During the first week (August 1-5) there will be 4 graduate level courses in Buenos Aires. Schedule.
Each course will consist of 5 classes of 60 to 90 minutes, which will include in some cases exercises to be worked out by the students. All lectures will be in English.
The first two topics are basic in algebraic geometry and have applications to physics, to dynamical systems, and to effective computations. The third topic is an emerging field, with applications to mirror symmetry, to degenerations of algebraic varieties, to algebraic statistics, etc. The fourth topic is an established application of the theory of algebraic curves over finite fields.
Michel Brion (Grenoble) is also a coorganizer of this School.
Lecturer: Joe Harris (Harvard)
The basics of intersection theory and its applications: cycles, rational equivalence, intersection products, pushforwards and pullbacks; applications to the geometry of Veronese varieties, Segre varieties and plane cubics.
Chern classes: motivation, definition and key properties; applications to linear spaces on hypersurfaces and singular elements of linear series.
Parameter spaces: the choice of parameter space in solving an enumerative problem; application to the 5-conic problem.
Projective bundles: Chow ring of a projective bundle and the classes of its tangent bundle; applications to the 8-conic problem and secant varieties to rational curves.
Excess intersection: examples in low dimension, proofs in special cases and a proof in general; applications to the Chow ring of a blow-up and the double point formula.
Possible additional advanced topics: Porteous' formula; the Grothendieck-Riemann-Roch formula.
Lecturer: Robin Hartshorne (UC Berkeley)
Abstract: This will be an introductory course, assuming only a basic knowledge of algebraic geometry in the language of schemes and cohomology. Topics will include the notion of a deformation as a flat family; first-order deformations of subschemes, abstract varieties, line bundles, vector bundles; higher-order deformations and the formal notion of pro-representable functors; the problem of global moduli and various extensions of the concept of scheme--algebraic spaces, stacks to accommodate the spaces that arise as moduli. Many examples will be given to illustrate these concepts.
Note: A Graduate Text book on "Deformation Theory" by Prof. Hatshorne is being published by Springer.
Lecturer: Diane Maclagan (Warwick)
Abstract: This course will provide an elementary introduction to the emerging field of tropical algebraic geometry, where a variety is replaced by a combinatorial shadow, known as the tropical variety. The lectures will define these objects, and explore what is preserved under this "tropicalization" procedure. No prior knowledge of algebraic geometry will be assumed. A preliminary outline of the course is as Ã‚Â follows:
Note: Professor Maclagan is preparing a book on Tropical Geometry, jointly with Bernd Sturmfels.
Lecturer: Peter Beelen (Copenhaguen)
Abstract: In this course we will introduce algebraic geometry codes, discuss their construction and their decoding. Since the introduction of these codes in the 1980's many developments took place. In the first place the underlying algebraic objects, namely algebraic curves and varieties with many points have been understood in a better and more explicit way. In the second place faster decoding algorithms have become available. In the course we will discuss some of these explicit constructions, among others asymptotically optimal towers of function fields. Also we will consider several decoding algorithms, among others the majority voting algorithm and the Guruswami-Sudan list-decoding algorithm. When time allows we will also mention some more recent applications to cryptography.