When you were younger, you learned about the counting numbers, 1, 2, 3, …. Eventually you learned about a new kind of number, the negative numbers, and then you learned about an even larger class of numbers, called rational numbers. Finally, you began to discover irrational numbers like $$\sqrt{2}$$ and $$\pi$$ and realize that the rational numbers live in the much larger world of real numbers, consisting of everything on the number line. What we’ll discover is that there’s a completely different kind of number, called a $$p$$-adic number. The rational numbers are all $$p$$-adic numbers, but it turns out that $$p$$-adic numbers are a completely different way of extending the rational number system than the real numbers! $$P$$-adic numbers arise when we try to make sense of expressions of the form $$...999$$, going infinitely far to the *left* instead of the the right. Come and learn about this strange new universe.

Let's say you have a string tied to poles on both ends and then you pluck the string. How will it move? You might imagine from everyday life that it would be something like a wave. It turns out that using some basic physics, we can express this as a differential equation. To solve it, we are led to the idea of Fourier series! These are a way of expressing a function as an infinite series of sines and cosines. We will explain what this has to do with the physics of a wave. If time permits, we will mention momentum and energy and what this has to do with quantum physics!

A proof has only finitely many steps. But did you know that you could use this fact (and that there are infinitely many prime numbers) to prove a fascinating theorem?! The Ax-Grothendieck Theorem says that a map from $$\mathbb{C}^n \to \mathbb{C}^n$$ given by $$n$$ polynomials in $$n$$ variables is onto if it is one-to-one. In this class we will present a proof of this theorem that uses a twisted idea. We will talk about mathematical logic, the underpinning of mathematical theories and axioms, and we will use something known as Godel's Completeness Theorem (not incompleteness!) to show that if this theorem is false, we can prove it is false using a certain set of axioms. We will then use the fact that such a proof can only have finitely many steps to conclude! (More precisely, we use the axioms $$p \neq 0$$ for every prime number $$p$$. But there must be some $$p$$ for which we do not use this axiom, so the theorem is false in some finite field. But a one-to-one map from a finite set to itself is onto so we are done.)

You probably heard in Algebra 2 that there's a cubic and quartic formula but no quintic formula. But you probably never dared to go past the quadratic one, especially given how nasty the others were! In this class, we'll explain why there's a cubic and quartic formula and how you can derive it, without getting too messy. In doing so, we'll introduce the beautiful idea of symmetry in polynomials. It turns out that this idea also motivates Galois theory, which explains why there is no quintic formula! Most of the material comes from Chapter 1 of these notes: http://homepages.warwick.ac.uk/~masda/MA3D5/Galois.pdf

A Diophantine equation (after Diophantus of Ancient Greece) is a polynomial equation in multiple variables where you're looking for integer or rational number solutions. If you find a solution, that's great, you're done! But what if there's no solution? A tried and true method to show that no solution exists is to show that it doesn't have a solution modulo some number. But what if it has a solution modulo everything? Sometimes you can use deeper ideas like quadratic reciprocity to do things that you can't do with modular arithmetic alone. We will survey this and possibly some other methods for showing that a Diophantine equation has no solution. It turns out that this is the beginning of a larger theory known as obstructions to rational points, and it figures prominently in my own graduate research!

When you were little, you learned about the counting numbers, 1, 2, 3, …. Eventually you learned about a new kind of number, the negative numbers, and then you learned about an even larger class of numbers, called rational numbers. Finally, you began to discover irrational numbers like $\sqrt{2}$ and $\pi$ and realize that the rational numbers live in the much larger world of real numbers, consisting of everything on the number line. What we’ll discover is that there’s a completely different kind of number, called a $p$-adic number. The rational numbers are all $p$-adic numbers, but it turns out that $p$-adic numbers are a completely different way of extending the rational number system than the real numbers! $P$-adic numbers arise when we try to make sense of expressions of the form $...999$, going infinitely far to the *left* instead of the the right. Come and learn about this strange new universe.