How has modern algebraic geometry affected other areas of math?

Question

I have a friend who is very biased against algebraic geometry altogether. He says it's because it's about polynomials and he hates polynomials. I try to tell him about modern algebraic geometry, scheme theory, and especially the relative approach, things like algebraic spaces and stacks, etc, but he still thinks it sounds stupid. This stuff is very appealing for me and I think it's one of the most beautiful theories of math and that's enough for me to love it, but in our last talk about this he asked me well how has the modern view of algebraic geometry been useful or given cool results in math outside of algebraic geometry itself. I guess since I couldn't convince him that just studying itself was interesting, he wanted to know why else he'd want to study it if he isn't going to be an algebraic geometer. But I found myself unable to give him a good answer that involved anything outside of algebraic geometry or number theory (which he dislikes even more than polynomials). He really likes algebraic topology and homotopy theory and says he wants to learn more about the categorical approaches to algebraic topology and is also interested in differential and noncommutative geometry because of their applications to mathematical physics. I know that recently there's been a lot of overlap between algebraic topology/homotopy theory and algebraic geometry (A1 homotopy theory and such), and applications of algebraic geometry to string theory/mirror symmetry and the Konstevich school of noncommutative geometry. However, I am far from qualified to explain any of these things and have only picked up enough to know they will be extremely interesting to me when I get to the point that I can understand them, but that's not a satisfactory answer for him. I don't know enough to really explain how modern algebraic geometry has affected math outside of itself and number theory enough to spark interest in someone who doesn't just find it intrinsically interesting.

So my question are specifically as follows:

How would one explain how the modern view of algebraic geometry has affected or inspired or in any way advanced math outside of algebraic geometry and number theory? How would one explain why modern algebraic geometry is useful and interesting for someone who's not at all interested in classical algebraic geometry or number theory? Specifically why should someone who wants to learn modern algebraic topology/homotopy theory care or appreciate modern algebraic geometry? I'm not sure if this should be CW or not so tell me if it should.

Answer 1

As others have suggested, your friend is getting it backwards. He's like a hammer asking what a carpenter is useful for.

Given a field (of mathematics, say), there are typically some fields that are more structured than it and others that are less structured. In mathematics, people often say the more structured ones are 'harder', and the less structured are 'softer'. For instance, in increasing order of hardness, we have sets, topological spaces, topological manifolds, differential manifolds, complex manifolds, complex algebraic varieties, algebraic varieties over the rational numbers, integral algebraic varieties. These are in a linear order, but if you throw in other subjects, you'll get a non-linear one. (p-adic algebraic geometry and Riemannian geometry immediately come to mind.)

(I think Gromov has some remarks at the end of an ICM address where he talks about this and gives other examples. Also, don't confuse 'harder' and 'softer' in this sense with what they mean in the sciences, which is essentially 'more precise' and 'less precise'. For instance, in science people say that biology is softer than chemistry. In fact, the two meanings are opposites because in science, more structured objects are less amenable to a precise analysis. But this typically isn't the case in mathematics.)

Now given a subject S and a harder subject H, it's usually true that most objects in S don't admit the structure of an object in H. For instance, most topological manifolds don't admit a complex structure. On the other hand, for the objects of S that do admit such a structure, their theory from the point of view of H is typically much richer than that from the point of view of S. For instance, the study of Riemann surfaces as topological spaces is less rich than their study as complex manifolds. You might say that softer subjects are broad and flexible and harder ones are rich and rigid. Mathematicians tend to view subjects that are softer than their specialty as general nonsense, and harder ones as excessively particular.

This is not to say that a soft field is easier or less interesting than a harder one. Even if it is true that the directly analogous question in the soft subject is easier (e.g. classify Riemann surfaces topologically rather than holomorphically), it just means that the people in the soft subject can move on and study more sophisticated objects. So they just get stuck later rather than sooner. For instance, over the past 50 years, a big fraction of the best number theorists have been studying elliptic curves over number fields. Now elliptic curves over the complex numbers are much easier (I think there hasn't been much new since the 19th century), so the algebraic geometers just moved on to higher genus or higher dimension and are grappling with the issues there, issues that are way out of reach in the presence of arithmetic structure.

