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, 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