Project Overview
The purpose of the project to study the intertwining of the analytic and algebraic properties of matrices and geometric objects associated to them. The objective is to formulate specific quantitative conditions and compute theoretical margins of error; if a matrix A satisfies these conditions then it must also satisfy a desired conclusion, within the predicted error. One such problem under consideration will be given an N-dimension matrix A to predict how well it can preserve n-dimensional information by studying its diagonal entries. A second problem concerns studying matrices B that are close to a k-simplex whose vertices lie in the unitary orbit of a self-adjoint matrix A with eigenvalues in [0,1]. In a later stage these discoveries will be put to the test as follows. If we demand that the same conditions are satisfied by a matrix that varies continuously over time can we then continuously track the previously predicted conclusions? The participating students will be exposed to the process of mathematical research through formulating questions, proving theorems, and communicating their results via presentations and articles.
Detailed Description
Rationale
The specific problems under consideration are inspired by operator theory and the theory of operator algebras, both of which are subfields of functional analysis. By stating analogous problems in a more tractable language the participants will be exposed to mathematical research in these areas in a more familiar environment.
Objective
The objective is to compute quantitative estimates related to two problems from matrix theory and their continuous counterparts.
Problem 1
The norm \|A\| of an m\times n matrix A is the maximum of |\langle Ax,y \rangle| over all unit vectors x and y of appropriate dimensions.
An N\times N matrix A is said to be a C-factor of an n\times n matrix B if there exist matrices L and R of appropriate dimensions so that B = LAR and \|L\|\cdot\|R\| is at most C.
Given an N\times N matrix A of norm one, a positive integer n, and a constant C we study necessary conditions to place on A so that if achieved then A is a C-factor of all n\times n matrices B of norm one.
Problem 2
An N\times N self-adjoint matrix with eigenvalues in the interval [0,1] is called a positive contraction.
An N\times N matrix B that represents A relative to some orthonormal basis is said to be in the unitary orbit of A.
For every positive integer k we study the matrices that are inside and close to a k-simplex whose vertices lie in the unitary orbit of a fixed positive contraction A.
Each of these problems can be enhanced by replacing the given stationary matrix A by a matrix function A=A(t) that varies continuously over time. In Problem 1 we then study conditions so that A(t) is a continuous C-factor of all nxn matrix functions of norm one. In Problem 2 we study k-simplices that continuously deform over time and continuous matrix functions that stay within a certain distance of the deforming simplex.
Tasks, Student Responsibilities, and Timeline
Throughout the project the participants will meet with supervisors twice a week. Furthermore, students are expected to hold independent meetings an additional two times per week. Each member will be asked to make contributions to all aspects of the project. They will be expected to spend time presenting their ideas on how to solve a given problem, to give a brief overview of the project so far, and to prepare written materials.
In weeks 1 to 3 the participating students will be guided through recalling and combining knowledge from linear algebra, analysis, combinatorics, and probability theory. In weeks 4-7 they will be asked to formulate and prove simplified partial statements and discuss possible improvements and ways to prove them. Step by step they will be guided towards asking the right questions and developing answers that may lead to gradual improvements. In the final 2 weeks of the project they will be tasked with preparing a slide presentation and poster that summarizes the conducted research and its conclusions as well as with preparing a short article.
Outcomes
The participating students will develop skills in cooperation, presentation, and independent thought. They will gain confidence and obtain leadership skills through taking on various roles that they might otherwise hesitate to take up voluntarily. Most importantly, they will be exposed to the process of pure mathematical research. They will learn that if the path to a final goal is not apparent then it is useful to formulate and solve a simpler problem and subsequently ask the right questions and develop answers that will yield gradual improvements.
Research article
