Andrew W. Herring
CV

I'm a fourth year mathematics PhD student at Western University. Chris Hall is my PhD supervisor.

I'm on the postdoctoral job market!

My current research is focused on Galois representations which arise in arithmetic dynamics. In the past, I studied Ramanujan graphs.

I grew up in Albuquerque, New Mexico, USA. When I'm not doing math, I enjoy rock climbing, playing drum set, and cooking.

If you're a human, feel free to contact me:

• Andrew W. Herring
• Middlesex College 245
• Western University
• aherrin6 "at" uwo "dot" ca

Research

Here is my research statement.

My research interests include:
• arithmetic dynamics
• arithmetic geometry
• spectral graph theory

"Galois representations in arithmetic dynamics?" You bet!

Suppose we're given a map f from the projective line over a field K to itself. Then we can iterate f, and studying the Galois group associated to periodic points leads us to consider the "dynatomic Galois representation".

We might start instead with a fixed initial point P and look at those points which are mapped to P after n-many iterations of f. On one hand, this collection of all "nth pre-images" is naturally a rooted (at P) tree. On the other, we generate a tower of Galois extensions of K by adjoining the nth pre-images. Then the Galois group of the tower acts on the tree: this action is called the "arboreal Galois representation."

In either case, it's a fascinating (and hard) question to ask about how large the Galois group can be in its codomain.

a gratuitous image of La Luz Trail in the Posu gai hoo-oo (Sandia Mountains)

"I thought you said you like Ramanujan graphs..." I do.

Ramanujan graphs are (in some precise sense) "optimal expander graphs"-optimally connected, yet sparse. While Ramanujan graphs are defined by forcing two eigenvalues to be as far apart as possible, the original constructions came from Cayley graphs of certain arithmetic groups, so they have their origins in number theory. (On the flip side, Hall, Ellenberg, and Kowalski deduced arithmetic results by observing that a family of graphs has nice expansion properties).

• I wrote part II of my comprehensive exam on the influential paper Interlacing Families I: Bipartite Ramanujan Graphs of all Degrees (2015) by Marcus, Spielman, and Srivastava in which they show that for a fixed degree of regularity, there are infinitely many Ramanujan graphs of that degree. A crucial ingredient was using (topological) 2-covers.
• In Ramanujan Coverings of Graphs (2018), Hall, Puder, and Sawin generalized the work of Marcus, et al. by showing that (the spirit of) the same argument goes through upon replacing 2-covers by d-covers for arbitrary d. To do so, they defined a new combinatorial generating function called the "d-matching polynomial."
• In A New [Combinatorial] Proof of the Commutativity of Matching Polynomials for Cycles (2018), Cochran, Groothuis, Rohatgi, Stucky, and I prove a functional equation involving the d-matching polynomial, and give a purely combinatorial proof that the matching polynomials of cycles commute under composition.
• An important tool in algebraic topology is the categorical equivalence between sets on which the fundamental group acts, and covering spaces. In my MS thesis (2016), I carefully wrote down the details of this equivalence in the particular context of graphs.

Teaching

Here is my teaching statement.

Here is a summary of my teaching evaluations.

Western University
Term Course Role Course Files and Evaluations
Fall 2020 Linear Algebra I discussion leader
Introduction to Abstract Algebra TA
Summer 2020 Linear Algebra I TA
Spring 2020 Linear Algebra II discussion leader
Group Theory TA evaluation
Fall 2019 First Year Calculus primary instructor
Summer 2019 Methods of Finite Mathematics TA evaluation
Spring 2019 Calculus II for the Mathematical and Physical Sciences TA evaluation
Linear Algebra I discussion leader evaluation
Fall 2018 Mathematical Structures TA evaluation
Linear Algebra I discussion leader evaluation
Spring 2018 Calculus II for the Mathematical Sciences TA evaluation
Elementary Number Theory I TA
Fall 2017 Fist Year Calculus TA evaluation

Albuquerque, New Mexico is down there

University of Wyoming
Term Course Role Course Files and Evaluations
Spring 2017 Multi-variable Calculus discussion leader
Fall 2016 Multi-variable Calculus discussion leader
Summer 2016 College Algebra primary instructor
Spring 2016 Calculus II primary instructor
Fall 2015 Calculus I primary instructor
Spring 2015 Multi-variable Calculus discussion leader
Fall 2014 Multi-variable Calculus discussion leader
University of New Mexico
Summer 2014 Trigonometry primary instructor syllabus
Spring 2014 Survey of Mathematics primary instructor syllabus
Fall 2013 Multi-variable Calculus discussion leader