Kiddie Colloquium

Similarities between Galois theory and the theory of covering spaces are so striking that algebraists use geometric language to talk of field extensions whereas topologists talk of Galois covers. I will explain A. Grothendieck's formulation of abstract Galois theory, which builds a bridge…

A Cayley graph G is a highly symmetric graph whose vertex set is a finite group Gamma. A rather surprising theorem, due to Payan, shows that, if Gamma is (Z/2Z)^n, then G cannot have chromatic number exactly 3. (In other words, if G is 3-colorable then G is also 2-colorable.) I'll show you…

Ergodic theory has applications to many different fields of mathematics, and it tends to look very different in each of these places. One of these fields is additive combinatorics. With an aim to demonstrate what ergodic theory looks like in this setting, I will present an ergodic-theoretic…

The cohomology of a variety is a sequence of vector spaces that captures the global structure of the variety, and it can be used to study various aspects of the variety, such as its connectivity, its Betti numbers, and its intersection theory." This is what GPT says, but what it doesn't say is…

I swear I've heard this before... Wait. Wait wait wait. No way! These drums sound exactly the same! And you're telling me you have a systematic way of constructing these?? That's craaaaaazy.

Complexity theory is an interesting subject. I will show you the fastest¹ algorithm for any² problem, as well as how to compute anything³ using only three bits of memory.⁴ If time permits, I will explain why these are (at least somewhat) useful.

How do the eigenfrequencies of a vibrating membrane change if one deforms the boundary? It turns out that this depends only on the behavior of the membrane at the boundary itself, as encapsuled in a famous formula of Jacques Hadamard from 1909. I will talk about this and some more odd facts…

When is a multivariate polynomial nonnegative (over the reals or some other semi algebraic set)? We’ll start with some easy methods to certify positivity, discussing Hilbert’s 17th problem and its positive resolution along the way.

The Riemann integral does not work well with limits, so naturally one wishes to make something better. Thus, every high school or undergraduate math student should attempt to develop a better integral before learning any measure theory. Naturally, they come up with many strange ways to do it,…

Ramsey and Turán numbers are both central quantities in graph theory. Both maximize some quantity — the number of edges (Turán) or independence number (Ramsey) — over n-vertex graphs containing no copy of a fixed forbidden subgraph. In this talk, I'll tell you about a quantity that combines the…