| About Mastermath |
| Courses & exams |
| Program & schedule |
| Registration & Evaluation |
| Links |
| Locations |
| Contact |
|
Toric Geometry
| Credits | 8 credit points |
| Instructors | Stienstra, Jan (Universiteit Utrecht) |
| J.Stienstra@uu.nl | |
| Aim | Introduction to geometry with emphasis on the coherence in a broad scala of geometric concepts without going to far into technical topics of specialized theories. |
| Description | Toric geometry deals with geometric objects on which a complex torus (i.e. a direct product of copies of C*) acts. This may take place in the context of, for instance, algebraic or symplectic geometry. The objects may arise from gluing charts, solving systems of equations or taking orbits for some group action. Via the character lattice of the torus many questions can be translated into questions about polytopes and linear algebra over the integers. Viewing polytopes and linear algebra as something concrete and easy this course aims at a lowbrow introduction to some basic principles and concepts in geometry. This includes, for instance, projective spaces, lattice polyhedra, subrings of polynomial rings and modules over these, vector bundles, functions/points-duality and the functor of points, differential forms, canonical bundles, Fano and Calabi-Yau varieties, Hamiltonian torus actions, moment maps, symplectic quotients, blowing-up and resolving singularities, McKay correspondence, Mirror Symmetry, hypergeometric functions. |
| Organization | In each class there will be three 45 minutes time slots with one devoted to exercises. |
| Examination | homework plus oral exam in January-February 2009 |
| Literature | If time permits lecture notes will be prepared and distributed during the course. Books on toric geometry (differing much in presentation of toric geometry ideas) are - Fulton: Introduction to Toric Varieties (Annals of Math. Studies 131; Princeton University Press, 1993) - Oda: Convex Bodies and Algebraic Geometry (Ergebnisse der Math. 3.Folge Band 15; Springer Verlag, 1988) - Guillemin: Moment Maps and Combinatorial Invariants of Hamiltonian T^n-spaces (Progress in Math. 22; Birkhauser, 1994) - Gelfand, Kapranov and Zelevinsky: Discriminants, Resultants and Multidimensional Determinants (Birkhauser, 1994) |
| Prerequisites | linear algebra and ring theory |