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Talk -- What is Algebraic Geometry and Why Should You Care?

01 May 2023 - Tags: my-talks

So an embarrassing amount of time ago (Feburary 17?) I gave a talk for the undergraduate math club titled “What is Algebraic Geometry, and Why Should You Care?”. I think it went quite well, and the audience seemed like they had a good time. I really wanted to have the talk recorded, since this is exactly the kind of talk I would have wanted to see as an undergrad and I think it should be available to more people. Unfortunately we weren’t able to make it happen, so we’ll have to wait until the next time I give this talk1.

I actually told the audience that I would have a blog post with the slides posted later that night, but uh… clearly that didn’t happen, haha. In my defense, I really wanted to add some sage code to this post in order to replicate some of the demos that I did during the talk, and to let readers play around with some of this stuff themselves. I never really built up the energy to write those demos, and I picked up two more projects along the way2 so the post never got made.

Well the other day I bumped into some students from the math club, and they teased me for never posting the slides! To be totally honest, I was surprised that they had noticed, haha. I’m happy to see that people were actually interested in reading them, so that interaction was exactly the motivation I needed to finally post this! The unfortunate fact, though, is that I’m still to busy to really make the demos as nice as I would like to… So we’re going to have to go without. You can see the kind of thing I would have made at an old blog post here, and you can imagine 3d versions of some of the pictures in the slides.

As a quick summary of what’s in the talk, I make an analogy to linear algebra (which is about as elementary as I think you can go while still giving an honest look into how the machinery works). Here we study a close connection between the algebra of linear equations and the geometry of linear subspaces! We can build a dictionary between these two worlds where, loosely:

So the geometers want powerful algebra to solve their problems, and conversely the algebraists want to make more general ideas of “geometry” that allow them to visualize the algebra they’re studying abstractly3. Then the geometers start studying these more complicated structures as geometry, and they ask for more algebra to study them, but then these newer more complicated algebraic gadgets work in more general settings, so we build a more general notion of “geometry” to visualize all of them at once, and so on.

Round and round we go, until we get to today, where algebraic geometry has a fearsome reputation for being incredibly abstract and challenging4. Our “geometric objects” are now things like schemes and topoi and stacks and they’re valued with points in an arbitrary ring (as opposed to the classical case of complex numbers), and it’s hard to believe that anyone can reasonably say they’re doing “geometry” when studying these. But of course, with practice, you really can come to visualize these objects! They really do deserve to be called geometric! And the benefit of taking the time to do this is that suddenly everything feels like it has geometric content, and the whole of math (well, a lot of it at least) becomes more fun!

Perhaps most surprisingly, though, is that these high abstraction notions of “geometry” really solve problems. This is something I emphasize at the end of the talk, since I think that beautiful math should solve problems, and it’s important to remember how these computations tether us to reality!

At the most basic level, the theory of gröbner bases allow us to computationally solve polynomial equations! This is useful almost everywhere in the real world, since polynomials arise naturally in physics, engineering, and really anywhere math is used to model the world!

Then we have “middle abstraction” tools from algebraic geometry, like schemes. These arise naturally as limiting cases of “honest” geometric objects, and they remember more information that the naive approach to these situations. Importantly this bonus information helps us solve problems! For instance, if we intersect the parabola $y = x^2$ with the line $y=0$, we get a single point $(x,y)=(0,0)$. However, this “single point” should really be counted twice if we want to make our formulas work properly.

Geometrically this is because if we wiggle the $y=0$ line slightly5 we actually get TWO intersection points ($y=\epsilon$ intersects $y = x^2$ at $\pm \sqrt{\epsilon}$ for each $\epsilon \neq 0$) and we want the number of intersection points to be continuous.

But even though this single point (viewed as a “classical” geometric object) doesn’t know it, if we take the intersection as schemes (the more advanced geometric object) we remember more about where we came from, and the scheme “has one point” but that point “counts twice”!

Indeed, the ring $k[x,y] \big (y = x^2, y = 0)$ gives the intersection, and this becomes $k[x] \big / (x^2 = 0)$. This has a single point6 yet it’s 2-dimensional (it has a basis ${1,x}$). The nilpotency of $x$ says our scheme sees something “infinitesimally bigger” than just the point $x=0$. In fact, it “remembers” that it came from the intersection of a curve with its tangent line (that is, we get not just a point of intersection, but an infinitesimally small “line segment” of intersection)!

Much of this “middle abstraction” level is learning to visualize these infinitesimals and other similar phenomena, which (as we’ve seen) contain the answer to many geometric problems7!

Finally the “high abstraction” machinery, like stacks allow us to solve problems for whole families of geometric objects at once. I won’t say more about this here, since I want this post to be fairly short, but the relevant buzzword is “moduli stack”. These are geometric objects where a single point can still “have symmetry”.

Separately, topoi are geometric objects which are characterized not by their points, but by their sheaves. Sheaves are certain objects associated to a geometric space, and we can rephrase a lot of geometry in terms of operations on sheaves. A (grothendieck) topos, then, is a category of sheaves that “looks like” a category of sheaves on some geometric space. Then we can pretend that it really is sheaves on some space, and use the fact that we can do geometry in terms of just sheaves in order to ask what properties this space must have!

This turns out to be extremely useful, even for surprisingly simple to state problems. After all, hard problems demand hard solutions, and sometimes easy to state problems are still hard to solve! Fermat’s Last Theorem, for instance, is famously easy to state, but requires the use of both stacks and topoi in order to complete the proof!

Lastly, I invite the people who have tuned out8 to tune back in for a more concrete ending to the talk. We show how to use a simple idea from algebraic geometry in order to classify all pythagorean triples! The fact that geometry should be useful here is surprising, and this is a very concrete example which forms the inspiration for the very fruitful subject of arithmetic geometry.

As usual, here is the title and abstract, and slides are available here.

What is Algebraic Geometry, and Why Should You Care?

Algebraic Geometry, at its core, is the study of solutions of polynomial equations. However in the 20th century it gained a reputation for both its abstraction and difficulty. In this talk we will outline what connects algebra and geometry, and explain what led to the explosion in abstraction that occurred in Grothendieck’s school. We will end the talk with some applications in both pure and applied math. We assume no prerequisites at all, except possibly some linear algebra.

  1. It went well enough that I’ll probably add it into my list of “talks I’m happy to give without much preparation”, so there’s likely to be more opportunities. 

  2. Both of which are super interesting problems, that I’ll likely talk about someday, but neither of which I’m comfortable talking about right now. 

  3. Since this can help them intuit what’s true! 

  4. Which is somewhat well deserved, but I would argue it’s no harder than what a lot of functional analysts get up to… 

  5. Grown-up mathematicians would probably prefer I say “perturb” instead of “wiggle”, but they can get over it. 

  6. If you’re in the know, this means “a single prime ideal”. See here, for instance. 

  7. In fact, this scheme-theoretic intersection still doesn’t always compute intersections properly… In some extreme situations it still “forgets too much”. This is one of the motivations behind the “high abstraction” notion of derived algebraic geometry, which is even harder to visualize (I certainly can’t… At least not yet) but which correctly solves even more problems

  8. And anyone who tuned out for some of this discussion of high abstraction tools would of course be completely forgiven!