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From 2D Hyperbolic Geometry to the Loday-Quillen-Tsygan Theorem

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Mahmoud Zeinalian
Long Island University/MPIM
Tue, 2018-06-12 14:00 - 15:00
MPIM Lecture Hall
The space of hyperbolic structures on an oriented smooth surface has a rich geometry. The study of the Hamiltonian flow of various energy functions associated with a collection of closed curves has led to a host of algebraic structures: first, on the linear span of free homotopy classes of non trivial closed curves (Wolpert-Godman-Turaev Lie bi-algebra) and then, more generally, on the the equivariant chains on the free loop spaces of higher dimensional manifolds under the umbrella of String Topology (Chas-Sullivan + Others). 
While it is understood that the most natural way to organize the relevant algebraic data would be through topological field theories whereby chains on the moduli space of Riemann surfaces with input and output boundaries label operations, various aspects of these structures are poorly understood. 
In this talk, I will provide some historical context and report on recent relevant joint work with Manuel Rivera on algebraically modeling the chains on the based and free loop spaces. One tangible result is that the long held simply-connected assumptions in various theorems, such as Adams’s 1956 cobar construction, can be safely removed once a derived version of chains is utilized. Towards the end of the talk I will report on a related ongoing joint work with Owen Gwilliam and Gregory Ginot based on the celebrated Loday-Quillen-Tsygan theorem which calculates the Lie algebra cohomology of the matrix algebras of very large size.
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