Seminar on
Cohomology Theories
(Working Seminar in
Mathematics and Statistics I & II, MATH 666/7. 200304).
Goal: Grothendieck topologies and etale and crystalline cohomologies; applications to zeta functions and the Weil conjectures; de Rham cohomology and its variation in families; calculation of cohomology groups. References we'd like to understand: Mumford/Picard groups of moduli problems; Grothendieck/Groupes de BarsottiTate et cristaux de Dieudonné;
Schedule:
September 17 
Gabriel Chenevert and Payman Kassaei 
Review of homological algebra, sheaf cohomology via derived functors and via Cech cohomology, some key theorems, examples of cohomology groups (in particular, Zariski is no good for constant coefficients). 

September 24 
Gabriel Chenevert and Payman Kassaei 
Continued. 


Andrew Archibald and Alex Ghitza 
Grothendieck topologies. The etale topology: flat and unramified morphisms. Etale morphisms and criteria. Lots of examples. Sheaves in the etale topology. The stalk of a sheaf. 

October 1 
Andrew Archibald and Alex Ghitza 
Continued. 

October 8 
Andrew Archibald and Alex Ghitza 
Continued 

October 10 
Elena
Mantovan 
The role
of the geometry of Shimura varieties in the 

October 15 
Eyal
Goren; 
Some
remarks on etale morphism; 

October 22 
Gil Alon 
Descent and a Grothendieck topology (Cont'd). 

October 29 
Pete Clark and MarcHubert Nicole 
de Rham cohomology and the Hodge to de Rham spectral sequence. Algebraic de Rham cohomology. The crystalline topology. Divided powers. The crystalline site. Lots of examples. 

November 5 
Pete Clark and MarcHubert Nicole (Cont'd) 


November 12 
Pete Clark and MarcHubert Nicole (Cont'd) 


November 19 
no meeting 


November 26, December 10, January 15 (2004), January 19, January 26. 
MarcHubert Nicole, Eyal Goren, Gabriel Chenevert, Ratnadha Kolhatkar 
Weil cohomology, zeta functions, cohomolgical interpretation, the zeta function of a curve and its Jacobian, variation of zeta functions, H0et and the zeta function of some zero dimensional schemes (including automorphicity), Grassmann varieties and the schubert calculus, the zeta function of Grassmann varieties. 
notes
(Nicole) 
February 2, February 9 
Matthew Greenberg 
Spectral sequences 
notes by Matt 
February 18, March 1, 8, 15, 22 
Andrew Archibald and David Savitt 
Back to basics  foundations of etale cohomology 

March 29, April 14. 
Alex Ghitza, Payman Kassaei, Shahab Shahabi 
Gauss and Jacobi sums and Fermat hypersurfaces; Estimates for Kloosterman sums using etale cohomology; Estimates for exponential sums using Crystalline cohomology. 
notes (Shahab) 




References
Berthelot 
Cohomologie cristalline des schémas de caractéristique $p>0$. LNM 407. 
Berthelot and Ogus 
Notes on crystalline cohomology. 
Freitag and Kiehl 
Étale cohomology and the Weil conjecture. 
Grothendieck 
Groupes de BarsottiTate et cristaux de Dieudonné. 
Hartshorne 
Algebraic Geometry 
Milne 
Etale cohomology 
Mumford 
Picard groups of moduli problems. 
Serre 
zeta and L functions 
Savitt 
Lecture notes on etale cohomologies 
SGA 4 1/2 
Cohomologie etale. LNM 569. 
Tamme 
Introduction to etale cohomology 
Katz 
An overview of Deligne's proof of the Riemann hypothesis for varieties over finite fields. Proc. Sympos. Pure Math., Vol. XXVIII 
Bosch, Lütkebohmert, Raynaud 
Neron Models 
Motives 
Proc. Symp. Pure. Math. Vol 52 III 
Jim Milne 

SGA 4 
Theorie des topos et cohomologie etale des schemas I, II, III, LNM 269, 270, 305 
Serre 

Katz and Messing 

Weston 
notes on spectral sequences (by popular demand... ) 
Participants: (email addresses are @math.mcgill.ca, unless otherwise specified)
Andrew Archibald 
archibal 
Benoit Arbour 
arbour 
Matthew Greenberg 
greenberg 
David Savitt 
savitt 
Gabriel Chenevert 
chenever 
Alexandru Ghitza 
ghitza 
MarcHubert Nicole 
nicole 
Payman Kassaei 
kassaei 
David Oury 
oury 
Shahab Shahabi 
shahabi 
Ratnadha Kolhatkar 
ratnadha 
Gil Alon 
alon 
Jonathan Pila 
pila 
Pete Clark 

Eyal Goren 
goren 
Adrian Iovita 
iovita at mathstat.concordia.ca 
Mak Trifkovic 
mak 
Robert Harron 
bobharron at hotmail.com 