Math 497A - Honors MASS Algebra

Hypercomplex numbers

Instructor:  Svetlana Katok, Professor of Mathematics
Teaching Assistant:

113 McAllister Building, MWRF 11:15 am - 12:05 pm

Description:  In this course we will study arithmetic, geometry and symmetry of hypercomplex numbers. The hypercomplex numbers are constructed by adding “imaginary units” to the real numbers. The complex numbers are a classical example of such a number system in dimension 2. It is easy to define addition, subtraction, and multiplication in each system of hypercomplex numbers, but the only dimensions in which there are hypercomplex numbers which allow division are 4 and 8. These hypercomplex numbers are called the Quaternions and the Cayley numbers (Octonions), respectively. With increases in dimension, some of the natural properties of the number systems cannot be maintained. In both the Quaternions and the Cayley numbers, multiplication is non-commutative, and in Cayley numbers multiplication is not even associative.

Reading:  I.L. Cantor and A.S. Solodovnikov. “Hypercomplex numbers. An elementary introduction to algebras”. Springer-Verlag, New York,, 1989.

Math 497B - Honors MASS Analysis

Approximation of functions and applications

Instructor:  Ludmil Zikatanov, Professor of Mathematics
Teaching Assistant:

113 McAllister Building, 10:10 am - 11:00 am

Description:  This is an introductory course on approximation of functions, a topic that has generated and still generates rich mathematical theory and is one of the driving forces in the modern scientific computing. Roughly, the problem of approximating an object (function or shape) consists of three main steps: (1) identify the possible approximants; (2) construct an approximation; (3) estimate the error. Two scenarios are often encountered in applications: first, when the function is explicitly known, and, second, when the function is a solution to another problem and cannot be found in explicit form. Classical examples for these scenarios include computing a best fit by a polynomial curve to given data points, or computing approximate solution to a differential equation modeling physical phenomena. The course covers classical results on approximation and interpolation with algebraic and trigonometric polynomials, numerical integration, and applications. The pool of applications touches upon: signal processing (the Fast Fourier Transform); iterative solution of linear systems (the method of Conjugate Gradients); numerical solution of models described by ordinary differential equations.

Reading:  Most of the material can be found in the book T. Rivlin. "An introduction to the approximation of functions". Dover Publications, Inc., New York, 1981 and in lecture notes provided by the instructor during the course. Materials on applications are found in E. Isaacson and H. Keller. "Analysis of numerical methods". Dover Publications Inc., New York, 1994.

Math 497C - Honors MASS Geometry

An introduction to dynamics from a topological geometric viewpoint

Instructor:  Federico Rodriguez Hertz, Professor of Mathematics
Teaching Assistant:

113 McAllister Building, 1:25 pm - 2:15 pm

Description:  We shall develop the basic theory of dynamics with an application to the study of homeomorphisms of surfaces. In the meantime, the basic properties of the hyperbolic plane and homotopy properties of surfaces will be discussed and applied.

Reading:  A. Casson and S. Bleiler. "Automorphisms of Surfaces after Nielsen and Thurston". Cambridge University Press, Cambridge, 1988.