Splash Biography

Finite fields are like the real, rational or complex numbers in that you can add, multiply, and divide, but very different in that they only have a finite number of elements. This means there are many nice counting questions one can ask about n-dimensional space over a finite field and matrices acting on it, and in many cases these counting questions mysteriously mirror analogous questions for finite sets, connecting to things you've seen like counting the number of $k$-element subsets of an $n$-element set (i.e. binomial coefficients).

We'll learn group theory in the context of reflections in Euclidean space (and, more generally, Coxeter groups). There are many beautiful connections between our algebraic and geometric understanding of these groups; in particular, we'll talk about how symmetries of Platonic solids connect to Dynkin diagrams, strange-seeming diagrams that mysteriously play a fundamental role across many different fields of mathematics.

SET is a fun card game which revolves around fast pattern-searching, but like many seemingly simple games, it has a lot more mathematical subtlety than meets the eye. In this class, we'll start by playing a game or two (beginners are welcome--the rules are easy to learn) and learning a clever trick to convince your amazed friends that you can count cards! This leads into a discussion of the mathematical life of SET, with connections to combinatorics, modular arithmetic, finite geometry, and group theory.

We’re going to talk a bit about a somewhat lesser-known cousin of the real numbers: the p-adics. These may be thought of as infinite geometric series of powers of a prime p, where we generously allow these series to retain their identity as independent ‘numbers’ rather than just throwing them away because they (often) diverge to infinity. Far from being a whimsical exercise in pretending things don’t diverge, they are actually extremely important in a wide variety of areas in math. They also have a lot of interesting properties—for instance, if two p-adic discs intersect at any point, one is contained in the other! We’ll define and discuss some of the fascinating ways in which the p-adic integers and p-adic rational numbers behave and their relation to the regular integers and rationals, as well as what being ‘cousin of the real numbers’ actually means in a rigorous sense.