Studying Algebraic Geometry in Bergen

Algebraic geometry is a rich area in mathematics which studies geometric objects that (locally) can be defined as zero sets of polynomials. The field combines different techniques of modern abstract algebra, topology and complex analysis. A solid background in at least algebra and topology is therefore required to complete a master's degree with specialization in this field.

Both MAT211 Real analysis, which should be taken in the 3rd semester and MAT220 Algebra, which should be taken in the 4th semester, are compulsory courses for admission to the Master in Algebraic Geometry. In addition you need to complete MAT224 Commutative algebra (offered in the Fall) and one between MAT243 Manifolds (offered in the Spring) and MAT242 Topology (offered in the Fall).  We strongly recommend that, if you have the possibility you take both MAT242 and MAT243. In MAT243 you will get an introduction to smooth (or differentiable) manifolds, which basically are "geometric objects" that locally look like Euclidean n-space. In MAT224 you get an introduction to commutative algebra, which forms the basis for the theoretical foundation of modern algebraic geometry. Algebraic geometry combines the geometric thinking in the courses MAT242 and MAT243 with algebraic techniques from MAT224, so it is important to have a good foundation and inspiration from these courses.

Now you are ready for an introduction to the field of algebraic geometry in the courses MAT229 Algebraic Geometry, MAT320 Introduction to Sheaves and Schemes, and MAT322 Algebraic Geometry II. Here one learns about the objects being studied in modern algebraic geometry, namely algebraic varieties (or algebraic manifolds) and schemes These geometric objects are similar to the smooth manifolds one has become familiar with in MAT243: however they are locally defined by polynomial equations in n-space where the real numbers are replaced by arbitrary fields, such as the complex numbers, whence the connection to the complex analysis, or even finite fields, and hence there are connections to number theory. A typical phenomenon appearing in algebraic geometry, and that would not appear in the more general study of smooth manifolds, are singularities, that is, points where the geometric object is not "smooth".

The courses MAT229, MAT320 and MAT322 are irregular and unfortunately rarely lectured. However, it is important to learn the contents of these courses as early as possible in the master's program.  You should contact the study coordinator Kristine Lysnes or  Andreas Leopold Knutsen if you are interested in these courses. An alternative is to do one year study as an Erasmus student in one of the universities offering such courses. Furthermore, it is also relevant to include in the study plan the  courses MAT214 Complex Analysis (offered every second fall), MAT244 Algebraic Topology and  MAT342 Differential Geometry.