## Mid-Term Report Summary - MODFLAT (Moduli of flat connections, planar networks and associators)

MODFLAT is a project at the interface between several fields of mathematics: geometry, topology, Lie theory and combinatorics. It also has links to path integrals and Feynman graphical calculus in theoretical physics. The project revolves around the formula which can be symbolically written as Z=XY. It acquires different meaning in different fields on mathematics. In topology, it defines the fundamental group of a 2-dimensional sphere with 3 holes. In Lie theory, it stands for the universal associative group law. In combinatorics, one can associate with connecting two planar networks to each other (as one does with electric circuits). The connection between different concepts in mathematics established by the Z=XY equation is deeper than it appears at the first glance. Study of this and other related links between these fields is one of the major goals of the project.

Significant new results obtained in the course of the project include:

- A proof of the 15 years old conjecture in homotopy algebra (the logarithmic formality conjecture) using Feynman graphical calculus and divergent integrals;

- An advance in non-commutative differential calculus which led to several versions of the non-commutative divergence;

- A surprising way of obtaining inequalities from Poisson brackets (similar to those used in mechanics to obtain equations of motion).

Among others, we are working on the following outstanding questions within the project:

- Is the information obtained by applying the non-commutative divergence to the equation Z=XY (known as the Kashiwara-Vergne problem) equivalent to the one hidden in the associator theory (this is the theory of re-bracketings of the type ((xy)(zw))--> ((xy)z)w)? Can Feynman graphs help in addressing this issue?

- For X,Y, Z close to 1, equation Z=XY leads to a simpler linear equation z=x+y. Is there an opposite universal (tropical) limit for X,Y,Z tending to infinity applicable to problems in algebra and geometry? Can it be applied in the associator theory?

- The equation Z=XY for X,Y, Z unitary matrices leads to a beautiful system of inequalities for their eigenvalues. Can they be interpreted in terms of planar networks (electric circuits of a special form)?

Our research group consists of the PI, a research collaborator, a postdoc and 3 Ph.D. students. Up to now, the project gave rise to 13 research articles and 1 Ph.D. thesis (defended in May 2016).

Significant new results obtained in the course of the project include:

- A proof of the 15 years old conjecture in homotopy algebra (the logarithmic formality conjecture) using Feynman graphical calculus and divergent integrals;

- An advance in non-commutative differential calculus which led to several versions of the non-commutative divergence;

- A surprising way of obtaining inequalities from Poisson brackets (similar to those used in mechanics to obtain equations of motion).

Among others, we are working on the following outstanding questions within the project:

- Is the information obtained by applying the non-commutative divergence to the equation Z=XY (known as the Kashiwara-Vergne problem) equivalent to the one hidden in the associator theory (this is the theory of re-bracketings of the type ((xy)(zw))--> ((xy)z)w)? Can Feynman graphs help in addressing this issue?

- For X,Y, Z close to 1, equation Z=XY leads to a simpler linear equation z=x+y. Is there an opposite universal (tropical) limit for X,Y,Z tending to infinity applicable to problems in algebra and geometry? Can it be applied in the associator theory?

- The equation Z=XY for X,Y, Z unitary matrices leads to a beautiful system of inequalities for their eigenvalues. Can they be interpreted in terms of planar networks (electric circuits of a special form)?

Our research group consists of the PI, a research collaborator, a postdoc and 3 Ph.D. students. Up to now, the project gave rise to 13 research articles and 1 Ph.D. thesis (defended in May 2016).