I disagree with the other comments. I can give you concrete reasons to care about algebraic geometry. I don't know enough of Grothendeick's vast generalizations

I disagree with the other comments. I can give you concrete reasons to care about algebraic geometry. I don't know enough of Grothendeick's vast generalizations yet to give you a good answer. I'm sure that I'll find applications of it, though ;) 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/rnoti-p380.pdf 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 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.