Math REU at Kent State University

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2013 Projects

Images of Polynomials on Matrices (Misha Chebotar)
Participants: David Buzinski, Michael Kaufman, Lillian Pasley, Robin Winstanley
Project Description: 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)
Participants: Riley Burkart, Kyle Meyer, Cody Stockdale
Project Description: We will study some questions in lattice point geometry in relation with toric codes. Here is a paper related to this project.

Minkowski length of 3D lattice polytopes, with Olivia Beckwith, Matthew Grimm, and Bradley Weaver, Discrete and Computational Geometry 48, Issue 4 (2012), Page 1137-1158.