It's that time of the year. Classes are starting up. You're nervous and excited to be taking some of your first "real" math classes called things like "Abstract Algebra" or "Real Anaylsis" or "Topology." It goes well for the first few weeks as the professor reviews some stuff and gets everyone on the same page.… Continue reading Surviving Upper Division Math

# Category: topology

## 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

## Topological Modular Forms

This will be my first and last post on this topic, since it will take us too far from the theme for this year which is arithmetic geometry. It took awhile for me to write this because something feels wrong in the last post and I wanted to correct it before doing this one. Unfortunately,… Continue reading Topological Modular Forms

## 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

## Gerbes 3: Another Example and Some Caution

This might be my last post on gerbes (explicitly for gerbe's sake), so as in my last 'stacks for stack's sake' post I'll try to clarify some things with more examples and then give some cautions. Last time I mentioned the classifying stack $latex {BA}&fg=000000$. Let's first actually construct it better than the quick idea… Continue reading Gerbes 3: Another Example and Some Caution

## Gerbes 2: The Motivation

I'm going to make another definition, but I may as well get to the punchline first or else anyone reading this that doesn't already know the punchline is going to skip reading it or tune out. If you have an abelian sheaf $latex {\mathcal{A}}&fg=000000$ on $latex {X}&fg=000000$, then there is a notion of $latex {\mathcal{S}}&fg=000000$… Continue reading Gerbes 2: The Motivation