The isoperimetric inequality has a long history, going back to the legend of Queen Dido. In this lecture, I will discuss the isoperimetric inequality and how it relates to the calculus of variations, the notion of mean curvature, and minimal surface theory.

The goal of spectral geometry is to understand the relation between geometric shapes and the frequencies at which they can vibrate. After providing a general introduction to the subject, I will discuss concrete examples of planar domains that are isospectral, or, more informally, sound the same.

It is an important fact that all finite-dimensional normed spaces are equivalent --- meaning that you can bound each norm |x|_1 and |x|_2 in terms of the other, c |x|_1 ≤ |x|_2 ≤ C|x|_1 for some 0 < c < C < infty. This is an important and useful fact in analysis. However, not all normed spaces are isometric, so that there is no invertible matrix A so that |Ax|_2 = |x|_1 for all vectors x. We will show that the finite-dimensional normed spaces ell^p_n (that is, R^n equipped with the l^p norm) are not isometric for any p =/= q unless {p, q} = {1, infty}. Even better, we will discuss and calculate a quantitative measure of how far apart these spaces are. On the way, we will pass briefly through probability and convex geometry, and these ideas give rise to non-obvious results in the theory of infinite-dimensional normed spaces.

I'll sketch the construction of an object that deserves to be called a "p-adic zeta function" and admire it a bit, starting with what "p-adic" means.

On Wednesday February 17 at 8:30pm, you are invited to participate in a zoom town hall meeting for Barnard/Columbia undergraduates who would like to discuss diversity, equality and inclusion (DEI) in mathematics. This is a chance for students to voice their perspectives, experiences and ideas. This event is cosponsored by the math department DEI committee and the Undergraduate Math Society.

Circa 300 BCE, Euclid proved for the first time that there are infinitely many prime numbers. Euler reproved this fact in 1737 using a completely different argument, and in 1955, Furstenberg gave a variant on Euclid's proof using elementary point-set topology. In my talk, I will start by reviewing these proofs, and then I will recast Furstenberg's proof in a way which, as far as I'm aware, is new, and which will shed some light on Euler's proof as well.

The famous Abel-Ruffini theorem asserts that there is no formula for expressing the roots of a general fifth-degree polynomial using only radicals. Rather than being the definitive end to a story, it turns out that there are many further aspects of root-finding left for more modern (and indeed future) mathematicians to discover. In this talk, I will discuss some more modern perspectives on root-finding that showcase the roles played by topology and by dynamics.

It is well-known that the set of modular forms is ring isomorphic to the set of complex polynomials in two variables. I will first go over the traditional method by which this is obtained. I then take a more geometrical approach to the result, viewing the objects involved in the traditional method as Riemann surfaces.

In Einstein's theory of general relativity, light rays weave together into a peculiar geometry measuring zero length. In this talk we will explore a class of these geometries and, in the presence of a Black Hole, use them to prove the famous positive energy theorem.

The three theorems of Sylow presented in an introductory group theory course have always been a mystery to me. That is until recently, when I learned of a fundamentally geometric interpretation of these theorems. I shall explain this, and also use this to draw some analogies and unify some seemingly disparate concepts.