Seminar on Topics in Arithmetic, Geometry, Etc.

STAGE is a seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.
Meetings are held on Fridays, 10am-12 in MIT room 2-139, unless indicated otherwise below. Talks will generally run only 90 minutes, however.
To receive announcements by email, add yourself to the STAGE mailing list.
If you are interested in giving one of the talks, please contact the organizers.

Fall 2018 topic: p-adic Hodge theory and diophantine applications.

References to be followed:

Grothendieck, On the de Rham cohomology of algebraic varieties, 1966 (first two pages).
Katz and Oda, On the differentiation of de Rham cohomology classes with respect to parameters, 1968 (up to Section 2).
Katz, Algebraic solutions of differential equations (p-curvature and the Hodge filtration), 1972 (GAGA (4.1.1.4) and Proposition 4.1.2).
Milne, Lectures on étale cohomology, 2013.
Chambert-Loir, Cohomologie cristalline: un survol, 1998.
Illusie, Cohomologie de de Rham et cohomologie étale p-adique, 1990, up to Section 2.2.
Berger, An introduction to the theory of p-adic representations, 2004.
Faltings, Finiteness theorems for abelian varieties over number fields, in: Cornell and Silverman (eds.), Arithmetic Geometry, Springer, 1986, Theorem 5.
Wüstholz, The finiteness theorems of Faltings, 1992 in Rational Points, Aspects of Math. vol. 6, Vieweg+Teubner Verlag, Section 2 up to the proof of Proposition 2.7.
Lawrence and Venkatesh, Diophantine problems and p-adic period mappings, arXiv, July 7, 2018.

Other references:

Some topics below might take more or less time than allotted. If a speaker runs out of time on a certain date, that speaker might be allowed to borrow some time on the next date. So the topics below might not line up exactly with the dates below. (This contrasts with what was done in previous semesters.)

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: Borys Kadets. Transcendence of period mappings (Lawrence-Venkatesh, Section 9)

December 14: Vishal Arul. Criteria for finiteness of S-integral points in higher dimension (Lawrence-Venkatesh, Section 10)

Past semesters: Fall 2009, Spring 2010, Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2016, Spring 2017, Fall 2017, Spring 2018

Organizers: Borys Kadets, Bjorn Poonen, Zhiwei Yun, Wei Zhang