# Poisson Brackets of Partitions of Unity on Surfaces

## TL;DR

This paper introduces a new method to analyze the Poisson bracket invariant of partitions of unity on surfaces, successfully proving a conjectured lower bound in dimension 2.

## Contribution

A novel approach to the Poisson bracket invariant that confirms Polterovich's conjecture for surfaces, advancing understanding of symplectic topology.

## Key findings

- Established a lower bound for the Poisson bracket invariant on surfaces.
- Validated Polterovich's conjecture in dimension 2.
- Provided new tools for studying partitions of unity in symplectic geometry.

## Abstract

Given an open cover of a closed symplectic manifold, consider all smooth partitions of unity consisting of functions supported in the covering sets. The Poisson bracket invariant of the cover measures how much the functions from such a partition of unity can become close to being Poisson commuting. We introduce a new approach to this invariant, which enables us to prove the lower bound conjectured by L. Polterovich, in dimension 2.