Prismatic cohomology and de Rham-Witt forms

For any prism $(A, I)$ we construct a canonical map $W_r(A/I)\to A/I\phi(I)\ldots\phi^{r-1}(I)$. This map is necessary for existence of a canonical base change comparison between prismatic cohomology and de Rham-Witt forms. We construct a canonical map from prismatic cohomology to de Rham-Witt forms and prove that it is an isomorphism in the perfect case. Using this we get an explicit description of the prismatic cohomology for a polynomial algebra over $A/d$.Comment: 22 page

Lower Resolvent Bounds and Lyapunov Exponents

We prove a new polynomial lower bound on the scattering resolvent. For that, we construct a quasimode localized on a trajectory \gamma which is trapped in the past, but not in the future. The power in the bound is expressed in terms of the maximal Lyapunov exponent on \gamma , and gives the minimal number of derivatives lost in exponential decay of solutions to the wave equation