Now my main point here is that soft subjects were typically invented to break up the study of harder ones into smaller pieces. (This is surely something of a creation myth, but one with a fair amount of truth.) For instance, the real numbers were invented to break up the study of polynomial equations into two steps: when a polynomial has a real solution and when that real solution is rational. I know very little about modern analysis, but I think that much of it was invented to do the same with differential equations. You first find solutions in some soft sense and then see whether it comes from a solution in the harder sense of original interest.

So the role of soft subjects is to aid in the study of harder ones---people usually don't ask for applications of partial differential equations to the study of topological vector spaces, but it's considered a mark of respectability to ask for the opposite. Similarly, no one talks about applications of engineering to mathematics. Since algebraic geometry is at the hard end of the spectrum above, there aren't many fields in which it is natural to ask for applications. Number theory, or arithmetic algebraic geometry, is harder and of course there are zillions of applications there, but that's not what your friend wants. Just about all mathematicians work in a subject that is softer than some and harder than others (and if you include non-mathematical subjects, then all mathematicians do). That's all good---it takes a whole food chain to make an ecosystem. But it's backwards to ask about the nutritional value of something that typically eats you.

[This picture of mathematics is of course simplistic. There are examples of hard subjects with applications to softer ones. See Donu Arapura's answer, for example. There are also applications of arithmetic algebraic geometry to complex algebraic geometry. For instance, Grothendieck's proof of the Ax-Grothendieck theorem, or the proof of the decomposition theorem for perverse sheaves using the theory of weights and the Weil conjectures. But I think it's fair to say that such applications are the exception---and are prized because of it---rather than the rule.]

Answer 2

I tend to shy away from questions like this, but for some reason I'm up at a strange hour with nothing better to do. And the news is too depressing. Actually I agree with your friend, "algebraic geometry" does sound a bit dull. We should have come up with a cooler name, but it's too late.

Anyway, here's something. Say you have a compact Riemann surface equipped with a positively curved metric. Then by Gauss-Bonnet, the Euler characteristic is positive. Therefore, modulo basic facts about Riemann surfaces, it must be the Riemann sphere. Now consider a higher dimensional version. Suppose that $X$ is a compact complex manifold with a Kähler metric with positive curvature in a suitable sense (i.e. positive bisectional curvature). Then Frankel conjectured that it must be a projective space. There are two proofs, one due to Siu and Yau uses harmonic maps and another due to Mori using algebraic geometry in positive characteristic. For the second proof, first observe that $X$ is projective algebraic by Kodaira's embedding theorem. Then the curvature condition implies that the tangent bundle is positive in the sense of algebraic geometry (i.e. ample). Mori proved that projective spaces are the only varieties with positive tangent bundle. Scheme theory is needed to move the problem into characteristic $p$, where the main argument takes place.

Answer 3

Usually it works the other way around: things appear in topology first and then people realize that those things may have analogs in algebraic geometry. Etale cohomology is perhaps the best known example.

But let me give a counter-example (I wrote something similar as a comment to a recent question but I can't find it now). In topology there is the Lefschetz formula, which expresses the alternating sum of the traces of the cohomology endomorphisms induced by a smooth self-mapping of a smooth manifold in terms of the local contributions of the fixed points, assuming those are non-degenerate. There is a generalization of this for self-maps of arbitrary finite CW-complexes. The contribution of each fixed point is local, i.e. it can be determined by looking at the map in an arbitrarily small neighborhood of the point. In particular, if there are no fixed points, the alternating sum of the traces is zero.

Inspired by this, Grothendieck proved in SGA 5 an algebraic version of the Lefschetz formula, without smoothness or completeness assumptions. It also works for more general sheaves than the constant sheaf. Inspired by this, Goresky and MacPherson gave a topological version of the formula, which, under some assumptions, allows one to calculate the contribution of each component of the fixed point set. See "The local contribution to the Lefschetz fixed point Formula", Inv. Math. 111, 1993, 1-33.

Answer 4

A lot of recent research in algebraic topology, particularly in stable homotopy theory, makes essential use of perspectives coming from algebraic geometry. It particularly aids in the conceptualization of computational results in the subject and the so-called "chromatic filtration".

