Załącznik do ogłoszenia konkursowego CAS/14/POKL Programs

Załącznik do ogłoszenia konkursowego CAS/14/POKL
Programs
Course Title: Abelian Varieties
By Yichao Tian (Princeton University), Weizhe Zheng (Columbia University)
Duration: June 28, 2010 - August 27, 2010
Course description:
The course should basically cover the following topics
(1) Generalities on group schemes over a field: Definition and basic properties and maybe
also Cartier's theorem on the smoothness of group schemes in characteristic 0.
(2) Definition of abelian varieties, rigidity lemma, rational maps between abelian varities.
(3) See-saw principle, Theorem of the Cube, consequences of the theorem of the Cube.
(4) Finite group schemes, Cartier duality of finite group schemes, quotients of a variety by a
finite group scheme.
(5) Picard variety of an abelian variety, dual abelian varieties, polarizations.
(6) Abelian varieties over the complex number field, Riemann forms.
(7) Endomorphism ring of abelian varieties, Tate modules, Weil pairing, Rosati involution.
(8) Introduction to advanced topics on abelian varieties.
This rough course plan might be slightly modified according to the speed of lectures.
Prerequisite:
The basic notions of algebraic geometry seem inevitable. The second chapter of Hartshorne's
<Algebraic Geometry> is a good reference for this part. In the proof of Theorem of the cube,
we will need some coherent cohomology of varieties. Some knowledge on elliptic curves will
be very helpful too.
Reference for the course:
[1] Ch. Birkenhake, H. Lange: Complex Abelian Varieties. Springer-Verlag, 2004.
[2] D. Mumford: Abelian Varieties. Tata Institute of Fundamental Research
[3] G. van der Geer, B. Moonen: Abelian Varieties. An electronic version is available at:
http://staff.science.uva.nl/~bmoonen/boek/BookAV.html
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Stypendia są współfinansowane przez Unię Europejską w ramach Europejskiego Funduszu Społecznego.
Course Title: Introduction to Arakelov Geometry
By Huayi Chen (Institut de Mathematiques de Jussieu, Paris), Xinyi Yuan (Harvard
University)
Duration: June 28, 2010 - August 27, 2010
Course description:
The course should basically cover the following topics
(1) Arithmetic curves (hermitian vector bundles, successive minima and the link with
transcendence problems)
(2) Arakelov's intersection of admissible divisors on surfaces;
(3) Basic definition of intersection theory of Soule-Gillet;
(4) Hilbert-Samuel formula;
(5) Volumes of big line bundles;
(6) Equidistribution of small points.
Prerequisite:
Basic notion in commutative algebra (projective module, local ring, valuation theory etc) and
algebraic number theory (Dedekind domain, elementary Galois theory). Some knowledge on
projective algebraic geometry as ampleness will be helpful.
Below is some reference for the prerequisite:
[1] Eisenbud, Commutative algebra with a view toward algebraic geometry, GTM 150.
[2] Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften
322.
[3] Lazarsfeld, Positvitiy in algebraic geometry, I, Ergebnisse der Mathematik und ihrer
Grenzgebiete 48
Definition of schemes, line bundles and intersections on algebraic surfaces will be useful. A
nice reference is Hartshorne's book chapter 2, 4, 5.
Reference for the course:
The articles by Chinburg and Silverman in the book “arithmetic geometry” by Cornell and
Silverman, More references will be posted as the lectures begin.
Course Title: Introduction to Automorphic Forms
By Chao Li (University of Toronto), Wei Zhang (Harvard University)
Course Duration: June 28, 2010 - August 27, 2010
Course description:
Strona 2 z 3
Stypendia są współfinansowane przez Unię Europejską w ramach Europejskiego Funduszu Społecznego.
The Langlands program has been far-reaching and influential in modern mathematics. It
conjectures some deep connection between number theory and the theory of automorphic
froms. In this course, we will give an introduction to the Langlands program. The emphasis of
our lectures is on the example of GL(2), through which we can take a glance for the general
cases. The course will cover the following topics:
(1) Tate's thesis
(2) Admissible representation of GL(2)
(3) Automorphic forms on GL(2)
(4) Introduction to the Langlands program
(5) L-functions and converse theorem
(6) Jacquet-Langlands correspondence
Prerequisite:
[1] p-adic numbers, Adeles and Ideles, Direchlet characters,etc, (related topics can be found in
"Algebraic Number theory" by S. Lang, GTM 110)
[2] Some basic theory of representations of groups. (Cf, "Linear Representations of Finite
Groups" by J.P. Serre, GTM 42.)
[3] A little experience with classical mordular forms will be helpful. (Cf, Chapter VII in "A
course in Arithmetic" by J.P. Serre, GTM 7)
Reference for the course:
[1] D. Bump, Automorphic Forms and Representations, Cambridge University Press.
[2] S. Gelbart, Automorphic forms on adele groups. Ann. of Math. Studies 83, (1975).
[3] A. Knapp, An Introduction to the Langlands Program, Proceeding of Symposia in Pure
Mathematics, Vol 61(1997), 245-302. http://www.math.sunysb.edu/~aknapp/pdf-files/245302.pdf
[4] J. Bernstein & S. Gelbart, An Introduction to the Langlands Program, Boston: Birkhauser,
(2003).
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Stypendia są współfinansowane przez Unię Europejską w ramach Europejskiego Funduszu Społecznego.