BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Xinyu Fang (Harvard University)
DTSTART:20250213T220000Z
DTEND:20250213T233000Z
DTSTAMP:20260422T220350Z
UID:STAGE/123
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/STAGE/123/">
 A tour of the Betti\, étale\, de Rham\, and crystalline cohomology</a>\nb
 y Xinyu Fang (Harvard University) as part of STAGE\n\nLecture held in Room
  2-449 in the MIT Simons Building.\n\nAbstract\nThis talk is a crash cours
 e on the properties of various cohomology theories that will be used in th
 e Lawrence-Venkatesh proof. These include the Betti cohomology\, étale co
 homology\, de Rham cohomology and crystalline cohomology. We review releva
 nt structures on each of these cohomology groups of a smooth proper variet
 y\, state some comparison theorems\, and explain how they come together to
  form the "big diagram" in the Lawrence-Venkatesh argument.\n\n[Update] I 
 uploaded my outline for the talk to (slides) below. It is only a skeleton 
 of the talk instead of a complete write-up\, so if you want to review the 
 material\, I recommend reading the first reference (it's only 2 pages)\, a
 nd if you would like more detail\, look into the other references.\n\nRefe
 rence: \n\n$\\bullet$ <a href="https://math.mit.edu/~poonen/papers/p-adic_
 approach.pdf">Poonen\, $p$-adic approaches to rational and integral points
  on curves</a>\, Sections 5 and 6.\n\nFor more details:\n\n$\\bullet$ <a h
 ref="https://link.springer.com/chapter/10.1007/978-3-540-38955-2_3">Delign
 e\, Hodge cycles on abelian varieties</a>\, Section 1 (for Betti and de Rh
 am cohomology).\n\n$\\bullet$ <a href="https://link.springer.com/article/1
 0.1007/s00222-020-00966-7">Lawrence and Venkatesh\, Diophantine problems a
 nd $p$-adic period mappings</a>\, Section 3 (to see how cohomology theorie
 s are going to be used).\n\n$\\bullet$ <a href="https://math.stanford.edu/
 ~conrad/papers/notes.pdf">Brinon and Conrad\, CMI summer school notes on $
 p$-adic Hodge theory</a>\, Section 9.1 (for details on the ring $B_{cris}$
  and the functor $D_{cris}$).\n\n$\\bullet$ <a href="https://www.math.mcgi
 ll.ca/goren/SeminarOnCohomology/Seminairecohomologie.pdf">Nicole\, Cris is
  for Crystalline (notes for a seminar talk on crystalline cohomology)</a> 
 (for the definition and motivations for crystalline cohomology).\n\n$\\bul
 let$ <a href="https://www.cambridge.org/core/books/hodge-theory-and-comple
 x-algebraic-geometry-i/A6E52939BA107FFCB5A901D5B5D88025">Voisin\, Hodge th
 eory and complex algebraic geometry I</a>\, Chapter II (for de Rham cohomo
 logy and the Hodge decomposition).\n
LOCATION:https://researchseminars.org/talk/STAGE/123/
END:VEVENT
END:VCALENDAR