All inquiries about our graduate program are warmly welcomed and answered daily:
All inquiries about our graduate program are warmly welcomed and answered daily:
The course is an introduction to Banach spaces, distributions, and operator theory. The aim of the course is to prepare students for a thorough understanding and appreciation of analysis and its applications. The course will cover the Hahn-Banach theorem, the Baire category theorem and its consequences, Frechet spaces, tempered distributions, inductive limits, topologies on locally convex spaces, bounded operators, spectral theory, and topics on unbounded operators.
This is the first semester of the basic graduate algebra course. It will include an introduction to groups, rings, and modules, culminating in the structure theorem for finitely generated modules over a principal ideal domain, with applications to the structure theory of linear operators on a finite dimensional vector space.
The equation y2 = x3 + ax + b, is the equation of an elliptic curve. Elliptic curves have a special position amongst all curves, both from a geometric and arithmetic view point. They also have many applications, for instance in cyrptography, and were also essential in the proof of Fermat's last theorem.
Elliptic curves have a very rich structure, in particular, there is an abelian group law on the points of an elliptic curve. This means given two points, they can be "added" to produce a third. We will be interested in the case where a, b are rationals, and we will be interested in the set of (x,y) satisfying the above equation. We will show that the group of rational points in finitely generated, so, to give all points with rational coefficients, it is essentially enough to just give a finite list of points, which generate the others. We will discuss the structure of this group, including mentioning the important Birch and Swinnerton-Dyer conjecture, about the rank of this group (i.e., the group has the structure G x Zr, where G is a finite group, and r is the rank).
We will follow the book by Cassels very closely, probably covering one chapter every 2 to 3 lectures, since the chapters are quite short. We will be taking an algebraic view point, and only very briefly mention analytic viewpoints (such as modular forms), if at all. The applications such as to cryptography, will not be covered, but after attending this course, the student will be in a good position to understand such applications. This course will fill in many of the details in the more advanced modular forms currently being given.
Grades will be determined via approximately 5 homework assignments.
This course is an introduction to algebraic number fields, that is, finite dimensional extensions of the rational numbers. We will cover all the standard topics like the finiteness theorems for the class number and the units. We will not only cover these topics from the classical abstract viewpoint, but students will also be introduced to the MAGMA computer package with which actual computations in, and about, number fields can be done. Note that this is the first of a two semester course. In the second semester we will move on to more advanced topics, in particular class field theory.
This is a seminar course in which the participants will lecture to each other. You have heard of the famous Riemann Hypothesis, which is a statement about the zeroes of the Riemann zeta function. This seminar is NOT devoted to finding a proof but if you have a good idea please tell me before you tell anyone else. Over the past century people have found many statements that are logically equivalent to the Riemann Hypotheses but which sound very different from Riemann's formulation. The idea is that each student will read, struggle with, and report on two or three of these papers. It is possible for two students to work together. Some of these papers require specialized aspects of real or complex analysis, some advanced algebra, and some involve elementary-sounding properties of matrices! I cannot specify in advance what you will need to learn. You should expect to visit the library frequently, to look in books you haven't seen before and to learn to pick out something relevant from the middle of an advanced book, skimming the beginning for definitions and assumptions but not reading line-by-line.
This is the introductory course in real analysis. The core of the course is Part One of Royden's book. Topics include Lebesgue measure and integration on the real line, convergence theorems, functions of bounded variation and absolute continuity, differentiation, and the classical Banach spaces. We may also study abstract Banach spaces and Hilbert spaces and abstract measure and integration. This course continues as Math 7312 in the spring semester.
A standard first course in functional analysis. Topics include Banach spaces, Hilbert spaces, Banach algebras, topological vector spaces, spectral theory of operators and the study of the topology of the spaces of distributions.
This course does not require previous knowledge of measure theory. In the first week I will give a brief review of elementary probability theory (Math 3355). In the following two weeks I will describe the construction of probability spaces and random variables. The probabilistic aspects of measure theory are developed along the way. Here is a list of topics to be covered:
This course will focus on the rich interplay between analysis, algebra, and geometry for which Lie groups are especially suited. Being manifolds they contain all the tools necessary for doing both differential and integral calculus. They are also fundamental when dealing with symmetry and homogeneity. Topics -- manifolds, Lie groups, Lie algebras, homomorphisms, the exponential map, Lie subgroups, Lie algebra homomorphisms, differentials, Taylor's formula, Campbell-Baker-Hausdorff formula, compact homogeneous spaces, semi-simple Lie algebras, root structure, and spherical functions.
This course provides an in-depth view of practical algorithms for solving large-scale linear systems of equations arising in the numerical implementation of various problems in mathematics, engineering and other applications. The dimension of such systems can easily exceed a million equations. On the other hand, many such systems are sparse, in the sense that each equation involves only a small number of unknowns. The course will focus on iterative methods, i.e. methods providing sequences of approximated solutions converging to that of the original problem.
The course will cover the following material:
The course will focus on both the theoretical aspects and the numerical implementation of these methods. Evaluation will be based upon both theoretical and numerical projects.
