MATH 431 Algebraic Geometry
Preliminary Notes
by
Ali Sinan Sertöz
January 2010

Mathematics can be described, in the most naive sense, as the art of making significant definitions and then attempting to classify all instances of that definition.

Generally we make a definition out of sheer curiosity.

What happens if we define an infinite dimensional vector space and put some inner product structure on it? This leads eventually to a better understanding of function spaces, solutions of differential equations and, most surprising of all, quantum mechanics.

What happens if we concentrate on the distance function and try to understand spaces through this local-to-global approach? This eventually leads to a revolutionary development in geometry and finds unexpected and certainly unintended applications in relativity theory.

What happens if we start defining classical geometric objects in a totally different way? Classically, conic sections are defined as the intersection of a cone with a plane perpendicular to a generator of the cone. Translating this to Cartesian coordinates, we see that a conic section is the set of points in the plane where a polynomial equation of degree two in the plane variables x and y vanish.

Now let us make a giant and courageous generalization and agree that geometry will study the set of points where a collection of polynomials each vanish. This of course should happen not necessarily in the plane but in n-space.

Does this lead to any meaningful theory?

It leads to Algebraic Geometry. I quote from Wikipedia:

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of polynomial equations in many variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations, as to find some solution; this leads into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique.

Does this apply anywhere? I again quote from Wikipedia:

Algebraic geometry now finds application in statistics, control theory, robotics, error-correcting codes, phylogenetics and geometric modelling. There are also connections to string theory, game theory, graph matchings, solitons and integer programming. Google scholar lists hundreds of more studies on algebraic geometry in biology, chemistry, economics, physics and of course other areas of mathematics. Recent work has suggested that algebraic geometry may be the key to solving the famous P vs NP problem.

 

For example, one of the crown theorems of Algebraic Geometry, the Riemann-Roch theorem, is applied in coding theory to actually construct good codes. We will study the meaning and implications of the Riemann-Roch theorem in the classification problem of curves from a purely geometric approach.

The plan of the course is as follows. We will first define the main objects of study. These are affine and projective varieties. Then we will define what we want to mean when two objects are the same. This requires that we describe what we want to mean by a map between varieties and when such maps detect the sameness of the two varieties, i.e. isomorphisms. We will then describe which properties of varieties are easy to detect so that we concentrate our attention on them. We will apply these techniques in understanding some special curves and surfaces. Finally we will attempt to say something about the totality of all curves. This is where we will be in a position to admire and appreciate the depth and beauty of the Riemann-Roch theorem.

The web page of the course can be reached by following the Math 431 link on my web page. Most announcements about the course will be posted there. I will use email to communicate with you outside class.

Homework is an essential part of learning. You can discuss the solution of homework problems with others but you should write your homework on your own. You are strongly encouraged to visit me during my office hours and ask questions including those questions which help you solve homework problems.

The grading policy will be a mixture of curve and catalogue systems; any grade average below 40% is an immediate F, while the other grades will be assigned a letter grade according to curve.

The only way to learn anything in any branch of mathematics is to start with the main concepts and to understand the meaning and scope of those concepts. Notice that there is no such thing as partially understanding something, and that will be reflected in my policy on giving partial credits to unsolved exam or homework problems.

If you come to class regularly, participate in class discussions, do your homework on time, then you will not only be successful in this course but, much to your own amazement, you will find yourself immensely enjoying the delicate gems of Algebraic Geometry. Anyone willing to continue an MSc in Algebraic Geometry is welcome, provided that I see you having fun with geometry.

Office: SA-121
Phone: 290 1490
Web Page: http://www.bilkent.edu.tr/~sertoz



Contact address is: