The course is largely based on Vakil's book in progress, which he titled The Rising Sea, after a quote from Grothendieck:

> I can illustrate the ... approach with the ... image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! . . .

> A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration ... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it ... yet finally it surrounds the resistant substance.

Here is, roughly speaking, a theorem that illustrates what modern algebraic geometry is good for.

Theorem. If you have two conic sections in the plane, which don't share a common component, then they intersect in exactly four points.

Here a "conic section" is just any quadratic equation in two variables, e.g. x^2 + y^2 + 3xy + 4x = 1. "Common component" means that the conic sections are identical or overlap in an entire line.

Now, here are some "examples".

Concentric circles, e.g. x^2 + y^2 = 1 and x^2 + y^2 = 4. They don't intersect at all, right? Ah, except you forgot to count points over the complex numbers, where they do.

Tangency, e.g. y = x^2 and x^2 + (y - 1)^2 = 1. These curves are tangent at (0, 0), so you have to count this point with multiplicity.

One major goal of modern algebraic geometry is to wrap these sorts of considerations into the foundations. So you have to work a lot harder to even say what a conic section is, or what it means for two of them to intersect. But, once you've laid the foundations in this manner, there are no special cases.

How is this possible? Are you saying that there are pairs of complex numbers (x,y) such that x^2 + y^2 = 1 = 4?

A correct example of what I had in mind is x^2 + y^2 = 1 and (x - 2)^2 + y^2 = 1.

That said, algebraic geometry takes care of this "bad example" also. This is done exactly as alcolade explained: consider "points at infinity" in projective space.

https://en.wikipedia.org/wiki/Gromov%E2%80%93Witten_invarian...

A couple of his contributions:

1. A far reaching generalization of the Riemann-Roch theorem, which is now called the Grothendieck-Riemann-Roch theorem. As with a great deal of his work, it's about drawing conclusions about global structure from local data:

https://en.wikipedia.org/wiki/Riemann%E2%80%93Roch_theorem

2. Creating the machinery used to solve the Riemann hypothesis for finite number fields.

https://en.wikipedia.org/wiki/Weil_conjectures

His work wasn't focused on solving particular problems so much as it was on finding the right language with which to describe problems. The philosophy is that, with the right language, your proofs should become obvious. This is somewhat in the same spirit as Leibniz's quest for the 'Universal Characteristic'.

https://en.wikipedia.org/wiki/Characteristica_universalis

https://www.youtube.com/watch?v=tX4H_ctggYo

Web site: https://bivector.net/

Algebraic geometry has to do with finding geometric structure in (the solutions of) systems of algebraic equations. Algebraic geometry is potentially useful whenever you're doing this, so cryptography, coding theory, optimization, ...

Algebraic geometry, broadly construed, is the study of solutions to polynomial equations, just as linear algebra is the study of linear equations. I hope we'll both agree that linear algebra is used literally /everywhere/. Algebraic geometry sees a fair bit of use, mostly through the use of objects called as a Grobner basis, which allows us to solve for large systems of polynomial equations efficiently. The book "Ideals, Varieties, and Algorithms" goes into detail about this sort of "effective algebraic geometry" that's used to solve systems.

One algebraic geometry tool I have personally used is known as cylindrical algebraic decomposition. It's quantifier elimination for real polynomials, so you can use it to ask questions about behaviours of polynomials and compute answers to them. So if you have a problem that can be phrased as "can polynomials do this" [think safety/liveness properties], you can use cynlindrical algebraic decompostion to answer such questions.

My understanding is that Grothendeick's vast generalizations and perspectives allows one to take "geometric insight" gained from thinking about polynomials, and translate them into many many more settings. He had visions of number theory, so he built a lot of (very general) machinery, with an eye towards number theoretic problems. But really, the machinery is so general, and casts deep geometric insight on such a basic object of mathematics (rings and their ideals, as well as "what is gemometry") that I can see myself using what AG I know many, many times in the coming years.

Then there is also a whole branch of math called as tropical geometry which transports a lot of the AG machinery into piecewise functions. This is incredibly useful, because the AG machinery is very powerful, while piecewise linear functions are ubiquitous. It's early days [to my understanding] of tropical geometry, and I know woefully little about it. But I feel that if I learn more, I can gain a lot of useful algorithms and insight from the field. For more, there is an AMS intro article: https://www.ams.org/publications/journals/notices/201704/rno...

For more topological flavoured answer, check out the book Elementary Applied Topology by robert ghirst: https://www.math.upenn.edu/~ghrist/notes.html. It chronicles large swathes of algebraic topology and related fields in a clean, accessible style with applications.