Divergent sums and the class number formula May 28, 2010
Posted by David Speyer in Number theory.4 comments
Over on MathOverflow, we’ve had a bunch of discussions about the class number formula. In particular, Keith Conrad pointed out a paper of Orde which gives a beautiful nonsense proof of the class number formula. This post is my attempt to understand why Orde’s argument works. Specifically, I am going to use an idea which I learned from Terry Tao’s blog: Arguments about divergent sums are often really arguments about the constant term in asymptotic expressions for smoothed sums.
This post assumes familiarity with the concepts and notations of a first course in algebraic number theory.
(more…)
Martin Gardner 1914-2010 May 23, 2010
Posted by David Speyer in Uncategorized.5 comments
Martin Gardner passed away yesterday, on Saturday, May 22. I’ve never met Gardner, but I grew up with his books and columns, ever since I discovered one of his Mathematical Games volumes in my local library.
Aha! Insight and Aha! Gotcha are my first recommendations for any bright kid who wants to stretch his mind. As an adult, whenever I want something new to think about, I can turn to the volumes of Scientific American columns. Every one is a clear presentation of an intriguing idea or puzzle. Gardner’s skeptical books helped teach me how to read and design a scientific study; and the adventures of Dr. Matrix showed how much fun it could be to discard skepticism and play with nonsense. And, if you enjoy Lewis Carroll, but have a feeling that many of the jokes are going over your head, you really owe it to yourself to pick up a copy of the Annotated Alice.
I hope Martin Gardner’s friends know how many people he helped and touched. For the rest of us, perhaps we could try to take something we love, and make it as exciting and accessible as Martin Gardner made mathematics for us.
Lattices and their invariants May 14, 2010
Posted by Scott Carnahan in linear algebra, Number theory.1 comment so far
This post started out as an exposition on the monster Lie algebra, but it grew out of control, so I’m hacking off a chunk. Here, I’ll describe the basics of integer lattices.
Lattices show up in many mathematical contexts, some of which may be unexpected to the uninitiated. These contexts include the study of optimal periodic sphere-packings, the topology of 4-manifolds (where lattices give a full classification in the simply connected case), algebraic number theory, finite group theory, and theoretical high-energy physics. I will say almost nothing about these applications, though.
The Weil Conjectures: The approach via the Standard Conjectures May 3, 2010
Posted by David Speyer in Algebraic Geometry, characteristic p, Number theory.7 comments
The aim of this post is to outline a strategy for proving the Weil conjectures, proposed by Grothendieck and others. This strategy is incomplete; at various stages, we will need to assume conjectures which are still open today.
Our aim is to prove:
Theorem:
Let 
In previous posts, we gave for a proof in characteristic zero and a proof in the case that
(more…)
