Course Descriptions: Spring 2002
This is primarily a course in field theory, including Galois theory. If time permits we will discuss transcendental extensions as well. Prerequisites: Basic knowledge of groups and groups acting on sets. The course will be mostly self-contained. Students with questions concerning this course should feel free to contact Dr. Perlis.
This is a basic introduction to algebraic geometry. The minimal requirement is facility with the concepts of abstract algebra (group, ring, ideal, field, module), which should be covered in Math 7200. Algebraic geometry is a subject that can be taught from many points of view. This time we will emphasize the computational aspect. This means that we will do an introduction to Groebner bases. Also, we will be using some software, principally Macaulay 2 and Singular. The main topics are varieties in affine and projective spaces, and their relation to rings and ideals. Morphisms and rational maps will be discussed. We will look at a lot of explicit examples. Depending on the background and interests of the students, more advanced topics, such as an introduction to schemes could be given.
This course will be concerned with those portions of linear algebra which are of particular interest in applications. Some topics to be covered include matrix norms, singular value decomposition, QR factorization, least squares problems, stability of algorithms such as least squares and Gaussian Elimination, eigenvalue computations, iterative algorithms for eigenvalues and for linear equations.
Algebraic Number Theory is a most classical topic. It is heavily researched.Gauss called it the Queen of Mathematics.
If one wants to answer questions about the rational integers, one is often naturally led to consider algebraic number fields and their integers; compare e.g. Fermat's Last Theorem.
We will teach a standard course and cover the three basic results: Rings of integers of number fields are Dedekind rings; Finiteness of class number; The Dirichlet Unit Theorem.
The course will focus on measure theory and integration. The goal is to learn some must-know machinery for real analysis. Topics will include: abstract measure and integration, convergence theorems for integration, Fubini's theorem, the Radon-Nikodym theorem, construction of Lebesgue measure, Riesz representation theorem, and duals of certain function spaces. Other topics may include complex measures, Hilbert and Banach spaces, and Fourier transforms.
This is a Core II course, and is an introduction to the theory and applications of ordinary differential equations. The topics 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 attempt to incorporate some applications in modern science and engineering through mathematical modeling and computer experiments. Finally, if time permits, we will give an introduction to optimal control theory.
This is an introductory course on Lie Groups and representation theory. Though we will discuss Lie groups in general, our main discussion will focus on linear Lie groups. This will give many of the most interesting examples and the concepts in this setting are more easily understood. Central examples include the linear groups SO(n) and SL(n,R). We develop the connection between the natural representations given by the actions of SO(n) on S(n-1) and Rn and spherical harmonics. Topics shall include:
Many processes in physics, the life sciences and technology that evolve in time are modeled by initial and boundary value problems for partial differential or integro-differential equations. The resulting mathematical theory of Evolution Equations involves tools from many fields of mathematical analysis, notably from functional analysis, operator theory and partial differential equations. This course is a self-contained introduction to the subject. It covers background materials like duality theory, the Bochner integral, vector-valued analytic functions, distributions and Fourier multipliers, vector-valued Laplace transforms, Tauberian theorems, asymptotic analysis, and spectral theory for closed operators on Banach spaces. The theory of Cauchy problems and operator semigroups will be developed completely in the spirit of Laplace transforms. Existence and uniqueness, regularity, numerical approximation, and asymptotic behavior of solutions will be covered and diverse applications to concrete problems will be given.
This course provides an introduction to the theory of composite materials with an emphasis on the mathematical methods necessary for describing the macroscopic behaviour of complex systems. The basic theory and relevant mathematics are presented in a self contained way and references to the research literature will be provided.
The course introduces the method of two scale expansions as a way to recover macroscopic transport equations from the microscopic description of a composite media. The notion of effective transport properties are introduced. Bounding techniques for characterizing extreme macroscopic behaviour are developed; these include the Hashin-Shtrikman variational principles and the translation method. Self consistent schemes for the estimation of effective properties of random media are presented; these include the effective medium approximation and the differential effective medium scheme. Examples where group theory and statistics are required for the characterization of macroscopic transport properties will be given.
Coverage:
Broadly speaking, topology is the study of space and continuity. Since topology includes the study of continuous deformations of a space, it is sometimes called rubber sheet geometry. To distinguish various spaces, notions from algebra are often introduced.
In Topology II, the student is introduced to some of the basic notions in the description of topology from the first paragraph: the most basic notion of deformation, homotopy and the most basic algebraic invariant to distinguish various spaces, the fundamental group. The central part of the course is the theory of the fundamental group and covering spaces. The theory of covering spaces attempts to clarify the geometric role of the fundamental group in a beautiful way that is analogous to Galois theory in the study of fields.
Topology II is useful in any field that uses a global geometric viewpoint, such as Lie groups, harmonic analysis, differential geometry, algebraic geometry, complex analysis, and, yes, topology.
The course title is Differential Geometry and Topology, but the content is a basic introduction to manifolds. When one studies analysis on Rn one is studying local phenomena; when one studies analysis on manifolds one is studying global phenomena. This subject is called global analysis. It is particularly useful for differential topology, differential equations, differential geometry, harmonic analysis, and mathematical physics. The prerequisites are advanced calculus, point set topology, basic group theory and linear algebra. The topics we plan to cover are
Associated to a chain complex over a field F with trivial homology, one associates an element ( called the torsion of the complex) which lies in the group of units of the field. If the chain complex is defined over a ring R, one may also define the torsion in a group denoted K_1(R). When one considers the chain complexes associated to acyclic topological spaces, one obtains topological invariants. We will use the free differential calculus to get a handle on chain complexes over interesting covering spaces. We will discuss the Alexander modules of knots from this point of view. We will also study the classification of lens spaces.
