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.

23more comments