abstract: We study twisted ergodic integrals for translation flows on surfaces of higher genus. The motivation is two-fold: on the one hand we want to understand the deviation of ergodic averages for a simple example of 3-dimensional translation flow given by the product of a translation flows on a higher genus surface and on a circle, on the other hand there is a well-known connection between twisted ergodic integrals and spectral measures of translation flows, already exploited in the work of Bufetov and Solomyak, and speed of weak mixing, which give an effective version of a result with Avila on weak mixing of translation flows.
The main new idea is a natural notion of twisted cohomology and a cocycle dynamical system over the Teichmueller geodesic flows acting on a twisted cohomology bundle. By introducing this cocycle we can then apply methods of Hodge theory similar to those applied in the study of the Kontsevich--Zorich cocycle. After reviewing notions related to the Hodge theory approach for the Kontsevich--Zorich cocycle, the talk will focus on these new cohomological tools and the overall strategy of the argument.