"The project lies at the crossroads between three different topics in Mathematics: moduli spaces of flat connections on surfaces in Differential Geometry and Topology, the Kashiwara-Vergne problem and Drinfeld associators in Lie theory, and combinatorics of planar networks in the theory of Total Positivity.The time is ripe to establish deep connections between these three theories. The main factors are the recent progress in the Kashiwara-Vergne theory (including the proof of the Kashiwara-Vergne conjecture by Alekseev-Meinrenken), the discovery of a link between the Horn problem on eigenvalues of sums of Hermitian matrices and planar network combinatorics, and intimate links with the Topological Quantum Field Theory shared by the three topics.The scientific objectives of the project include answering the following questions:1) To find a universal non-commutative volume formula for moduli of flat connections which would contain the Witten’s volume formula, the Verlinde formula, and the Moore-Nekrasov-Shatashvili formula as particular cases.2) To show that all solutions of the Kashiwara-Vergne problem come from Drinfeld associators. If the answer is indeed positive, it will have applications to the study of the Gothendieck-Teichmüller Lie algebra grt.3) To find a Gelfand-Zeiltin type integrable system for the symplectic group Sp(2n). This question is open since 1983.To achieve these goals, one needs to use a multitude of techniques. Here we single out the ones developed by the author:- Quasi-symplectic and quasi-Poisson Geometry and the theory of group valued moment maps.- The linearization method for Poisson-Lie groups relating the additive problem z=x+y and the multiplicative problem Z=XY.- Free Lie algebra approach to the Kashiwara-Vergne theory, including the non-commutative divergence and Jacobian cocylces.- Non-abelian topical calculus which establishes a link between the multiplicative problem and combinatorics of planar networks."