Rataj Lecture: Kenneth Ribet
Ken Ribet, UC Berkeley professor and President of the American Mathematical Society, gave a talk on Fermat's Last Theorem. When an alternate title could be "Andrew Wiles Bests Gauss and Euler," you know there's got to be a compelling story in there somewhere. Indeed, Dr. Ribet gave a fascinating historical account of the story, from antiquity to the modern day solution. A brief recap follows below. (Browse the photos for some familiar faces among the eighty students and faculty who attended, and ask them for further details.)
Some Pictures from the Talk and Reception
Fermat's Last Theorem involve equations that can be viewed as generalizations of the Pythagorean Theorem: \(x^2+y^2=z^2\). A Pythagorean triple is a triple \((a,b,c)\) for which \(a^2+b^2=c^2\). It is easy to come up with such triples: e.g., \((a,b,c)=(1,1,\sqrt2)\). It is harder to come up with examples where \(a,b\) and \(c\) are all integers: e.g., \((a,b,c)=(3,4,5)\). The Greeks knew how to parametrize effectively all examples of integer Pythagorean triples: namely, let \(p\) and \(q\) be any integers and construct the Pythagorean triple \((a,b,c)=(p^2-q^2,2pq,p^2+q^2)\) . Try for yourself! In particular you can generate infinitely many integer Pythagorean triples this way.
The stunning observation of Fermat is that if we simply increase the exponent appearing in the Pythagorean Theorem, then there are effectively no integer solutions!
Fermat's Last Theorem (FLT): for every integer \(n>2\), there is no positive integer triple \((a,b,c)\) satisfying \(a^n+b^n=c^n\).
Fermat famously made this observation in 1647 as a note written in a volume of ancient mathematics, claiming his proof wouldn't fit in the margin. Until the 20th century progress towards a proof of FLT proceeded by tackling the infinitely many \(n\)'s involved one case at a time. (Along these lines, Sophie Germain and Ernst Kummer had some key insights in the mid 19th century that later could be used to prove Fermat's statement for \(n < 4\times10^6\). We like to think of 4 million as a very big number, but it's only a drop in the bucket... what is our National debt right now?)
The first key step toward a complete proof came from Gerhard Frey's idea of associating to a hypothetical solution \((a,b,c)\) of a case if FLT an elliptic curve \(E_{abc}\). If one could show this elliptic curve would simply be too wacky to exist, then one could prove there was no such triple \((a,b,c)\) in the first place. But how to do this? It had long been conjectured that elliptic curves corresponded to modular forms, so conjecturally there was some modular form \(f_{abc}\) associated to this elliptic curve \(E_{abc}\). We thus reduce FLT to showing (1) this modularity conjecture is true, and (2) that this modular form \(f_{abc}\) would be too wacky to exist. Ribet himself was able to show (2) by proving a "level-lowering" conjecture of Jean-Pierre Serre's. It took Andrew Wiles a good seven years of attic-dwelling reclusion, along with some very significant last-minute help from Richard Taylor, to prove just enough of the modularity conjecture to guarantee us that modular form \(f_{abc}\) associated to our hypothetical FLT triple \((a,b,c)\).
Only then was Fermat's Last Theorem proved at last!
Ken Ribet, UC Berkeley professor and President of the American Mathematical Society, gave a talk on Fermat's Last Theorem. When an alternate title could be "Andrew Wiles Bests Gauss and Euler," you know there's got to be a compelling story in there somewhere. Indeed, Dr. Ribet gave a fascinating historical account of the story, from antiquity to the modern day solution. A brief recap follows below. (Browse the photos for some familiar faces among the eighty students and faculty who attended, and ask them for further details.)
Some Pictures from the Talk and Reception
Fermat's Last Theorem involve equations that can be viewed as generalizations of the Pythagorean Theorem: \(x^2+y^2=z^2\). A Pythagorean triple is a triple \((a,b,c)\) for which \(a^2+b^2=c^2\). It is easy to come up with such triples: e.g., \((a,b,c)=(1,1,\sqrt2)\). It is harder to come up with examples where \(a,b\) and \(c\) are all integers: e.g., \((a,b,c)=(3,4,5)\). The Greeks knew how to parametrize effectively all examples of integer Pythagorean triples: namely, let \(p\) and \(q\) be any integers and construct the Pythagorean triple \((a,b,c)=(p^2-q^2,2pq,p^2+q^2)\) . Try for yourself! In particular you can generate infinitely many integer Pythagorean triples this way.
The stunning observation of Fermat is that if we simply increase the exponent appearing in the Pythagorean Theorem, then there are effectively no integer solutions!
Fermat's Last Theorem (FLT): for every integer \(n>2\), there is no positive integer triple \((a,b,c)\) satisfying \(a^n+b^n=c^n\).
Fermat famously made this observation in 1647 as a note written in a volume of ancient mathematics, claiming his proof wouldn't fit in the margin. Until the 20th century progress towards a proof of FLT proceeded by tackling the infinitely many \(n\)'s involved one case at a time. (Along these lines, Sophie Germain and Ernst Kummer had some key insights in the mid 19th century that later could be used to prove Fermat's statement for \(n < 4\times10^6\). We like to think of 4 million as a very big number, but it's only a drop in the bucket... what is our National debt right now?)
The first key step toward a complete proof came from Gerhard Frey's idea of associating to a hypothetical solution \((a,b,c)\) of a case if FLT an elliptic curve \(E_{abc}\). If one could show this elliptic curve would simply be too wacky to exist, then one could prove there was no such triple \((a,b,c)\) in the first place. But how to do this? It had long been conjectured that elliptic curves corresponded to modular forms, so conjecturally there was some modular form \(f_{abc}\) associated to this elliptic curve \(E_{abc}\). We thus reduce FLT to showing (1) this modularity conjecture is true, and (2) that this modular form \(f_{abc}\) would be too wacky to exist. Ribet himself was able to show (2) by proving a "level-lowering" conjecture of Jean-Pierre Serre's. It took Andrew Wiles a good seven years of attic-dwelling reclusion, along with some very significant last-minute help from Richard Taylor, to prove just enough of the modularity conjecture to guarantee us that modular form \(f_{abc}\) associated to our hypothetical FLT triple \((a,b,c)\).
Only then was Fermat's Last Theorem proved at last!