Rather than me writing at length, you should see Mike Hopkins' ICM lecture for a readable introduction to this connection.

Answer 5

Atiyah and Singer proved their famous "index theorem" in topology using ideas and methods directly inspired by Grothendieck's proof of his huge generalization of Riemann-Roch's theorem, a proof that makes a prototypical use of several of the newest aspects of modern (that is Grothendieck's) algebraic geometry, for example the insistence in working in a relative rather than absolute situation, that is for morphisms rather than simply for objects.

Answer 6

My favorite such example is Stanley's proof of McMullen's conjecture characterizing the number of faces that a simplicial polytope can have.

I summarize the proof below (although Stanley's article is not much longer) in order to illustrate the extent to which "modern algebraic geometry" is used:

(1) Build a toric variety from the dual polytope

(2) Count points over a finite field. Since the toric variety is built by replacing each face of the dual polytope by an algebraic torus of this dimension, the number of points over a field with $q$ elements is

$$\sum_{faces} (q-1)^{\mbox{dimension of face}}$$

where we include the enclosed face "inside" the polytope

(3) The original polytope being simplicial, hence the dual polytope being simple, means the toric variety is rationally smooth. In particular, its cohomology = its intersection cohomology

(4) Purity of intersection cohomology, hence of cohomology = intersection cohomology, together with the point count above, implies (by the Weil conjectures) that the variety has no odd cohomology and that its $2i$'th Betti number is the coefficient of $q^i$ above.

(5) Intersection cohomology obeys Hard Lefschetz, giving a condition on this Poincare polynomial, translating into a condition on the faces. (polytopes with arbitrary face numbers obeying this condition had already been constructed.)

Example: For a polygon, we have $$(q-1)^2 + (\#faces)(q-1) + (\#vertices) = q^2 + (\#faces-2) + 1$$ So by the Hard Lefschetz theorem, every lattice polygon has at least 3 sides.

Answer 7

This is not likely to satisfy the OP, but the study of the dynamics of billiards in polygons has a lot to do with things like variations of Hodge structures and slopes of divisors on Moduli space.

The connection is via Teichmuller curves and related objects.

Answer 8

My own knowledge of algebraic geometry is not yet even at a "cocktail party" level, but I'd also love to learn a bit about why (and how) I should care about AG. But I have two points to mention here.

1. Lior Pachter and Bernd Sturmfels have edited a book called Algebraic Statistics for Computational Biology, and in that book they argue how the language of algebraic geometry offers some help for tackling problems in computational biology + statistics.

2. Another exciting perspective might be offered by the work of Ketan Mulmuley on Geometric complexity theory, where Mulmuley is using algebraic geometric ideas to tackle the P vs NP problem.

Answer 9

The size of Fourier coefficients of modular forms can only be studied (so far) via the use of very sophisticated tools from Algebraic Geometry. Of course, one could argue that modular forms are part of Number Theory, but this is not how they arose and they appear in many other branches of mathematics (combinatorics or theoretical physics for example).

Answer 10

If you study representation theory of groups or algebras, then representation varieties are useful.

Answer 11

The notion of (Grothendieck) topos is coming right from algebraic geometry, yet they are very useful in homotopy theory, see for example Lurie’s book "Higher Topos Theory".
In particular, the homotopy category is the archetypal example of $(\infty,1)$-topos.

Answer 12

Here are some recent papers in discrete geometry where algebraic methods are used, in particular some famous theorems of algebraic geometry, like Bezout's or Milnor-Thom theorem, are frequently applied in this area.

http://arxiv.org/abs/1011.4105

http://arxiv.org/abs/0812.1043

http://arxiv.org/abs/1102.5391

http://arxiv.org/abs/0905.1583

Answer 13

The Gelfand-Naimark theorem shows that every compact Hausdorff space can be regarded as an affine scheme determined up to homeomorphism by its scheme-structure. See Introduction to Algebraic Geometry, chapter 4.

The idea is as follows: If X is a compact Hausdorff space and C(X) is the ring of continuous, complex-valued functions, then X is homeomorphic to specm C(X) (with the Zariski topology).