## Short course: A Novice's Road Map of Research Trends in Contemporary Algebraic Geometry

Karen E. Smith (Keeler Distinguished Professor, University of Michigan, Ann Arbor and Vieraileva Professori, University of Jyväskylä)

17.-19. May 2010, University of Helsinki

### Time and place

The lectures will be held in Exactum (Kumpula, University of Helsinki) as follows:

Monday 17. May 10 - 12 o'clock in lecture room CK112

Tuesday 18. May 12 -14 o'clock in lecture room CK112

Wednesday 19. May 12 - 14 o'clock in lecture room CK112

### Scope

2 credit points (lectures + extra course work). A set of Exercises for the extra course work is found here. **Please note**: it is not neccessary to complete all exercises in order to receive credits.

### Level

Advanced studies

### Abstract

Algebraic geometry, the study of solutions sets for systems of polynomial equations, is one of the oldest branches of mathematics, beginning already with the Greeks' deep studies of conic sections. Yet is also one of the most central and active areas of modern mathematics, with connections and applications to many areas of mathematics and beyond. The lectures will introduce students to the basic definitions and questions of algebraic geometry, using many concrete examples. The main goal is to instill in the listener an understanding and appreciation of some of the main research directions in algebraic geometry today.

**Lecture 1**: What is Algebraic Geometry?

(Topics: Definition of algebraic variety, projective varieties, maps of varieties, many examples. The place of algebraic geometry in mathematics: Connections with algebra, differential geometry, applied math, representation theory, arithmetic geometry and number theory, complex geometry).

**Lecture 2**: Research trends: resolution of singularities and birational geometry.

(Topics: resolutions of singularities, rational varieties, birational classification of varieties)

**Lecture 3**: Moduli spaces

(Topics: parametrizing families of varieties, compactifying parameter spaces, moduli spaces of compact Riemann surfaces)