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Legesgue measure theory. Measurable sets and measurable functions. Legesgue integration, convergence theorems. Lp spaces. Decomposition and differentiation of measures. Convolutions. The Fourier transform. MATH 501 Real Analysis I (3) This course develops Lebesgue measure and integration theory. This is a centerpiece of modern analysis, providing a key tool in many areas of pure and applied mathematics. The course covers the following topics: Lebesgue measure theory, measurable sets and measurable functions, Lebesgue integration, convergence theorems, Lp spaces, decomposition and differentiation of measures, convolutions, the Fourier transform.
Complex numbers. Holomorphic functions. Cauchy's theorem. Meromorphic functions. Laurent expansions, residue calculus. Conformal maps, topology of the plane. MATH 502 Complex Analysis (3) This course is devoted to the analysis of differentiable functions of a complex variable. This is a central topic in pure mathematics, as well as a vital computational tool. The course covers the following topics: complex numbers, holomorphic functions, Cauchy's theorem, meromorphic functions, Laurent expansions, residue calculus, conformal maps, topology of the plane.
Banach spaces and Hilbert spaces. Dual spaces. Linear operators. Distributors, weak derivatives. Sovolev spaces. Applications to linear differential equations. MATH 503 Functional Analysis (3) This course develops the theory needed to treat linear integral and differential equations, within the framework of infinite-dimensional linear algebra. Applications to some classical equations are presented. The course covers the following topics: Banach and Hilbert spaces, dual spaces, linear operators, distributions, weak derivatives, Sobolev spaces, applications to linear differential equations.
MATH 515: Classical Mechanics and Variational Methods
Classical Mechanics and Variational Methods
Introduction to the calculus of variations, variational formulation of Lagrangian mechanics, symmetry in mechanical systems, Legendre transformation, Hamiltonian mechanics, completely integrable systems.
Approximation and interpolation, numerical quadrature, direct methods of numerical linear algebra, numerical solutions of nonlinear systems and optimization. MATH 523 Numerical Analysis I (3) 1. Approximation and interpolation. Weierstrass theorem, Bernstein polynomials, Jackson theorems, Lagrange interpolation, least squares approximation, orthogonal polynomials, piecewise Lagrange and Hermite interpolation, spline interpolation, the Fast Fourier Transform.2. Numerical quadrature. Newton-Cotes rules, Peano Kernel Theorem, Euler-Maclaurin expansion, Romberg integration, Gaussian quadrature, adaptive quadrature.3. Direct methods of numerical linear algebra. Gaussian elimination with pivoting, backward error analysis, conditioning of linear systems.4. Numerical solution of nonlinear systems and optimization. One-point iterations, Newton's and quasi-Newton's method, Broyden's method, unconstrained optimization, line-search methods.
Matrix decompositions. Direct method of numerical linear algebra. Eigenvalue computations. Iterative methods. MATH 524 Numerical Linear Algebra (3) This course provides a graduate level foundation in numerical linear algebra. It covers the mathematical theory behind numerical algorithms for the solution of linear systems of equations and eigenvalue problems. Specific topics include: matrix decompositions, direct methods of numerical linear algebra, eigenvalue computations, iterative methods.
This course provides an overview of the fundamental concepts of Geometric and Algebraic Topology and presents examples of calculations of principal topological invariants. It starts with review of general topology and covers the following topics: fundamental group, homology theories, index theory, CW complexes, and examples of calculations.
Smooth manifolds, smooth maps, Sard's theorem. The tangent bundle, vector fields, differential forms, integration on manifolds. Foliations. De Rham cohomology; simple applications. Lie groups, smooth actions, quotient spaces, examples. MATH 528 Differentiable Manifolds (3) This course covers the foundations of differential geometry, developing the theory of differentiation and integration on manifolds. It provides tools for the study of nonlinear problems, combining techniques in analysis and geometry. Concepts and tools from differential geometry have found wide use in different areas of mathematics, including nonlinear differential equations, control and optimization problems, and numerical analysis. The goal is to cover the most important techniques of differential geometry in a concise way. The course will appeal not only to students who plan to do research in geometry, but also to those interested in analysis, or applied and computational mathematics, as well. It covers the following topics: smooth manifolds, smooth maps, Sard’s theorem, the tangent bundle, vector fields, differential forms, integration on manifolds, foliations, de Rham cohomology, Lie groups, smooth actions, quotient spaces, examples.
Groups. Sylow's theorems. Rings. Ideals, unique factorization domains. Finitely generated modules. Fields. Algebraic and transcendental field extensions, Galois theory. MATH 536 Abstract Algebra (3) This course covers fundamental concepts, needed toward the study of advanced areas in abstract algebra. The course covers the following topics: groups, Sylow’s theorems, rings, ideals, unique factorization domains, finitely generated modules, fields, algebraic and transcendental field extensions, Galois theory.
Comparison of projections, traces, tensor products, ITPFI factors and crossed products, the Jones index, modular theory, free probability. MATH 584 Introduction to von Neumann Algebras (3) A concise introduction to von Neumann algebra theory, beginning with the basic definitions and proceeding through modular theory. The currently important subjects of index theory and free probability theory will be introduced.