Dept. of Math Sciences
Kent State University
Math & CS Building
Summit Street, Kent OH 44242
reu [at] math.kent.edu
FAX: (330) 672-2209
Images of Polynomials on Matrices (Misha Chebotar)
Participants: David Buzinski, Michael Kaufman, Lillian Pasley, Robin Winstanley
Let f(x,y)=xy-yx be a polynomial in two non-commutative
variables x and y. If we plug in different n by n matrices for x and y,
then what kind of matrices can we get? A celebrated theorem in linear
algebra asserts that we can get all matrices of trace 0. The purpose of
this project is to study the images of polynomials in several
non-commutative variables. This area contains many important open
questions, so this project can be considered as a starting point for
serious research activity.
Prerequisites: Advanced Linear Algebra or Theory of Matrices.
David Buzinski and Robin Winstanley On multilinear polynomials in four variables evaluated on matrices , Linear Algebra and its Applications 439, Issue 9, November 2013, Pages 2712-2719.
Michael Kaufman and Lillian Pasley On commutators of matrices over unital rings, Involve 7 (2014), No. 6, 769?772.
How to estimate the size of a polynomial (Andrew Tonge)
Participants: Rachel Carleton, Dorothy Klein, Hope Snyder, Ryann, Cartor
Project Description: Polynomials are important objects in many areas of pure and applied
mathematics. In particular, polynomials of one or more real variables
are essential building blocks in calculus. It is theoretically
and practically important to be able to estimate how big polynomial values can be
on domains of interest.
Exact computations of polynomial maxima are difficult or even impossible
for high degree polynomials of a single variable - and even for low
degree polynomials of several variables. However, for many purposes, a
close approximation is good enough. The objective of this project is to
identify simple but effective ways to estimate polynomial maxima in
terms of their coefficients.
Toric codes and lattice point geometry (Jenya Soprunova)
Graduate students: Michelle Cordier, John Hoffman, Matt Alexander.
Participants: Riley Burkart, Kyle Meyer, Cody Stockdale
We will study some questions in lattice point geometry in relation with toric codes. Here is a paper related to this project.