Current MSCS Graduate Courses ( Spring 2013, Fall 2013, Spring 2014 )
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:
- Textbook- Linear Analysis: An Introductory Course (Cambridge Mathematical Textbooks), by Bela Bollobas. The course will cover some of the most fundamental topics of linear functional analysis: Hahn-Banach extension theorem, Banach-Steinhauss boundedness principle, open mapping theorems, weak and weak^* topologies, Tihonov's compactness theorem, Alaoglu's theorem. We will also discuss Hilbert spaces, construction of orthonormal bases, general operator theory, spectral theory of compact operators, fixed point theorems. Homework assignments will be given on the bi-weekly basis, one midterm, and final.
MATH 548
Algebraic Topology II
(L. Kauffman)
- 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. We will use Hatcher's book and will start with a review of fundamental group and homology. In particular we will start with section 2.3 (on categories and functors) and the additional topics in Chapter 2. There may be some overlap with the first term. We will then do Chapters 3 and 4 with some additional topics. Time permitting, we will discuss exact couples and spectral sequences, the Serre spectral sequence and its applications to homotopy groups of spheres.
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 570
Topics in Teichmuller Theory and Geometric Structures
(D.Dumas)
- PREREQUISITES:
- Math 535 and Math 549, or consent of the instructor
- DESCRIPTION:
- Geometric structures on surfaces and their deformations: Riemann surfaces, Teichmuller space, hyperbolic structures, character varieties, complex projective structures, the Schwarzian derivative,grafting, twisting, earthquakes, connections to 3-dimensional hyperbolic geometry.
MATH 575
Integral Equations and Applications
(J. Bona)
- PREREQUISITES:
- Math 411 and Math 417 and Math 481; or consent of instructor.
- DESCRIPTION:
- This course will develop the elementary theory of Nonlinear Functional Analysis. After a review of important points from linear functional analysis, the elements of calculus in Banach spaces will be introduced, including Frechet Derivatives, higher-order derivatives, Taylor expansions, analysis of extrema and implicit-function theorems. A selection of fixed-point theorems will be derived and elementary degree theory introduced. The Euler-Lagrange equations will be derived and constrained-extrema and Lagrange multipliers studied. Lower- and upper- semicontinuity of nonlinear operators will be investigated with an eye toward establishing the existence of extrema. While knowledge of linear functional analysis will be helpful, the results needed from the linear theory will be introduced in context and written notes provided. Measure theory will not be needed.
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.
Fall 2013
Courses for this term have not been posted yet.
Spring 2014
Courses for this term have not been posted yet.