Michael Bradford Williams

Research

My primary research interests are differential geometry, geometric analysis. This includes geometric flows such as Ricci flow and harmonic map flow, and the coupling of the two. I have studied the stability and convergence of solutions of these systems in various contexts. I am also interested in the Ricci flow on homogenous Riemannian manifolds (usually nilpotent and solvable Lie groups), especially as it relates to Ricci soliton inner products and the Lie bracket flow.

I graduated in May, 2011 from UT Austin, under the supervision of Dan Knopf.

My Ph.D. dissertation:

Analysis of geometric flows, with applications to optimal homogeneous geometries

My research papers:

Explicit Ricci solitons on nilpotent Lie groups, Journal of Geometric Analysis 23 (2013), no. 1, 47-72.
Results on coupled Ricci and harmonic map flows, submitted.
Stability of solutions of certain extended Ricci flow systems, submitted.
Solvsolitons associated with Heisenberg algebras, submitted.

Other Stuff

Here are a few math-related things.

Here is an expository paper that I wrote for a recent class, concerning groupoids.

	Here is another expository paper that I wrote for a recent class, this time concerning collapse in Riemannian geometry.

Here are notes from my recent oral candidacy exam. I presented the main ideas from this paper, which uses the Ricci flow to classify certain types of manifolds.

	What is infinity? How many infinities are there? Infinitely many! I wrote another short note that introduces the concept of cardinality and proves a few facts about it. It is written at a basic level, and does not assume much knowledge of mathematics beyond basic facts about sets and functions.

Quick! What's the spectrum of a self-adjoint compact linear operator on a Hilbert Space? If you have to think for more than 3 seconds before answering, then you should check out my worksheet (pdf) on such operators.

	Even better, what's the universal cover of this topological space? Click for the answer.

I wrote a short note that proves a few interesting facts about the Fibonacci sequence. It should be understandable to anyone who knows anything about calculus and basic linear algebra.

	Finally! Your chance to learn all about super linear algebra! I prepared a few background notes for a paper I was reading recently. Also included is info on vector bundles. These are in no way comprehensive.

Richard P. Feynman is ~~approximately~~ the man. Here's an amazing speech given to the National Academy of Sciences in 1955.

	Also the man: Paul Erdos, the Kevin Bacon of Mathematics. Here is perhaps the first published reference to the Erdos number, appearing in the American Mathematical Monthly in 1969. Erdos himself replied to the short article by, unsuprisingly, doing real mathematics with the Erdos number. Sadly, my own Erdos number is still undefined.
I wrote another short note that explains my favorite number, Khinchin's constant. This number is the geometric mean of the denominators in the continued fraction representation of almost every real number!

	I recently participated in transcribing a series of lectures on Topological Quantum Field Theory and the Cobordism Hypothesis, give by Jacob Lurie. Videos of the lectures, which were part of the Persepctive in Geometry lecture series, are also available.