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Specific topics include: Fourier series and trigonometric interpolation, well-posedness of PDEs, finite difference schemes for parabolic and hyperbolic equations, stability and convergence theory for difference methods. Riemann surfaces are widely used in mathematical physics finite-gap solutions of nonlinear equations, random matrices and orthogonal polynomials , fluid mechanics, elasticity, and electromagnetics. Then the course will concentrate on applications of the theory of Riemann surfaces. We will discuss periodic solutions to the Korteweg - de Vries equation, a generalized hodograph method for multiply connected domains and supercavitating flow, and Wiener-Hopf matrix factorization in fracture mechanics and diffraction of electromagnetic waves.

The course begins with an introduction to the theory and problems associated with Laplace's equation, the heat equation, and the wave equation. Then continues with a treatment of nonlinear first-order PDE including the Hamilton Jacobi equation and an introduction to conservation laws. The course finishes with an introduction to Sobolev spaces and the existence of weak solutions to second order elliptic equations. In this course, we shall discuss several important applications of Lie groups in number theory, quantum mechanics and invariant theory:. The purpose of this course is to introduce the theory of linear and integer programming.

The focus will be the mathematical theory rather than that actual numerical computation. Main topics of this course will include: linear inequalities, the structure of polyhedra, linear programming and its algorithms, well solved integer programs, other algorithms for integer programming cutting plane, branch-and-bound, etc. This course is a preparation course for the Core I examination in topology. It will cover general point set topology, the fundamental group, and covering spaces. A good supplementary reference is Chapter 1 of Algebraic Topology by Allen Hatcher, available online.

While MATH developed the theory of fundamental groups and MATH developed homology theories for topological spaces the focus of this course will be cohomology theory which is dual to homology theory. However, one of the advantages of developing cohomology theory of spaces is that they are naturally equipped with a ring structure. The course will be mainly based on Hatcher's book Algebraic topology , and additional material that we will hand out.

We will study differential manifolds with an emphasis on topological results and tools used in topology. We will study transversality, vector bundles, intersection numbers etc. We will study knot invariants arising from Lie algebras including the proof of the MMR conjecture stated by Melvin and Morton and elucidated further by Rozansky saying that the Alexander-Conway polynomial of a knot can be read from some of the coefficients of the Jones polynomials of cables of that knot i.

In many areas of mathematics, such as topology and representation theory, algebraic objects called chain complexes and their homology arise when studying invariants or obstructions.

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This class will study these objects in general, and see why the arise so frequently when studying a mathematical object with an abelian category attached to it. This class will start with an introduction to categories and abelian categories, and then develop the machinery of homological algebra. There will be an emphasis on examples, explicit computation and classroom interaction, and homework will be assigned. Most examples will be from algebra categories of modules or representations , with less emphasis on the examples from topology categories of sheaves , though the material can be tailored to the students.

We will study the basics of first-order logic, and then move into model theory. Topics include: the satisfaction of a first-order formula by a structure; Goedel's completeness theorem every sentence true in all structures is provable ; the compactness or "finiteness" theorem a set S of sentences possesses a model if and only if every finite subset of S does ; the Lowenheim-Skolem Theorem every infinite structure possesses a countable, "elementary" substructure; in particular, if the Zermelo-Fraenkel axioms for set theory are consistent, then they have a countable model, even though ZF proves the existence of uncountable sets ; saturated structures; ultraproducts; model complete theories; and complete theories.

This course provides a basic introduction to the qualitative theory of ordinary differential equations at the beginning graduate level. It will focus on topics from Chapters and 17 of the text. This will be a rigorous, proof-oriented theory course. Possible topics may include existence and uniqueness theory, Peano's theorem, applications of Gronwall's inequality, dependence of solutions on initial conditions, linear systems, stability of equilibrium points, Lyapunov functions, and phase plane analysis.

## Analytic Number Theory: An Introductory Course - P. T. Bateman, Harold G. Diamond - Google книги

The exact coverage will depend on the progress of the class. Functional analysis is the study of topological vector spaces and provides fundamental tools needed for research in pure and applied mathematics. Topics include: Normed vector space; Banach spaces and their generalizations; Baire category, Banach-Steinhaus, open mapping, closed graph, and Hahn-Banach theorems; duality in Banach spaces, weak topologies; Hilbert Spaces; and other topics such as commutative Banach algebras, spectral theory, and Gelfand theory.

