NAA: Really Big Numbers

10 July 2010

Today at Not About Apples, the disconnect between the our inituitive grasp of reasonable-sized numbers and our confusion over huge numbers, as motivated by this quotation from Ron Graham.

The trouble with integers is that we have examined only the very small ones.  Maybe all the exciting stuff happens at really big numbers, ones we can’t even begin to think about in any very definite way.  Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed.  Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions.

Includes a math gem you’ve almost surely never heard before, which you almost surely won’t believe.

Permalink: Really big numbers.

CW: The integers are a topological space?

10 July 2010

This time on Cap’s Whiteboard, some remarks on an undeservedly-little-known proof that there are infinitely many prime numbers, which works by treating \mathbb{Z} as a topological space, a bit of ingenuity due to Furstenburg (as far as I know).

Permalink: Idle remarks on the Furstenburg topology

CW: Counting Problems in Apollonian Circle Packings

8 July 2010

The first “real” post on Cap’s Whiteboard. I talk a little about the counting problem that kicked off my study of Apollonian circle packings.

It all starts with this picture.

Apollonian circle packing with root quadruple (-1,2,2,3)Apollonian circle packing with root quadruple (-1,2,2,3)

Permalink: Counting Problems in Apollonian Circle Packings

New blog: Cap’s Whiteboard

8 July 2010

One of my favorite things to do as a mathematician is to drop in on one of my colleagues and ask them about what they’re working on at the moment. Mathematics is a big place, and what’s second nature to someone else may be quite novel to you.

That’s the idea behind Cap’s Whiteboard. This blog is offered as your way to drop in on me and hear about the mathematics that I’m learning, that I’m thinking about, that I’m doing. Think of these posts as what might happen if you asked me about my office blackboard, or one of my home whiteboards. Expect some combination of my latest research, problems I’m chewing on, particularly interesting articles I’ve read, and my idiosyncratic perspectives on “familiar” mathematics.

This is intended as blog written by a mathematician for other mathematicians, and I’m not making any promises that individual posts will be self-contained. Some stuff will be suitable for strong math majors, some for graduates, and likely some will only be for number theorists. Please feel free and encouraged, though, to ask me (by email or, preferably, in comments) for references if you want some background on anything you see here.

Posts are on no set schedule, and the amount of time between posts will probably have a high variance, depending on what else is going on for me mathematically. I expect to average about a significant post a week, not counting links and the like.