IGL project: p-adic properties of sequences and finite state automata

Overview

Project description

A powerful approach to understanding sequences of numbers is to study them p-adically. For instance, one can reduce a sequence modulo a prime p and consider the resulting values. For an interesting class of sequences, work of Furstenberg, Deligne, Denef and Lipshitz implies that, for fixed p, these values can be produced by a finite state automaton. These are very basic computers that can be represented visually as a graph. Interesting properties of our sequence can then be read off from these automata.

On the other hand, recent work of Rowland and Yassawi demonstrates that these automata can be obtained automatically (meaning that a computer can do the work for us). This allows us to experimentally go through lists of interesting sequences, compute the automata for some small primes, and deduce p-adic properties of the sequences (which, in the past, had to be proved by hand). We aim to find automatic proofs of known results and, possibly, interesting new congruences.

Depending on the background and taste of participants, we will spend variable amounts of time on familiarizing ourselves with using Mathematica, refreshing our modular arithmetic, getting to know finite state automata and, above all, computing with lots of examples. It is hoped that, at the end of this project, you are inspired and ready to use computer algebra, and that you have seen a glimpse of the experimental nature of mathematical research.

Presentations

All IGL teams gather at two big meetings throughout the semester, one in the middle of the semester and one towards the end. Each team is allowed five minutes to present their progress.

Pujan delivered our midsemester presentation on Tuesday, October 28, and Arian the final presentation on Thursday, December 4.

At the IGL open house on Thursday, December 11, our team will present the poster below.

Brochure