I will discuss a new method for estimating Lyapunov exponents for hypoelliptic diffusions from below using quantitative local regularity estimates on the stationary measure of the lift to the projective bundle. For damped driven truncations of nonlinear conservative SPDE in fluctuation dissipation scaling, I will outline a general strategy for proving positivity of the top Lyapunov exponent that can be applied to Galerkin truncations of the stochastic Navier-Stokes equations on T^2. The proof in the Navier-Stokes case involves verifying transitivity properties of a certain matrix Lie algebra associated with the linearized Euler equations using tools from algebraic geometry. The techniques are quite general and can be applied more broadly to Galerkin truncations of many other evolution type SPDE. This work is joint with Jacob Bedrossian and Alex Blumenthal.