# Events

The Maseeh Mathematics and Statistics Colloquium Series*

presents

**Structure and randomness in functional graphs of polynomials**

** over finite fields **

*Abstract:*** **

*f* be a polynomial with integer coefficients. For a finite field Fp, we form a (directed) graph that describes the action of *f* on Fp by drawing a vertex for each element of Fp and drawing a (directed) edge between the vertices *x* and *y* if *f(x)=y*. For certain special polynomials like *f(x)=x ^{n}*, the graphs are very structured and easy to describe. For most polynomials, various aspects of their functional graphs resemble the graphs of functions chosen at random. We investigate this relationship and prove that, for some families of polynomials, the number of cycles of any length behaves in a way that is as "random" as possible. This is joint work with Derek Garton.

*Bio:*

My research interests are mainly in number theory and algebraic geometry, particularly in arithmetic dynamics. I study arithmetic and algebraic properties of dynamical systems, such as the arboreal Galois representations attached to preimage fields of rational maps and the cycle structure of polynomials in finite fields. I am also very interested in the connections between automata theory and number theory.

Andrew Bridy earned his Ph.D. in Mathematics from the University of Wisconsin-Madison in 2014. His dissertation was The Artin-Mazur Zeta Function of a Rational Map in Positive Characteristic.

**Fariborz Maseeh Hall room B128
1855 SW Broadway**

**Light refreshments served**

**The faculty host of this speaker is Dr. Derek Garton**