Overview

Course design, motivation, and objective.


Design

The project “Graphs, Lie Algebras and Differential Geometry” is written to meet the criteria of NREUP for summer’12, and is based on the previous year summer research. It is designed to increase the math knowledge of a group of six sophomore and junior students, and to give them basic research problems (expansion on the last year’s results) to work on for a period of six weeks.

Following our previous experience, we plan to have two phases of the project. The first phase (the first three weeks) is devoted to teaching the participants topics of Linear Algebra, and basics of Differential Geometry with emphasis on Lie algebras and Lie groups. This would provide the students with the necessary knowledge for understanding the problems and doing research on them. During the first phase, a gradual introduction to the problems of the project will be done as well. The second phase (the last three weeks) will be devoted to active research. The students we are planning to work with this year are at average better prepared for their task, due to the junior we accepted in the project. We will build upon their knowledge, and will teach them the subject in a more rigorous way.


Motivation

Symmetry is one of the most important notions in mathematics. In many situations, typically when studying Physics problems, the objects under consideration have many symmetries. An extremely important case, with far reaching applications, is when the symmetries can be described as a family parametrized by a smooth manifold, called Lie group. A common example is the motions in the Euclidean plane/space. Among the nice features of Lie groups is the fact that they can be studied, through their corresponding Lie algebras, by using linear algebra methods. Lie groups and Lie algebras are fundamental notions, and belong to the toolkit of any research mathematician and theoretical physicist.


Objective

This project’s goal is to teach a group of six selected sophomore and junior students with background in math and physics the basic notions of Lie algebras with emphasis on those which arise from simple graphs. The construction of these Lie algebras is elementary, using only the definition of graphs and wedge products of elements of a vector space. Due to fact that they provide a link between simple graphs on one hand and geometry of Lie groups on the other, these Lie algebras have recently enjoyed a strong interest among the mathematicians.