Introductory description

Algebraic geometry studies solution sets of polynomial equations by geometric methods. These are called algebraic varieties and arise in innumerable instances in physics as space-times, energy surfaces, phase spaces; in economics as indifference surfaces; in population genetics and other parts of biology; and in general, as the geometries of the “backgrounds” or “state-spaces” of mathematical or general scientific models.
This type of equations is ubiquitous in mathematics and much more versatile and flexible than one might as first expect (for example, every compact smooth manifold is diffeomorphic to the zero set of a certain number of real polynomials in R^N). On the other hand, polynomials show remarkable rigidity properties in other situations and can be defined over any ring, and this leads to important arithmetic ramifications of algebraic geometry.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

In the lectures, we will introduce the category of (quasi-projective) varieties, morphisms and rational maps between them, and then proceed to a study of some of the most basic geometric attributes of varieties: dimension, tangent spaces, regular and singular points, degree. Moreover, we will present many concrete examples, e.g., rational normal curves, Grassmannians, flag and Schubert varieties, surfaces in projective three-space and their lines, Veronese and Segre varieties etc.

Learning outcomes

By the end of the module, students should be able to:

• A first module in algebraic geometry is a basic requirement for study in geometry, number theory or many branches of algebra or mathematical physics at the MSc or PhD level. Many MA469 projects are on offer involving ideas from algebraic geometry.

Subject specific skills

Students will learn in the module how to analyse polynomial equations both quantitatively and qualitatively and get acquainted with several of the algorithms originating in the field, many of which are today extensively used in various types of computer algebra systems and other software. Since the subject is used in the various other disciplines mentioned above, these subject specific skills will often be transferable, too, and lead to employment opportunities in the respective fields.

Transferable skills

Another important skill, beneficial to personal development and elsewhere, communicated to students in most mathematics modules is the point of view and approach to problems provided by the scientific method: you should believe, in science and mathematics, as well as profitably quite a few other disciplines, logic and arguments, carefully drawn, and not authorities. Moreover, Algebraic Geometry in particular is also an excellent example where purely curiosity-driven research, free from corporate agendas, led to the breakthroughs and fundamental insights in the field that were later used in unforeseen ways in the above areas of applications that people profit from today.”
In addition, the field has also found extensive recent applications to robotics, computer-aided graphic design and cryptography.