Hands-on algebraic geometry

Description

Algebraic geometry is a branch of mathematics that studies geometric objects defined by polynomial equations and their properties. It deals with spaces called algebraic varieties, which are sets of solutions to polynomial equations in several variables. Since polynomial equations appear in many mathematical models, algebraic geometry has natural applications to various scientific fields, including physics, computer science, cryptography, and robotics.

Despite the high usefulness of these methods, the contemporary theory of algebraic geometry relies on very abstract notions of commutative algebra, algebraic topology and category theory. For this reason, the field is sometimes regarded as particularly technical, daunting for beginners, and even dubbed as elitist. But lots of algebro-geometric objects such as plane curves and quadratic forms, can be also explored without the technical machinery above, and a different, “hands-on” approach to these object is not only possible but sometimes even desirable both for the applications and as a motivation to learn more of the abstract theory.

This point of view has been championed by great algebraic geometers such as Miles Reid and Igor Shafarevich, resulting in beautiful books that introduce the subject at the undergraduate level. In this project, the students will start their exploration from these texts and will then focus on a specific object or application that will lead them to develop their unique perspective on this fascinating theory.

As the solutions of a system of polynomial equations can be searched in different rings, algebraic geometry also comes in different flavours: complex algebraic geometry, real algebraic geometry, finite field algebraic geometry, arithmetic geometry, etc. Each of these fields has distinct properties and students can choose to explore one or more of them.

Essential modules

It is essential that students have taken Algebra II and Complex Analysis II. It is suggested (but not essential) that students take Geometry III during the course of the project.

Resources

“Undergraduate Algebraic Geometry” by Miles Reid [Link: https://homepages.warwick.ac.uk/staff/Miles.Reid/MA4A5/UAG.pdf]
“Basic Algebraic Geometry” by Igor R. Shafarevic [Library link]