A reader points me to this recent Topology paper by Nabutovsky and Weinberger, which probably contains the biggest numbers to have ever arisen naturally in mathematics. Specifically, the authors show that, if we maximize the kth Betti number (for k≥3) over all groups whose presentation has size N (while keeping the number finite), then it grows like the “super-duper Busy Beaver function” (that is, Busy Beaver with an oracle for the halting problem with an oracle for the halting problem).

The spiked magazine survey I blogged about earlier has finally been published. (Warning: Spouters ahead.)

1. John Sidles Says:

It’s hard to take seriously a “greatest innovation list” that doesn’t include Kähler manifolds and model order reduction.

Especially on this sad day, which brings news that Herbert F. Kornfeld has passed away. Those of us who cherished mathematical essays like Kornfeld’s It Wuz Always ‘Bout Tha Numbahs will miss him … even this musical essay on dealing with life’s mystery cannot fully comfort us. :;

2. anonymous Says:

Funny that you know Nabutovsky. I used your notes about ZF to help me study for his undergrad logic course last semester.

3. Andy D Says:

I think that, to be fair in the biggest-number competition, we need to create separate ‘weight classes’. The heavyweight class would contain problems where the big numbers are clearly corollaries of Turing-universality/RE-completeness of associated decision problems. From what little I know about word problems in group theory, and from the nature of the bounds cited, this paper looks like a heavyweight.

In the middleweight category you’d have the Ramsey-type numbers that are computable but grow faster than all functions provably total in some system of bounded arithmetic. I’m still trying to understand these problems’ link to computation, but they are clearly ‘large for a reason’.

Finally, you’ve got the scrappy, wild-card numbers whose largeness seems self-made and unreliant on known links to logic or computation. Their existence may be frustrating, but these are the big numbers I root for.