Courses 2001-02            Winter

Topics in Geometry and Topology II  (Introduction to Algebraic Geometry) 189-707 A 

Time: Monday 13:30-15:30, Wednesday, 9:30-10:30.
Lecture Room: BURN 1205

Course syllabus: This is an introductory course in Algebraic Geometry. It develops from scratch the theory of algebraic varieties over algebraically closed fields, including morphisms, sheaves and cohomology. The course presupposes basic commutative algebra. The topics to be studied are: affine and projective algebraic varieties; regular functions and morphisms; singularities, normalization and blow-up; sheaves, Cech co-homology and the Riemann-Roch theorem for curves. The examples of curves, and Grassmannian manifolds are studied in detail. The course consists of three weekly hours (two meetings per week). It is given in the first term. The final grade will be based on assignments and a take home exam.
(1) Kempf, George R.: Algebraic varieties. London Mathematical Society Lecture Note Series, 172.
(2) Hartshorne, Robin: Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag. (Main Text)
(3) Mumford, David: The red book of varieties and schemes. Lecture Notes in Mathematics, 1358, Springer-Verlag.
(4) Shafarevich, Igor R.: Basic algebraic geometry. 1. Varieties in projective space. Springer-Verlag, Berlin, 1994.
(5) Harris, J.: Algebraic Geometry: A First Course, Springer Graduate Texts in Mathematics 133.

Course Structure: The Monday meeting will be a full 2 hour lecture. The Wednesday meeting will early on in the semester be devoted to lectures given by the students.
Course pre-requisites: General knowledge of algebra at the undergraduate level (in particular, fields, algebraically closed fields, rings, prime and maximal ideals etc.) and rudimentary topology (open and closed sets, continuous functions, separation axioms, compact spaces -- all at a very basic level). During the course the students will need to catch-up with lots of algebra. Hand-out and references will be provided. It is recommended that students taking this course take (or at least audit) Prof. Russell's course "Higher Algebra I, 189-570A".

Course requirements:

Seminar topics:
Multilinear Algebra Khanh Huynh September 19
Direct and Projective Limits Neil Kennedy September 26
Groebner Bases I Melisande Fortin-Boisvert Ocotber 3
Groebner Bases II  Dimitry Zuchowski October 10
Toric Varieties Pavel Dimitrov October 17
Projection from a Point Benoit Arbour October 24
Flag Varieties Dmitry Chtcherbine October 31
Sheaves on Topological Spaces Carlos Philips November 14
Algebraic Groups I Maxim Samsonov November 21
Algebraic Groups II Roman Tymkiv November 28
Galois Theory for Curves Matthew Greenberg December 5
Goppa Codes Sebastien Loisel December 17
Points on Curves over Finite Fields Gabriel Chenevert December 17
The Hilbert Polynomial Vasilisa Chramtchenko December 17
Integral Morphisms and Normalization Alexandru Stanculescu December 17


Maple/Macaulay examples.



Topics in Geometry and Topology III (Vector Bundles on Curves) 189-708 B

First Meeting Handout

There is a graduate seminar accompanying this course, held jointly with Prof. Darmon's course on Automorphic forms. In the seminar we shall study parts of the book: O. Forster/Lectures on Riemann Surfaces.GTM 81.

Lectures in the Seminar:
Matthew Greenberg: Holomorphic functions on a Riemann surface.
Melisande Fortin-Boisvert: The degree of a morphism.
Benoit Arbour: Extending unramified coverings of punctured Riemann surfaces.
Vasillisa Chramtchenko: Linear differential equations on a Riemann surface I.
Maxim Samsonov: Linear differential equations on a Riemann surface II.
Gabriel Chenevert: Linear differential equations on a Riemann surface III.
Marc-Hubert Nicole: Triviality of vector bundles over an a non-compact Riemann surface.

Final Projects (all copyrights belong to the authors) :
Gabriel Chenevert: The Riemann-Hilbert Problem.
Matthew Greenberg:
Melisande Fortin-Boisvert:
Benoit Arbour:
Vasillisa Chramtchenko:
Maxim Samsonov:
Marc-Hubert Nicole: