Reading Seminar on Local Systems and Mapping class groups

We will meet on  Wednesdays 12:30-2:00pm  in room 2-361 (except on Feb 19, where we will meet in 2-255), for a 1 hour talk followed by a 30 minute question and discussion session. 

The main goal of this reading seminar is to understand the work of Landesman-Litt on the classification of a special class of local systems on surfaces called "canonical representations". By definition, those are representations (up to conjugation) of the fundamental group of a surface with finite orbit under the mapping class group action.  Despite its root in surface topology, Landesman-Litt's approach to this problem uses ideas from non-abelian Hodge theory, the Langlands program and arithmetic geometry. 

Tentative topics in this seminar include: background in local systems and variation of Hodge structures; Simpsons' non-abelian Hodge theory; the work of Esnault-Groechenig on cohomologically rigid local systems; Landesman-Litt's proof that MCG finite implies finite image and its applications. If time permits, and depending on participants' interest, we may also go into arithmetic aspects of local systems, such as the recent paper of Lam-Litt on non-abelian p-curvature conjecture. 

If you have questions, suggestions, or want to sign up for a talk, please feel free to email one of the organizers:

Zihong Chen (zihongch@mit.edu), Yonghwan Kim (yonghkim@mit.edu), Sasha Petrov (alexander.petrov.57@gmail.com)

Main reference:

Landesman-Litt, Canonical representations of surface groups (link) 

Other references:

Landesman-Litt, Geometric local systems on very general curves and isomonodromy (link)

Lam-Litt, Algebraicity and integrality of solutions to differential equations (link)

Esnault-Groechenig, Cohomologically rigid local systems and integrality (link)

Katz, Algebraic solutions to differential equations (link)

Simpsons, The ubiquity of variations of Hodge structures (link)

Schedule

Week 1  (Feb 12)     Overview.  (Zihong, notes)

Week 2  (Feb 19)          Review of variation of Hodge structures (VHS) and some results in non-abelian Hodge theory. (Important: this talk will be in Room 2-255)  (Kenta, notes)

Week 3  (Feb 26)         Review of Mapping class groups (Birman exact sequence, Teichmuller space and M_{g,n}) +  Section 2 of Landesman-Litt's paper.  (Zihong, notes)

Week 4  (Mar  5)        Overview of Landesman-Litt's result and structure of the proof. (Aaron, notes 1 and 2)

Week 5  (Mar  12)      Character variety, the Hitchin map and Simpson's theorem (Anne, notes)

Week 6  (Mar 19)     Period map and unitary representations (Section 5 of Landesman-Litt). (Sasha, notes)

(MIT Spring break)

Week 7   (Apr 2)      Cohomological rigidity results (Section 6 of Landesman-Litt). (Ben, notes)

Week 8   (Apr 9)     Other examples of MCG-finite representations (with infinite monodromy), from TQFT or Kodaira-Parshin construction.   (Yonghwan, notes)

Week 9 (Apr 16)     Overview of p-curvature and the result of Lam-Litt (Sasha, notes)

Week 10 (Apr 23)   Some results on linear differential equations (Section 3 of Lam-Litt). (Ben, notes) 

Week 11 (Apr 30)    (Daniel's talk at Harvard)

Week 12 (May 7)     Computation of p-curvature of the horizontal foliation on M_dR (Shaoyun, notes)

Week 13 (May 14)   (Kenta)