Seminar on
Cohomology Theories
(Working Seminar in
Mathematics and Statistics I & II, MATH 666/7. 2003-04).
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 Barsotti-Tate 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). |
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September 24 |
Gabriel Chenevert and Payman Kassaei |
Continued. |
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|
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. |
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October 1 |
Andrew Archibald and Alex Ghitza |
Continued. |
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October 8 |
Andrew Archibald and Alex Ghitza |
Continued |
|
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October 10 |
Elena
Mantovan |
The role
of the geometry of Shimura varieties in the |
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October 15 |
Eyal
Goren; |
Some
remarks on etale morphism; |
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October 22 |
Gil Alon |
Descent and a Grothendieck topology (Cont'd). |
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October 29 |
Pete Clark and Marc-Hubert 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. |
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November 5 |
Pete Clark and Marc-Hubert Nicole (Cont'd) |
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November 12 |
Pete Clark and Marc-Hubert Nicole (Cont'd) |
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November 19 |
no meeting |
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November 26, December 10, January 15 (2004), January 19, January 26. |
Marc-Hubert 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 |
|
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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) |
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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. |
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Grothendieck |
Groupes de Barsotti-Tate et cristaux de Dieudonné. |
|
Hartshorne |
Algebraic Geometry |
|
Milne |
Etale cohomology |
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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 I-II |
|
Jim Milne |
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SGA 4 |
Theorie des topos et cohomologie etale des schemas I, II, III, LNM 269, 270, 305 |
|
Serre |
|
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Katz and Messing |
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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 |
|
Marc-Hubert 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 |