# Kiddie Colloquium

Organizers: Andrew Lin & Shengtong Zhang

## Past Events

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…

I will explain the proof of the following statement: given n vectors in a vector space, let a_k be the number of sets of k of the n vectors which are linearly independent. Then the sequence a_0, a_1, a_2, ... is unimodal. The proof is an application of the Hodge index theorem in algebraic…

An “abstract polyhedron” means, roughly, a graph that “might be the edges and vertices of a polyhedron”. When can we promote “might be” to “is”? This question is answered by a beautiful theorem about circle packings on the sphere. I will explain the proof of this theorem, as well as some…

We will show how to differentiate computer programs (lambda-expressions, Turing machines, etc) by encoding them in a new system called linear logic that endows the space of programs/proofs with the structure of a differential k-algebra. We will discuss this theory from the perspective of the…

This talk plans to design an immersive game for people who are still kids at heart to experience learning mathematics from the very beginning, but in a completely non-traditional way. We will start analysis without \epsilon-\delta, start algebra without writing operators and laws, start topology…

Calculus is hard. In most textbooks and calculus classes, the chain rule (f∘g)′(x) = f′(g(x))∘g′(x) is either not proved, or only partially proved. The reason is that the proof requires knowledge of topology not covered in the first two years of university, and most importantly, the result fails…