Problems of the guiding of electromagnetic, acoustic, and elastic waves by material structures pervade mathematical physics, and their analysis penetrates a rich swath of classical and modern mathematics. This course will concentrate on a number of specific problems for which techniques in complex and functional analysis will be developed and applied. The spectral theory for self-adjoint operators in Hilbert space is realized in a concrete way in this problem. The continuous spectrum is associated with extended fields--those resulting from the scattering of a guided wave by the obstacle in the guide.

The eigenvalues are frequencies of trapped modes--those fields whose energy is exponentially trapped around the obstacle. Eigenvalues embedded in the continuous spectrum are unstable with respect to perturbations of the structure. This instability causes anomalous scattering behavior, which we will analyze by means of complex-analytic perturbation techniques. Exact solutions for waves in split waveguides can be obtained by sophisticated methods of complex variables called Wiener-Hopf techniques. They involve Tauberian and Abelian theorems of Fourier analysis.

The monodromy matrix, which transfers field data across one period, is the fundamental object of analysis. Through it, one develops the spectral theory for periodic media, known as the Floquet theory, which is the Fourier transform of the subgroup of translations in the real line. The salient phenomenon is the existence of stop-bands--frequency intervals in which the coherent scattering by the periodic structure prohibits the propagation of waves.

Localized defects admit trapped modes with frequency in a stop-band. High resonant amplification of fields makes the study of nonlinearities necessary. We will see how even weak nonlinearity has pronounced effects, such as bistability, in the presence of resonance. See the course webpage for a more in-depth description of the course. Instructor: Prof. Prerequisites: Math , Math or consent of the instructor. Text: Typed Lecture Notes will be provided The aim of the course is to learn the basic theory of stochastic differential equations.

The course starts with a study of Wiener processes and martingales. Stochastic integration with respect to a Wiener process will be developed with full details. Next, the course will focus on topics such as i connection between SDEs and partial differential equations, ii large deviations principle for diffusion processes, and iii invariant measures and ergodicity. Throughout the course, several examples will be given to illustrate the theory.

## Analytic Number Theory

Prerequisites: Math and Matroid Theory or permission of the department. Text: Matroid Applications edited by Neil White Chapter 6: The Tutte Polynomial and its Applications by Thomas Brylawski and James Oxley The theory of numerical invariants for matroids is one of many aspects of matroid theory having its origins within graph theory. Most of the fundamental ideas in matroid invariant theory were developed from graphs by Veblen, Birkhoff, Whitney, and Tutte when considering colorings and flows in graphs.

This course will introduce the Tutte polynomial for matroids and will consider its applications in graph theory, coding theory, percolation theory, electrical network theory, and statistical mechanics.

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Hatcher A fundamental problem in topology is that of determining whether or not two spaces are topologically equivalent. The basic idea of algebraic topology is to associate algebraic objects groups, rings, etc. These algebraic objects are invariants of the space, and provide a means for distinguishing between topological spaces. The focus of this course is one such class of invariants, namely the homology groups of a topological space.

Topics include simplicial, singular, and cellular homology, computations of homology groups and applications such as the Brouwer Fixed point theorem and generalizations of the Jordan curve theorem. Warner is also good. We will cover basic concepts such as: manifolds, submanifolds, tangent vactors, vector fields, vector bundles, transversality, Lie groups and other topics.

Farber Topological robotics, part of what one might call applied algebraic topology , is a relatively new area studying topological problems inspired by robotics and engineering, as well as problems of robotics requiring topological tools. Topics we will pursue in the course include configuration spaces of various types e. This last problem motivates a homotopy type invariant, the topological complexity of a space.

We will investigate this invariant for spaces including some of the configuration spaces mentioned previously.

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This course will be devoted to the structure theorems on automorphisms of 2-dimensional manifolds and the 3-dimensional manifolds, their inter-relationships and computation of invariants that distinguish between two such objects. Central example will be 3-manifolds that fiber over a circle e. To aid in the computation of invariants, various computational programs will be introduced: Twister for surface diffeomorphisms , Dror Bar-Natan's KnotTheory Mathematica package and Snappea.

The course will also introduce the use of William Stein's SAGE package as a common interface supporting coercion of output between packages, a notebook interface for saving computations and native graphical and group theoretical computations. The course will include both the presentation of the theoretical development of the ideas and the concomitant computation aspects of the various structure theorems. Student projects may be theoretical, computation or both. Graduate Course Outlines, Summer Spring Contact All inquiries about our graduate program are warmly welcomed and answered daily: grad math.

Prerequisite: Math Text: Online Test Bank.

Instructor: Dr. Welters, Prof. Lipton and Prof. Shipman Prerequisites: Audience: Graduate and undergraduate students; those interested in partial differential equations, spectral theory, and applications.