September 7: Vishal Arul. Algebraic de Rham cohomology and the Gauss-Manin connection: Define hypercohomology and algebraic de Rham cohomology, and use GAGA to prove that for a smooth projective complex variety it agrees with classical de Rham cohomology (Grothendieck, first two pages). Recall the definition of connection, explain the algebraic definition of the Gauss-Manin connection in terms of differentiating k-forms (Katz-Oda up to Section 2), prove that relative Betti cohomology is horizontal (see Katz 1972, around (4.1.1.4) and Proposition 4.1.2). |
September 14: Hyuk Jun Kweon. Review of étale cohomology (Milne). The étale site, definition of étale cohomology and l-adic cohomology, statement of comparison with Betti cohomology for a smooth projective complex variety, Weil conjectures, bad behavior of p-adic cohomology of varieties in characteristic p. |
September 21: NO MEETING (MIT student holiday, and AGNES at Brown) |
September 28: Atticus Christensen. Crystalline cohomology (Chambert-Loir, I.1 and I.2) |
October 5: Dmitry Kubrak. Summary of p-adic Hodge theory (Illusie up to 2.2, Berger II.4) |
October 12: Siyan Daniel Li. Faltings finiteness for Galois representations (Faltings Theorem 5, Wüstholz section 2 up to the proof of Prop. 2.7, and Lawrence-Venkatesh section 2.3) |
October 19: Borys Kadets. p-adic Period Mappings and the S-unit equation (Lawrence-Venkatesh, Section 3) |
October 26: Aleksandr Petrov. (Lawrence-Venkatesh, Section 4) |
November 2: Siyan Daniel Li. Argument Outline For Mordell's Conjecture (Lawrence-Venkatesh, Section 5) |
November 9: Alex Best. (Lawrence-Venkatesh, continued) |
November 16: Kevin Lin. (Lawrence-Venkatesh, continued) |
November 23: NO MEETING (Thanksgiving Break) |
November 30: Nicholas Triantafillou. (Lawrence-Venkatesh, continued) |
December 7: (Lawrence-Venkatesh, continued) |
December 14: (Lawrence-Venkatesh, continued) |
Organizers: Borys Kadets, Bjorn Poonen, Zhiwei Yun, Wei Zhang