Non-commutative Hilbert modular symbols

Modular symbols for some classes congruence subgroups of SL(2,Z) were introduced by Manin and Birch in relation to the Birch and Swinnerton-Dyer Conjecture. About ten years ago Manin constructed a noncommutative modular symbol for congruence subgroups of SL(2,Z), using Chen's iterated integrals. In this talk I will present a generalization of Manin's noncommutative modular symbol for congruence subgroups of SL_2 over the ring of integers in a real quadratic field. Most of the talk will be based on my paper ``Non-commutative Hilbert modular symbols" in Algebra and Number Theory. I will also present some of the current progress towards constructing a nonabelian 2-nd cohomology class of the noncommutative Hilbert modular symbol(s), which is in collaboration with two algebraic topologists - Mahmoud Zeinalian and Thomas Tradler.