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 Thursdays, 5:15-7:15pm (that is, 5:20pm sharp) in MIT room 2-135, 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.

Spring 2018 topic: Nonabelian Chabauty.

References to be followed:

McCallum and Poonen, The method of Chabauty and Coleman, pp. 99--117 in: Explicit methods in number theory; rational points and diophantine equations, Panoramas et Synthèses 36, Société Math. de France, 2012. We will cover Sections 1-5 and 7-8.
Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics 163, Springer, 1970. We will cover up to 2.21 on page 13.
Berthelot, Cohomologie rigide et cohomologie rigide à supports propres, Prépublication IRMAR 96-03 (1996). We need to go up to the definition of the inverse image functor on isocrystals on p. 71.
Chiarellotto and Le Stum, F-isocristaux unipotents, Compositio Math. 116 (1999), 81-110. We need just the equivalence of categories in Proposition 2.41.
Deligne and Milne, Tannakian categories, corrected version (August 15, 2012) of pp. 101-228 in Hodge cycles, motives, and Shimura varieties Lecture Notes in Math. 900, 1982, Springer. We will cover the statement of Theorem 2.21.
Deligne, Le groupe fondamental de la droite projective moins trois points, pp. 79-297 in: Ihara Y., Ribet K., Serre J.-P. (eds.), Galois groups over Q, MSRI Publications 16, Springer, 1989. We will cover Sections 9 and 10.
Besser, Heidelberg lectures on Coleman integration, pp. 3-52 in: Stix J. (ed.), The arithmetic of fundamental groups, Contributions in Mathematical and Computational Sciences 2, Springer, 2012. We will cover up to p. 25. (This is an exposition discussing of some of the results of the 2002 Besser paper below.)
Wojtkowiak, Cosimplicial objects in algebraic geometry, pp. 287-327 in: Algebraic K-theory and algebraic topology (Lake Louise, AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 407, Kluwer, 1993. Constructs the Hodge filtration on the de Rham fundamental group. (The papers of Faltings and Hadian below sketch an alternative approach to the Hodge filtration, using Tannakian formalism.)
Olsson, Towards non-abelian p-adic Hodge theory in the good reduction case, Memoirs of the AMS 210 (2011), no. 990.
Kim, The unipotent Albanese map and Selmer varieties for curves, J. Europ. Math. Soc. 45 (2009), 89-133.

Other references:

Serre, Lectures on the Mordell-Weil theorem, 3rd edition, Springer, 1997. Chabauty's theorem is in Section 5.1.
Faltings, Mathematics around Kim's new proof of Siegel's theorem, pp. 173-188 in: Diophantine geometry, CRM Series 4, Ed. Norm., Pisa, 2007.
Hadian, On a motivic method in Diophantine geometry, pp. 127-146 in: Stix J. (ed.), The arithmetic of fundamental groups, Contributions in Mathematical and Computational Sciences 2, Springer, 2012.
Hadian, Motivic fundamental groups and integral points, Duke Math. J. 160, no. 3 (2011), 503-565.
Besser, Coleman integration using the Tannakian formalism, Math. Ann. 322, 19-48.
Nekovar, On p-adic height pairings, pp. 127-202 in: Séminaire de Théorie des Nombres, Paris, 1990-91, Progr. Math. 108, Birkhäuser, 1993. Section 1 contains a crash course in (abelian) p-adic Hodge theory.

February 8: Waqar Shah. Abelian Chabauty (McCallum and Poonen, Sections 1-5 and 7-8)

February 15: David Corwin. Connections (Deligne, Équations, up to 2.21 on page 13)

February 22: Borys Kadets. Rigid analytic spaces, formal schemes, and tubes (Berthelot, Chapters 0 and 1, skipping most proofs)

March 1: Zijian Yao. Overconvergent isocrystals (Berthelot, Chapter 2 up to p. 71, skipping most proofs; and Chiarellotto and Le Stum, statement of Proposition 2.4.1)

March 8: NO SEMINAR (Arizona Winter School)

March 15: Alex Best. Neutral Tannakian categories (Deligne and Milne: definitions, statement of Theorem 2.11, and examples) and (pro-)unipotent algebraic groups (Deligne, Le groupe fondamental, Section 9)

March 22: Atticus Christensen. Fundamental groups, part 1 (Deligne, Le groupe fondamental, Section 10)

March 29: NO SEMINAR (Spring Break)

April 5: Renee Bell. Fundamental groups, part 2 (Deligne, Le groupe fondamental, Section 10)

April 12: Sachi Hashimoto. p-adic iterated integrals (Besser, Heidelberg Lectures, pp. 7-25)

April 19 at 5:35pm sharp: Nicholas Triantafillou. Hodge filtration on the de Rham fundamental group (Wojtkowiak)

April 26: David Corwin. The Local Albanese Map and the Hodge Filtration (Wojtkowiak)

May 3: Wei Zhang. Unipotent p-adic Hodge theory (Olsson)

May 10: Jingwei Xiao. The unipotent Albanese map and nonabelian Chabauty, part 1 (Kim)

May 17: Qirui Li. The unipotent Albanese map and nonabelian Chabauty, part 2 (Kim)

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

Organizers: Bjorn Poonen, Nicholas Triantafillou, Isabel Vogt (with advice from Jennifer Balakrishnan, Ishai Dan-Cohen, Netan Dogra, Jan Vonk, and Wei Zhang)