**Instructors:** Francis Brown (Oxford) and Richard Hain (Duke)

** Lectures:**

- (Brown) Overview: Projective line minus 3 points, I:
- iterated integrals and the KZ connection
- multiple zeta values (MZV)
- unipotent fundamental groups, Tate case

- (Hain) Moduli of elliptic curves:
- the unipotent fundamental group of a punctured elliptic curve and its MHS
- moduli spaces of elliptic curves: \(\mathcal{M}_{1,1}\), \(\mathcal{M}_{1,\vec{1}}\), and \(\mathcal{M}_{1,2}\)
- the local system \(\mathbb H\)

- (Brown) Projective line minus 3 points, II:
- Tannakian considerations: mixed Hodge structures, motives, and motivic periods
- Computation of action of motivic Galois group on \(\pi_1^{\mathrm{un}}(\mathbb{P}^1-\{0,1,\infty\},\vec{1})\)

- (Hain) The elliptic KZB connection and its limit MHS:
- the elliptic KZB connection
- limit MHSs and regularized periods (notes)
- the limit MHS on \(\pi_1^{\mathrm{un}}(E_{\partial/\partial q}',\vec{v})\)

- (Brown) Iterated Shimura integrals:
- Eichler-Shimura
- a universal connection
- the non-abelian cocycle

- (Hain) Relative completion of \(\mathrm{SL}_2(\mathbb{Z})\):
- definition and basic properties
- construction of the MHS (sketch)
- variations of MHS over \(\mathcal{M}_{1,1}\)

- (Brown) Action of the motivic Galois group on \(G^{\mathrm{rel}}\):
- review of the case of \(\mathbb{P}^1-\{0,1,\infty\}\)
- conjectural shape of the fundamental group of mixed modular motives
- the Galois action on \(G^{\mathrm{rel}}\)

- (Hain) What are universal mixed elliptic motives?
- Eisenstein variations of MHS
- limit MHS at \(\partial/\partial q\)
- the Eisenstein quotient of \(G^{\mathrm{rel}}\)

- (Brown) Multiple modular values:
- period computation using Rankin-Selberg
- transference
- the Pollack relations

- (Hain) Select from:
- modular inverter(s)
- Goldman Turaev story

**References:**

- F. Brown:
*Mixed Tate motives over \(\mathbb Z\)*, Ann. of Math. (2) 175 (2012), 949-976, arXiv:1102.1312 - F. Brown:
*Multiple Modular Values and the relative completion of the fundamental group of \(\mathcal{M}_{1,1}\)*, arXiv:1407.5167 - D. Calaque, B. Enriquez, P. Etingof:
*Universal KZB equations: the elliptic case*, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, 165-266, Progr. Math. 269, Birkhäuser, 2009, arXiv:math/0702670 - K.-T. Chen:
*Iterated path integrals*Bull. Amer. Math. Soc. 83 (1977), 831-879. - B. Enriques:
*Elliptic associators*, Selecta Math. (N.S.) 20 (2014), 491-584, arXiv:1003.1012 - R. Hain:
*The geometry of the mixed Hodge structure on the fundamental group*, Algebraic geometry, Bowdoin, 1985, 247-282, Proc. Sympos. Pure Math., 46, 1987, (pdf) - R. Hain:
*Lectures on moduli spaces of elliptic curves*, Transformation groups and moduli spaces of curves, 95-166, Adv. Lect. Math. (ALM), 16, Int. Press, 2011, arXiv:0812.1803 - R. Hain:
*Notes on the Universal Elliptic KZB Connection*, PAMQ, to appear, arXiv:1309.0580 - R. Hain:
*The Hodge-de Rham theory of modular groups*, Recent advances in Hodge theory, 422-514, London Math. Soc. Lecture Note Ser., 427, Cambridge Univ. Press, 2016, arXiv:1403.6443 - A. Levin, G. Racinet:
*Towards multiple elliptic polylogarithms*, arXiv:math/0703237 - Y. Manin:
*Iterated Shimura integrals*, Mosc. Math. J. 5 (2005), 869-881, arXiv:math/0507438 - Y. Manin:
*Iterated integrals of modular forms and noncommutative modular symbols*, Algebraic geometry and number theory, 565-597, Progr. Math., 253, Birkhäuser Boston, Boston, MA, 2006, arXiv:math/0502576 - S. Zucker:
*Hodge theory with degenerating coefficients. \(L_2\) cohomology in the Poincaré metric*, Ann. of Math. (2) 109 (1979), 415--476. (JSTOR)

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