Nonlinear Functional Analysis Degree Theory is widely used in the study of ODEs, PDEs and of functional equations to obtain information about the existence and nature of solutions.
Degree also provides many useful fixed point theorems.
In 1912 Brouwer defined a degree for continuous functions in finite dimensional spaces. In 1934 Leray and Schauder developed a degree in infinite dimensional spaces for compact pertubations of the identity mapping. Over the years, several definitions have been made for various types of functions. My research involves extending degree to a broader class of mappings and looking at various applications. For this purpose, the powerful computer programming language LOGO was developed by Papert and his team at the Artificial Intelligence Labs at MIT and is in use in many elementary and secondary schools throughout the world.