**Title:** Vector spaces, Matrices, and Linear Recurrences over Finite Fields

**Abstract:** We will discuss a number of questions concerning basic objects in algebra, such as polynomials, matrices, and vector spaces in the setting of a finite ground field. Here is a sampling of some of the simpler questions that we may ask.

- What is the maximum possible order of an element of the general linear group over a given finite field, and how many elements there are of this order?
- What is the probability that two randomly chosen polynomials of a given positive degree with entries in a given finite field are relatively prime?

We will discuss, in particular, a question of Niederrieter that was open since 1995 (and until recently) and the recent progress on it. Connections and applications to cryptography via the so-called linear feedback shift registers (LFSRs) or in other words, homogeneous linear recurrences over finite fields, will also be discussed.