Cohomology in algebra, geometry, physics and statistics

In my talk   I shall explain   our    with Kaoru Ono   construction   of  Floer-Novikov  cohomology  groups $HFN^* (M^{\Gamma_\xi \times H},\xi, Q)$ defined on a regular covering $M^{\Gamma_\xi \times H}$ of a  compact   symplectic  manifold   $(M, \omega)$ with  transformation group  $\Gamma_\xi \times  H$  and associated  to  a    locally symplectic isotopy ${\{\varphi_t\}}$ of $(M, \omega)$ with  flux $\xi \in H ^1 (M, R)$. Then  $H$ acts naturally on $HFN^* (M^{\Gamma_\xi \times H},\xi, Q)$.  For a subgroup $G \subset H$  denote  by $(HFN^* (M^{\Gamma_\xi \times H},\xi,  Q))^G$  the   subgroup of $HFN^* (M^{\Gamma_\xi \times H}, \xi, Q)$  consisting   of the fixed  points of the $G$-action.  We  prove that  the   rank   of  $(HFN^* (M ^{\Gamma_\... more