Rose-Hulman Institute of Technology
Professor
Available through 2030

This Section Visitor is available to present on the following topics at Section meetings:

The Rotational Trace Abstract
Combing ideas from geometry, calculus, complex numbers, and linear algebra, we can find a simple way to characterize the tendency of solutions of a 2x2 linear system of differential equations to rotate in an "averaged sense." We define a simple scalar quantity that we call the rotational trace. The sign of the rotational trace predicts the direction of rotation for rotational solutions, and extends to solutions that do not truly rotate by giving geometric information about the eigenvectors or generalized eigenvectors of the corresponding coefficient matrix.

The rotational trace connects ideas from seemingly disparate areas of the undergraduate mathematics curriculum. The rotational trace is related to well-known invariants of 2x2 matrices such as the trace under rotational transformations. The rotational trace connects to vector calculus by finding finite approximations that compute the flux and rotation of the system of differential equations exactly while using as few as three points and provide a simple proof of the right-hand rule for cross-products. This is joint work with Tim All.

What exactly is half a derivative anyway?
The idea of (and how to compute) a derivative is well-known to any student who has taken calculus. However, to make sense of a non-integer order derivative takes considerably more work. Tools are needed from complex analysis, harmonic analysis and linear algebra to understand a half derivative. In this talk, we will begin by investigating what it means to take the square root of a matrix, and viewing a derivative as a "really large matrix" we can begin to make sense of a half derivative. With these simple tools, we can make sense of even crazier objects such as derivatives of imaginary order!

For more information about the Section Visitor program, please visit the Section Visitor webpage.