Kiddie Colloquium

In 1859, Riemann investigated ζ(s) and made the famous hypothesis (RH) that all nontrivial zeros of ζ(s) must lie on the line Re(s)=1/2. Since then, many variations of ζ(s) have been developed. While some of them are believed to satisfy the analog of RH, there are cases where RH is false.
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This talk is a live performance in mathematical illustration using Inkscape, a free and open-source vector graphics editor. I will draw topological concepts in real time, working through path editing, node manipulation, and frequent undo/redo operations. Rather than treating these as…

Mason is a flooring installer who’s just landed a huge gig: the president has tasked him with tiling his new 90000-square-foot ballroom! Upon arrival, he finds out that the room is shaped like a hexagon and he’s been given millions of rhombus tiles to put on the ground however he likes. Mason is…

Why are certain number theorists seemingly obsessed with Shimuravarieties? Starting from modular curves, I'll give a gentleintroduction to the theory of Shimura varieties. Then I'll discuss therole Shimura varieties play in the Langlands program.

We explore how contractible pieces inside 4-manifolds can magically change smooth structures when removed and reglued. Starting from the Mazur cork, we take a tour through our 4-manifold zoo and meet some interesting exotic pairs that differ only by a cork twist. There may also be infinite…

The simplest cipher is just a permutation of the alphabet, which can be easily broken from a (long enough) encrypted piece of text using frequency analysis. Now suppose that I encrypt my text such that each subsequent letter is encrypted using a different permutation, out a list of 10^16 unique…

We will explore the classification of compact flat manifolds, talk about crystallographic groups, and, of course, relate all this to the eigenfunctions of the Laplacian.

What happens if you take two knots, attach a band, and refuse to think too hard? Sometimes you get a new knot. Sometimes you get more knots. Sometimes you accidentally wander into four dimensions.Of course, one might want to think harder and ask why knots are band summed in the first place. In…

Suppose you are given a sequence {a_n} of integers which obeys a linear recurrence relation. Can you decide whether this sequence has a zero? It turns out that this problem is hard. I will attempt to explain why. There might be p-adic analysis, transcendental number theory, and/or finite…

The p-adic numbers form a topological field, like the real or complex numbers—so we should be able to do geometry with them. But a naive definition of a p-adic manifold runs into the problem that the topology is “too discrete,” making the theory quickly uninteresting. Starting from the basics of…