Cohomology in algebra, geometry, physics and statistics

The search for integrable partial differential systems in four independent variables (4D) is a longstanding problem in mathematical physics. In the present talk we address this problem by introducing a new construction for integrable 4D systems which are dispersionless (a.k.a. hydrodynamic-type) using nonisospectral Lax pairs that involve contact vector fields. In particular, we show that there is significantly more integrable 4D systems than it appeared before, as the construction in question produces new large classes of integrable 4D systems with Lax pairs which are polynomial and rational in the spectral parameter. For further details please see A. Sergyeyev, New integrable (3+1)-dimensional systems and contact geometry, Lett. Math. Phys. 108 (2018), no. 2, 359-376, https://arxiv.org/abs/1401.2122