Description: The study of polynomials plays an important role in many areas of mathematics, and much of our intuition and tools for working polynomials comes from study polynomial with complex coefficients through the lens of complex geometry. This course will provide an accessible introduction to complex geometry through the study of Riemann surfaces, complex spaces of dimension $1$ like the complex plane. We will begin by developing the basic language of modern geometry (like the definition of a manifold and the definition of a sheaf) and end by developing powerful tools for study geometry like the Riemann--Hurwitz theory and the Riemann-Roch formula. As applications, we'll see results like a short proof of the polynomial analogue of Fermat's Last Theorem (I sketched the proof at the Graduate Colloquium in Fall).
This course should be of interest to anyone specializing in algebraic geometry, commutative algebra, or algebraic geometry as well as people outside these fields hoping to gain an appreciation for the topic. Provided there is sufficient enrollment, there will be a continuation of the course in spring semester. potential topics.