Our objective will be to study the elements of stochastic differential equations through applications to problems arising from finance. Topics in finance will include the general Black-Scholes type formula, interest rate and credit products. On the stochastic side we will study the Ito formula and certain specific stochastic differential equations.
What is the essence of the similarity between forests in a graph and linearly independent sets of columns in a matrix? Why does the greedy algorithm produce a spanning tree of minimum weight in a connected graph? Can one test in polynomial time whether a matrix is totally unimodular? Matroid theory examines and answers questions like these.
The study of matroids is a branch of discrete mathematics with basic links to graphs, lattices, codes, transversals, and projective geometries. Matroids are of fundamental importance in combinatorial optimization and their applications extend into electrical engineering and statics.
This course will provide a comprehensive introduction to the basic theory of matroids. This will include discussions of the basic definitions and examples, duality theory, and certain fundamental substructures of matroids. Particular emphasis will be given to matroids that arise from graphs and from matrices.
Topology contains at least three (overlapping) subbranches: general (or point-set) topology, geometric topology and algebraic topology. General topology grew out of the successful attempt to generalize some basic ideas and theorems (e.g., continuity, open and closed sets, the Intermediate Value and Bolzano-Weierstrass Theorems) from Euclidean spaces to more general spaces. The core of this course will be a thorough introduction to the central ideas of general topology (Chapters 2 - 5 of Munkres). This material is fundamental in much of modern mathematics. Time permitting, we will also look briefly at the ideas of homotopy and the fundamental group (Chapter 9), subjects that belong to the algebraic-geometric side of topology, and will be covered in greater depth in 7512.
Geometric Topology: Geometric Structures in Infinite Groups
This course will survey application of the fundamental group in areas of current research interest in low dimensional topology within our department. The emphasis will be on geometric methods in the study of infinite groups with applications to theory of braids (and other arrangements), knots and links, three dimensional manifolds and the mapping class group of surfaces and related moduli spaces. Concepts studied will include aspects of infinite group theory such as the lower central series, word and conjugacy problems, and the Fox differential calculus. Other (related) topics include braid group representations and classical knot and link invariants (e.g., Alexander polynomials). Of particular interest will be the lower central and solvable series of a group and computations for semi-direct products.
The fundamental group rules in low dimensions. (Closed) two manifolds are classified by their genus and also by their fundamental group. In three manifold theory, Mostow's theorem declares that homotopy equivalent hyperbolic three manifolds are homeomorphic. Likewise, the fundamental group system of a knot complement determines the isotopy class of the knot (although not the fundamental group alone).
The applications chosen for this incarnation of the course will have little overlap (primarily braid theory) with the Fall 2003 course taught by Prof. Cohen although there will be a significant overlap in tools -- particularly the free differential calculus and the lower central series.
Finitely Presented Groups, Finite State Automata and Normal Forms
The word problem in group theory, deciding whether a word formed from a prescribed generating set represents the identity element, is the entryway for the appearance of formal language theory and finite state automata (theoretical computers) in low dimensional topology.
Free Calculus and Intersection Theory on Covering Spaces
In dimension two, the intersection theory of paths on covering spaces can be equivariantly computed using the free differential calculus and the associated subgroup of the fundamental group.
Combinatorial Structures on Surfaces
Returning to more combinatorial ideas, we introduce combinatorial maps and construct moduli spaces. If time permits, we will develop the connection between braid theory and the absolute galois group of the rational field.
The topics in this course will be different from the Fall 2003 version of the course, although, in spirit, the focus of this course will be very similar.
An expanded syllabus is available.
This course is a continuation of the fall 2004 Algebraic Number Theory course. If you have had a basic Algebraic Number Theory course at another time you are welcome to join the class. The course will start with an introduction to complete fields (the p-adics) and move on to other advanced topics including some class field theory. We will again do some computations using the MAGMA computer program.
The purpose of this course is to provide a thorough grounding in some of the basic tools of real analysis. The material is standard: sigma-algebras and measurable functions, measures, integration, limit theorems, Fubini's theorem, Lp-spaces, convexity, the Radon-Nikodym theorem, the elements of Hilbert, Banach, and Topological Vector Space theory.
This course is an introduction into the theory and applications of Ordinary Differential Equations (ODEs). The first topics to be covered are standard, and include existence and uniqueness of solutions, dependence on initial conditions, linear theory, stability theory, and aspects of dynamical system theory. We shall also incorporate applications in modern science and engineering through mathematical modeling and computer experiments. The final part of the course will be an introduction to dynamical systems and chaos.
Classical electrodynamics will be studied in two different mathematical contexts. First, we introduce the Maxwell equations with an understanding of how they arise from physical experiment. We will concentrate mostly on the linear equations and their behavior in periodic media (photonic crystals). The mathematical context for this is the spectral theory of unbounded differential operators and compact integral operators. Second, we will go into the fundamental physical principles axiomatically and understand the Maxwell-Lorentz space-time relation. The mathematical context for this is differential geometry.
