MATH5535 is a Mathematics Special Topics in Pure Mathematics Honours or Master course. See the course overview below. The Topic for 2022 will be **Introduction to Algebraic Geometry**.

**Units of credit:** 6

**Prerequisites: ** MATH3701 and MATH3711.

Familiarity with modules and basics of category theory, introduced in MATH5735 is highly recommended.

**Cycle of offering: **Term 2 2022

**Graduate attributes:** The course will enhance your research, inquiry and analytical thinking abilities.

**More information:** A Course outline will be provided here closer to Term 2.

The course outline contains information about course objectives, assessment, course materials and the syllabus, but only if a course actually running with this course code.

If you are currently enrolled in MATH5535, you can log into UNSW Moodle for this course.

#### Course Overview

The aim of this course is to put the theories of rings, fields and modules in a geometric context. This perspective is of fundamental importance in various branches of mathematics, including number theory, representation theory, differential geometry and mathematical physics. In algebraic geometry varieties play the role of manifolds in differential geometry.

Our first aim is the study of affine and projective varieties and maps between them. Next, we will go through local properties of these objects and discuss what is meant by singularity. An important goal for this course is the study of the famous Riemann-Roch theorem for curves, for which we will need the notions of Divisors and Differential Forms. Riemann-Roch traces a deep connection between topology and complex analysis, through purely algebraic methods. As such it provides a showcase for the power of algebro- geometric techniques. We will end the course by discussing a few applications of this theorem and briefly review some more advanced topics in the field.