algebra, analysis, math, physics, topology

Surviving Upper Division Math

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

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algebra, math, physics

Mathematical Reason for Uncertainty in Quantum Mechanics

Today I'd like to give a fairly simple account of why Uncertainty Principles exist in quantum mechanics. I thought I already did this post, but I can't find it now. I often see in movies and sci-fi books (not to mention real-life discussions) a misunderstanding about what uncertainty means. Recall the classic form that says… Continue reading Mathematical Reason for Uncertainty in Quantum Mechanics

algebra, computer science, math

How Hard is Adding Integers for a Computer?

In our modern world, we often use high level programming languages (Python, Ruby, etc) without much thought about what is happening. Even if we use a low level language like C, we still probably think of operations like $latex {1+1}&fg=000000$ yielding $latex {2}&fg=000000$ or $latex {3-2}&fg=000000$ yielding $latex {1}&fg=000000$ as extremely basic. We have no… Continue reading How Hard is Adding Integers for a Computer?

algebra, algebraic geometry, manifolds, math, number theory, 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

algebra, algebraic geometry, algebraic topology, manifolds, math, number theory, topology

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

algebra, algebraic geometry, math, number theory

Newton Polygons of p-Divisible Groups

I really wanted to move on from this topic, because the theory gets much more interesting when we move to $latex {p}&fg=000000$-divisible groups over some larger rings than just algebraically closed fields. Unfortunately, while looking over how Demazure builds the theory in Lectures on $latex {p}&fg=000000$-divisible Groups, I realized that it would be a crime… Continue reading Newton Polygons of p-Divisible Groups

algebra, algebraic geometry, math, number theory

More Classification of p-Divisible Groups

Today we'll look a little more closely at $latex {A[p^\infty]}&fg=000000$ for abelian varieties and finish up a different sort of classification that I've found more useful than the one presented earlier as triples $latex {(M,F,V)}&fg=000000$. For safety we'll assume $latex {k}&fg=000000$ is algebraically closed of characteristic $latex {p>0}&fg=000000$ for the remainder of this post. First,… Continue reading More Classification of p-Divisible Groups