, Wedderburn theorems.
MATH 535 Complex Analysis I (L. DeMarco)
PREREQUISITES:
MATH 411 required; MATH 445 recommended.
DESCRIPTION:
Analytic functions as mappings. Cauchy theory. Power series. Partial fractions. Infinite products.
MATH 537 Introduction to Harmonic Analysis I (M. Greenblatt)
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 539 Functional Analysis I (R. Shvydkoy)
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 548 Algebraic Topology II (B. Shipley)
PREREQUISITES:
MATH 547.
DESCRIPTION:
Cohomology theory, universal coefficient theorems, cohomology products and their applications, orientation and duality for manifolds, homotopy groups and fibrations, the Hurewicz theorem, selected topics.
MATH 549 Differentiable Manifolds I (K. Whyte)
PREREQUISITES:
Math 445; and Math 310 or Math 320 or the equivalent.
DESCRIPTION:
Smooth manifolds and maps, tangent and normal bundles, Sard's theorem and transversality, embedding, differential forms, Stokes' theorem, degree theory, vector fields.
MATH 553 Algebraic Geometry II (L. Ein)
PREREQUISITES:
Math 552.
DESCRIPTION:
Divisors and linear systems, differentials, Riemann-Roch theorem for curves, elliptic curves, geometry of curves and surfaces.
MATH 555 Complex Manifolds II (L. Ein)
PREREQUISITES:
MATH 517 and 535.
DESCRIPTION:
Dolbeault Cohomology, Serre duality, Hodge theory, Kodaira vanishing and embedding theorem, Lefschetz theorem, Complex Tori, Kahler manifolds.
MATH 570 Topics in Teichmuller Theory and Geometric Structures (D.Dumas)
PREREQUISITES:
Approval of the department.
DESCRIPTION:
MATH 575 Integral Equations and Applications (J. Bona)
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 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.