Factorization of continuous matrix functions on a multidimensional domain

Research Area: Functional analysis

Project description:

The purpose is to study the connections between topological K-theory and the behaviour of a matrix function A defined on a subset U of a Euclidean space of dimension at least two and takes values in a space of nxn matrices. Topological K-theory is a field that is traditionally established from a scope of algebraic topology. To each compact topological space, an abelian group is assigned using the notion of a vector bundle. A variation of this approach uses continuous matrix functions instead. By studying the topological properties of a domain U, the final goal is to distill information regarding how well a continuous matrix function defined on it preserves certain information.

Student responsibilities:

Throughout the duration of the project the student researcher will meet with the faculty mentor approximately twice a week. They will initially be tasked with studying bibliography and research papers relevant to the topic. In direct collaboration with the faculty mentor, they will explore connections between the tools developed in K-theory and factorizing continuous matrix functions of a high-dimensional domain. Step by step they will be guided towards asking the right questions and developing answers that may lead to gradual improvements. In the end of the project they will be asked to write a report or a short article.

Desired background/skills:

  • The student researcher needs to be familiar with the following subjects, in addition to being skilled in Latex.
  • Differential Calculus (Math 1300 and Math 1310)
  • Linear Algebra (Math 1021 and Math 2022)
  • Real Analysis (at least Math 2001 and desirably Math 3001)
  • Algebra (Math 3021) Vector Calculus (Math 2310 and desirably Math 3010)
  • Complex Variables (Math 3410)
  • Topology (Math 4081)