Mathematics 562 - Intro to Algebraic Geometry

Algebraic geometry provides a precise language for describing geometric shapes defined by systems of polynomial equations - algebraic varieties - with applications ranging from configuration spaces in physics to parametric models in statistics. This versatile language is used across mathematics and has fueled recent advances not only within algebraic geometry itself but also in representation theory, number theory, symplectic geometry, and other fields. Breakthroughs in algebraic geometry have been recognized with Fields Medals in nearly every award cycle, contributing to surging interest in the subject among young mathematicians worldwide. The goal of the course is to introduce both the theoretical foundations and the range of applications of algebraic geometry. Topics include basic examples such as elliptic curves and low-degree algebraic surfaces; computer implementation of algorithms for ideals in polynomial rings; enumerative geometry of projective varieties; singularities of algebraic varieties; and applications to semidefinite optimization, maximum-likelihood estimation, and machine learning.

Math 411