Since my blog claims to talk about physics sometimes and I just finished teaching multivariable calculus, I thought I'd do a post on one form of Gauss' law. As a teacher of the course, I found this to be an astonishingly beautiful "application" of the divergence theorem. It turned out to be a touch too… Continue reading Gauss’ Law

# Category: manifolds

## An Application of p-adic Volume to Minimal Models

Today I'll sketch a proof of Ito that birational smooth minimal models have all of their Hodge numbers exactly the same. It uses the $latex {p}&fg=000000$-adic integration from last time plus one piece of heavy machinery. First, the piece of heavy machinery: If $latex {X, Y}&fg=000000$ are finite type schemes over the ring of integers… Continue reading An Application of p-adic Volume to Minimal Models

## Volumes of p-adic Schemes

I came across this idea a long time ago, but I needed the result that uses it in its proof again, so I was curious about figuring out what in the world is going on. It turns out that you can make "$latex {p}&fg=000000$-adic measures" to integrate against on algebraic varieties. This is a pretty… Continue reading Volumes of p-adic Schemes

## Classical (Lagrangian) Mechanics

It turns out that because I work with Calabi-Yau varieties I often encounter various ideas and terms from physics. In particular, quantum field theory is a something that comes up a lot. I took a lot of physics as an undergrad, and I've pieced together a tiny bit about what is meant by "quantum field… Continue reading Classical (Lagrangian) Mechanics

## Classical Local Systems

I lied to you a little. I may not get into the arithmetic stuff quite yet. I'm going to talk about some "classical" things in modern language. In the things I've been reading lately, these ideas seem to be implicit in everything said. I can't find this explained thoroughly anywhere. Eventually I want to understand… Continue reading Classical Local Systems

## Mirror Symmetry A-branes

I started writing this post this past weekend, but got stuck really quickly and then kept putting it off. I don't want to leave anyone following this hanging with no idea what the A-model is. This is harder for me to describe than the A-model for some reason. Mostly I'm running into the problem of… Continue reading Mirror Symmetry A-branes

## Stacks 2: An example

This will hopefully be a short, yet enlightening post in which the concept of a stack starts to make more sense than the abstract nonsense of the last few posts. Recall that we formed a category of line bundles on a manifold $latex {L(X)}&fg=000000$ and had a natural forgetful functor: $latex {L(X)\rightarrow \text{Top}(X)}&fg=000000$. If one… Continue reading Stacks 2: An example