Celebratio Mathematica

It gives me great pleas­ure that I have been asked to de­scribe to you some of the very beau­ti­ful math­em­at­ics which res­ul­ted in the award­ing of the Fields Medal to Vaughan F. R. Jones at ICM ’90.

In 1984 Jones dis­covered an as­ton­ish­ing re­la­tion­ship between von Neu­mann al­geb­ras and geo­met­ric to­po­logy. As a res­ult, he found a new poly­no­mi­al in­vari­ant for knots and links in 3-space. His in­vari­ant had been missed com­pletely by to­po­lo­gists, in spite of in­tense activ­ity in closely re­lated areas dur­ing the pre­ced­ing 60 years, and it was a com­plete sur­prise. As time went on, it be­came clear that his dis­cov­ery had to do in a be­wil­der­ing vari­ety of ways with widely sep­ar­ated areas of math­em­at­ics and phys­ics, some of which are in­dic­ated in Fig­ure 1. These in­cluded (in ad­di­tion to knots and links) that part of stat­ist­ic­al mech­an­ics hav­ing to do with ex­actly solv­able mod­els, the very new area of quantum groups, and also Dynkin dia­grams and the rep­res­ent­a­tion the­ory of simple Lie al­geb­ras. The cent­ral con­nect­ing link in all this math­em­at­ics was a tower of nes­ted al­geb­ras which Jones had dis­covered some years earli­er in the course of prov­ing a the­or­em which is known as the “In­dex The­or­em”.

My plan is to be­gin by dis­cuss­ing the In­dex The­or­em and the tower of al­geb­ras which Jones con­struc­ted in the course of his proof. After that, I plan to re­turn to the chart in Fig­ure 1 in or­der to in­dic­ate how this tower of al­geb­ras served as a bridge between the di­verse areas of math­em­at­ics which are shown on the chart. I will re­strict my at­ten­tion throughout to one very spe­cial ex­ample of the tower con­struc­tion, and so also to one spe­cial ex­ample of the as­so­ci­ated link in­vari­ants, in or­der to make it pos­sible to sur­vey a great deal of math­em­at­ics in a very short time. Even with the re­stric­tion to a single ex­ample, this is a very am­bi­tious plan. On the oth­er hand, it only be­gins to touch on Vaughan Jones’ schol­arly con­tri­bu­tions.

[Ed­it­or’s note: The above text is the ab­stract of Joan Birman's talk on Vaughan Jones’ math­em­at­ics on the oc­ca­sion of his be­ing awar­ded a Fields Medal in 1990. The en­tire text can be viewed at the PDF link at the top right of this page and is made avail­able with the kind per­mis­sion of the Math­em­at­ic­al So­ci­ety of Ja­pan.]