### October 2007

I’ve been a bit busy lately, and so I missed last week posting. So what I’ve decided to do is to take the talks I’ve given in various graduate student seminars over the last year or so and convert them into posts. This one is a particularly tough prospect, as the talk didn’t go very well. I’m following a paper of Hamilton‘s titled “Ricci Flow on Surfaces” and only present the high genus case. Comments and (especially!) corrections are encouraged.

Hi everybody,

Well, I’m finally getting around to talking about the Hamiltonian formalism of classical mechanics. Of course, if you’ve been keeping up with The Everything Seminar, the good people over there have discussed this very topic recently (with some of the Lagrangian formalism thrown in), but I feel it never hurts to have someone go over it again. Besides, I promised in an earlier post that I would talk about this, so here I go: (more…)

Hi everybody,

In my last post, I promised that I would talk about the Hamiltonian and Lagrangian formalism of classical mechanics. In order to do that, we need to first develop some topological ideas, specifically in the area of symplectic topology. (It appears that the nice folks over at The Everything Seminar has beaten me to it, but I had this mostly written, so I decided to post it anyway.) The outline of these notes comes from a mathematical physics course I took last year, while the content is drawn primarily from Ana Cannas da Silva’s Lectures on Symplectic Geometry; other sources will be mentioned as they come up. (more…)

So, on Thursday, I posted some background stuff, which I think I might continue to do in the future. Just in general if I want to do a real post which there’s some basic technical stuff that needs to be mentioned to do properly, I’ll post it on Thursday, but otherwise, I’m going to stick to Monday posts. So this week, I’m going to generalize the example from last week and talk about Elliptic curves in general.

Ok, I’m writing this now specifically so that I don’t have to do it in my Monday post, because I need it to even define what I’m talking about. Today’s topic of interest are locally ringed spaces, and they are rather important in general. In fact, any geometric or topological object can be interpreted in this language, so let’s get started.

This week, we’re going to talk about some of the more important ideas in algebraic geometry, and in particular we will be picking a polynomial and describing its geometry in detail. This polynomial is not taken at random in any way, shape or form, but we’ll go in pretending that we have no clue how nice it is. Ready? Here’s the polynomial: $f(x,y)=y^2-x^3+x$.
(more…)

Hi everybody,

I was thinking that I would do a series of posts about classical mechanics from a mathematician’s standpoint. This post will be an overview of the Newtonian formalism of the subject, with some important examples so the usual physics jargon can be introduced. Future posts will cover the Hamiltonian and Lagrangian formalisms with all of the geometry involved. (more…)

Today I’m going to talk about parallel parking. We’ve all had to do it at some point (at least, those who drive) and certainly we’ve all noticed how much of a pain it is to get into a small space. Well, as it happens, if your car has length $L$, then for any $\epsilon>0$, it is possible to parallel park, assuming some things like that the driver can make arbitrarily small movements.