Algebraic geometry is a pillar of modern mathematics. Classically, it is the study of the zero level sets of systems of polynomial equations; such simultaneous solution sets are known as algebraic varieties. With applications to robotics, cryptography, enumerative combinatorics and more, the subject has significant utility in the contemporary world. However, algebraic geometry also has a host of intersections with pure mathematics, notably with number theory and topology.

Within the study of algebraic geometry, toric varieties form a class of algebraic variety whose rich combinatorial structure allows one to develop a greater intuition for certain highly abstract geometric spaces, such as projective space. Particularly nice toric varieties admit a model as polytopes (n-dimensional polygons), yielding a bridge between convex and algebraic geometry. This bridge allows toric varieties to be studied and understood from a more grounded perspective, and even allows one to read off aspects of their geometry from (comparatively) simple data.

This talk is intended to be a crash course on toric varieties, and will assume cursory knowledge of topology (i.e., a  familiarity with open sets), some experience with complex numbers, as well as an awareness of the fundamentals of groups and rings. However, much of the relevant material will be quickly refreshed (and at worst may be black boxed), and so those who are inquisitive but unfamiliar should feel explicitly invited to attend.