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71
votes
7answers
7k views

What is the field with one element?

I've heard of this many times, but I don't know anything about it. What I do know is that it is supposed to solve the problem of the fact that the final object in the category of schemes is ...
10
votes
3answers
2k views

ubiquitous quantum cohomology

Manin stressed that every projective scheme should have a quantum-cohomology structure. I'd like to know more about that. And since the varieties considered in texts about monodromy resp. vanishing ...
49
votes
3answers
8k views

What's the “Yoga of Motives”?

There are some things about geometry that show why a motivic viewpoint is deep and important. A good indication is that Grothendieck and others had to invent some important and new ...
5
votes
3answers
518 views

Solving “a, b, a+b have given divisors” problem

I've read an interesting article, math.NT/0409456 where you're just trying to solve a simple problem: For a given (finite) set of primes S find all solutions to an equation ...
11
votes
0answers
659 views

Kato's log motives

What are they and what are their intended uses? Does anyone have notes/slides of this talk? I am curious about "log motives" because there seems to exist a "log motivic yoga" among experts in ...
22
votes
4answers
2k views

Spectrum of the Grothendieck ring of varieties

Here's a problem that may ultimately require just simple algebraic-geometry skills to be solved, or perhaps it's very deep and will never be solved at all. From the comments, some literature and my ...