The Rankin Lectures 2008

The Rankin Lectures 2008, My Favorite Numbers, were given by John Baez of the University of California at Riverside. Baez is well-known for, among other things, his long-running web column This Week's Finds in Mathematical Physics and the research blog The n-Category Café. News:  Streaming videos of the lectures are now available. (To choose which lecture to watch, use the menu to the right of the video screen.) The slides from the lectures are also available. Discussions of the talks are here: 5, 8, and 24. The Herald had a feature on the lectures (6 September 2008, page 5).

 

Outline

 

Click for poster The lectures took place in the Department of Mathematics of the University of Glasgow. Date     Title Monday 15 September, 4pm (suitable for a general audience)     5 Wednesday 17 September, 4pm     8 Friday 19 September, 4pm     24 The first talk was for a general audience. It was attended by non-mathematicians and schoolchildren, as well as mathematicians. The Rankin Lectures are supported by the Glasgow Mathematical Journal Trust. They are a celebration of the achievements and influence of the Scottish mathematician Robert Rankin, who spent most of his career at Glasgow. The previous Rankin Lectures (2006) were given by Persi Diaconis.

 

5

 

Click for poster For a general audience Different numbers have different personalities. The number 5 is quirky and intriguing, thanks in large part to its relation with the golden ratio, the "most irrational" of irrational numbers. The plane cannot be tiled with regular pentagons, but there exist quasiperiodic planar patterns with pentagonal symmetry of a statistical nature, first discovered by Islamic artists in the 1600s, later rediscovered by the mathematician Roger Penrose in the 1970s, and found in nature in 1984. The Greek fascination with the golden ratio is probably tied to the dodecahedron. Much later, the symmetry group of the dodecahedron was found to give rise to a 4-dimensional regular polytope, the 120-cell, which in turn gives rise to the Poincaré homology sphere and the root system of the exceptional Lie group E8. So, a wealth of exceptional objects arise from the quirky nature of 5-fold symmetry.

 

8

 

Click for poster The number 8 plays a special role in mathematics due to the "octonions", an 8-dimensional number system where one can add, multiply, subtract and divide, but where the commutative and associative laws for multiplication — ab = ba and (ab)c = a(bc) — fail to hold. The octonions were discovered by Hamilton's friend John Graves in 1843 after Hamilton told him about the "quaternions". While much neglected, they stand at the crossroads of many interesting branches of mathematics and physics. For example, superstring theory works in 10 dimensions because 10 = 8+2: the 2-dimensional worldsheet of a string has 8 extra dimensions in which to wiggle around, and the theory crucially uses the fact that these 8 dimensions can be identified with the octonions. Or: the densest known packing of spheres in 8 dimensions arises when the spheres are centered at certain "integer octonions", which form the root lattice of the exceptional Lie group E8. The octonions also explain the curious way in which topology in dimension n resembles topology in dimension n+8.

 

24

 

Click for poster The numbers 12 and 24 play a central role in mathematics thanks to a series of "coincidences" that is just beginning to be understood. One of the first hints of this fact was Euler's bizarre "proof" that 1 + 2 + 3 + 4 + … = -1/12 which he obtained before Abel declared that "divergent series are the invention of the devil". Euler's formula can now be understood rigorously in terms of the Riemann zeta function, and in physics it explains why bosonic strings work best in 26=24+2 dimensions. The fact that 12 + 22 + 32 + … + 242 is a perfect square then sets up a curious link between string theory, the Leech lattice (the densest known way of packing spheres in 24 dimensions) and a group called the Monster. A better-known but closely related fact is the period-12 phenomenon in the theory of "modular forms". We shall do our best to demystify some of these deep mysteries.

This page was last changed on 25 February 2009. Contact: Tom Leinster (T.Leinster#maths.gla.ac.uk, with # changed to @).