… in an ideal world, people would learn this material over many years, after having background courses in commutative algebra, algebraic topology, differential geometry, complex analysis, homological algebra, number theory, and French literature.
In a high school algebra class, you learn how to solve polynomial equations in one variable, and systems of linear equations. You might reasonably ask “So when do we combine these and learn to solve systems of polynomial equations?” The answer would be “Maybe years from now, but most likely never.” There are systematic ways to solve systems of polynomial equations, but you’re unlikely to ever see them unless you study algebraic geometry.
Here’s an example from . Suppose you want to find the extreme values of
on the unit sphere using Lagrange multipliers. This leads to the following system of polynomial equations where λ is the Lagrange multiplier.
There’s no obvious way to go about solving this system of equations. However, there is a systematic way to approach this problem using a “lexicographic Gröbner basis.” This transforms the problem from into something that looks much worse but that is actually easier to work with. And most importantly, the transformation is algorithmic. It requires some computation—there are numerous software packages for doing this—but doesn’t require a flash of insight.
The transformed system looks intimidating compared to the original:
We’ve gone from four equations to eight, from small integer coefficients to large fraction coefficients, from squares to seventh powers. And yet we’ve made progress because the four variables are less entangled in the new system.
The last equation involves only z and factors nicely:
This cracks the problem wide open. We can easily find all the possible values of z, and once we substitute values for z, the rest of the equations simplify greatly and can be solved easily.
The key is that Gröbner bases transform our problem into a form that, although it appears more difficult, is easier to work with because the variables are somewhat separated. Solving one variable, z, is like pulling out a thread that then makes the rest of the threads easier to separate.
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 David Cox et al. Applications of Computational Algebraic Geometry: American Mathematical Society Short Course January 6-7, 1997 San Diego, California (Proceedings of Symposia in Applied Mathematics)
“Tate helped shape the great reformulation of arithmetic and geometry which has taken place since the 1950’s.” — Andrew Wiles
At the Heidelberg Laureate Forum I had a chance to interview John Tate. In his remarks below, Tate briefly comments on his early work on number theory and cohomology. Most of the post consists of his comments on the work of Alexander Grothendieck.
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JT: My first significant work after my thesis was to determine the cohomology groups of class field theory. The creators of the theory, including my thesis advisor Emil Artin, didn’t think in terms of cohomology, but their work could be interpreted as finding the cohomology groups H0, H1, and H2.
I was invited to give a series of three talks at MIT on class field theory. I’d been at a party, and I came home and thought about what I’d talk about. And I got this great idea: I realized I could say what all the higher groups are. In a sense it was a disappointing answer, though it didn’t occur to me then, that there’s nothing new in them; they were determined by the great work that had already been done. For that I got the Cole prize in number theory.
Later when I gave a talk on this work people would say “This is number theory?!” because it was all about cohomology groups.
JC: Can you explain what the great reformulation was that Andrew Wiles spoke of? Was it this greater emphasis on cohomology?
JT: Well, in the class field theory situation it would have been. And there I played a relatively minor part. The big reformulation of algebraic geometry was done by Grothendieck, the theory of schemes. That was really such a great thing, that unified number theory and algebraic geometry. Before Grothendieck, going between characteristic 0, finite characteristic 2, 3, etc. was a mess.
Grothendieck’s system just gave the right framework. We now speak of arithmetic algebraic geometry, which means studying problems in number theory by using your geometric intuition. The perfect background for that is the theory of schemes. ….
Grothendieck ideas [about sheaves] were so simple. People had looked at such things in particular cases: Dedekind rings, Noetherian rings, Krull rings, …. Grothendieck said take any ring. … He just had an instinct for the right degree of generality. Some people make things too general, and they’re not of any use. But he just had an instinct to put whatever theory he thought about in the most general setting that was still useful. Not generalization for generalization’s sake but the right generalization. He was unbelievable.
He started schemes about the time I got serious about algebraic geometry, as opposed to number theory. But the algebraic geometers classically had affine varieties, projective varieties, … It seemed kinda weird to me. But with schemes you had a category, and that immediately appealed to me. In the classical algebraic geometry there are all these birational maps, or rational maps, and they’re not defined everywhere because they have singularities. All of that was cleared up immediately from the outset with schemes. ….
There’s a classical algebraic geometer at Harvard, Joe Harris, who works mostly over the complex numbers. I asked him whether Grothendieck made much of a difference in the classical case — I knew for number theorists he had made a tremendous difference — and Joe Harris said yes indeed. It was a revolution for classical algebraic geometry too.
Related: Applied number theory