This course presents the mathematical analysis and the numerical implementation of the Finite Element method for the numerical solution of Partial Differential Equations.
The syllabus includes some refreshers on the theory of interpolation in one dimension, on basic functional analysis and partial differential equations. Then, we will study the concepts of composite multidimensional interpolation, and its application to the finite element method. On then numerical side, I will present PETSc, the Portable and Extensible Toolkit for Scientific Computing, as well as some pre- and post-processing tools.
The evaluation is based on theoretical and numerical project works. As part of their project, the students are expected to develop a parallel solver for Poisson's equation on arbitrary domains, and to study its convergence.
The class will be held in parts in a traditional classroom, and in a computer lab.
This course provides a basic introduction to finite dimensional, continuous time, deterministic control systems at the graduate level. The course is intended for PhD students in applied mathematics and engineering graduate students with a solid background in graduate real analysis and nonlinear ordinary differential equations. It is designed to help students prepare for research at the interface of applied mathematics and control engineering. This will be a rigorous, proof-oriented systems theory course emphasizing controllability and stabilization. Select this link for additional course information.
Description: The objective of the course is to build the theory of stochastic integration and stochastic differential equations. A careful study of Brownian motion, martingales and Poisson random measures as stochastic integrators will the initial aim of the course. Properties of solutions of stochastic differential equations, and applications of the theory will also be studied.
This course will concentrate on the interplay between asymptotic analysis and the theory of distributions. We will cover the following topics: basic ideas in asymptotics, introduction to the theory of distributions, the moment expansion, Laplace's asymptotic formula, Fourier type integrals, expansions in several variables, applications in number theory, the average expansion of generalized functions, expansion of pseudo-differential operators, applications to Fourier series, applications in spectral analysis, applications to differential equations, series of delta functions.
In the last 25 years a theory has emerged that deals with the non spatial localization of the Fourier transform when the frequency is well localized. Wavelet theory provides a framework to construct an orthonormal basis that has localization properties both in space and frequency. It uses translation and dilation to zoom into a given part of the spatial variable. Wavelet theory lies in the intersection among harmonic analysis, signal processing, image processing, and scientific calculation.
This course will present an overview of wavelet theory and its applications. It will deal with the following topics: introduction to Fourier series and integrals; Gabor analysis, or the windowed Fourier transform; Hilbert spaces, orthonormal basis and Riesz basis; the continuous wavelet transform in one and several dimensions; multi-resolution approximations; Haar wavelets and Daubechies wavelets; construction of wavelets in one dimension; and wavelet sets.
To each topological space, we will associate a group, called the fundamental group of the space. We study the basic properties of the fundamental group. We will give such applications of this group as the Brouwer fixed point theorem, the fundamental theorem of calculus, and the Jordan curve theorem. We will learn to calculate this group, and relate it to the theory of covering spaces. Time permitting we will study surfaces and their classification.
To each topological space, we will associate a sequence of abelian groups, called the homology groups of a space. We will study the basic properties of the these groups. We will give such applications as the Brouwer fixed point theorem and a generalization of the Jordan curve theorem in all finite dimensions. We will learn to calculate this group. We will discuss how these groups can be used to study knots. Time permitting, we will discuss the cohomology of a space. This has the additional structure of a graded ring. The study of the homology of topological spaces is a good place to first encounter some homological algebra which now permeates much of modern mathematics.
Course contents include the following, with emphasis accorded as best suits the audience. Topological manifolds and smooth manifolds (basic definitions and examples), tangent and cotangent bundles, abstract vector bundles. The inverse function theorem and its corollaries; submanifolds, quotients of manifolds by group actions, embeddings of smooth manifolds in Euclidean space, approximation of continuous maps by smooth ones. Tensors, Riemanian metrics, differential forms, Stoke's theorem, deRham cohomology. Integral curves and flows of vector fields. Lie groups and Lie algebras.
There are three parts of the course. The first gives an introduction into standard semigroup theory with view to the solution of linear time-independent initial-value problems, in particular parabolic and hyperbolic linear partial differential equations (PDE) with time-independent coefficients. The second gives an introduction into abstract linear evolution equations of the `hyperbolic' type, which also covers the case of time-dependent coefficients. The final part introduces into the solution of non-linear (quasi-linear) evolution equations of the hyperbolic type, which in particular allow the treatment of quasi-linear PDE, thereby covering the majority of PDE relevant in physics and engineering. Lecture notes for the whole course will be online available during the class. For the first part also any standard text on semigroup can be used, like Pazy, A. 1983, Semigroups of linear operators and applications to partial differential equations, Springer, New York.
For the second and third part also Tosio Kato, Abstract evolution equations, linear and quasilinear, revisited, in: Komatsu. H. (ed.) 1993, Functional analysis and related topics,1991, Proceedings of the International conference in memory of professor Kosaku Yosida held at RIMS, Kyoto University, Japan, July 29-Aug. 2, 1991, Lecture Notes in Mathematics 1540, Springer, Berlin can be used.