IPDF Talk
Speaker: Supriyo Jana, IITMadras
Host: G.K. Srinivasan
Title: Dynamics and integrability of polynomial vector fields on certain compact manifolds
Time, day and date: 11:00:00 AM - 12:00:00 PM, Tuesday, May 19
Venue: Ramanujan Hall
Abstract: We present a study of polynomial vector fields on algebraic manifolds, including the n-dimensional sphere and product of spheres. We provide a complete algebraic characterization of polynomial vector fields on $S^n$ of arbitrary degree, and investigate their first integrals and invariant algebraic sets. Sharp bounds on the number of invariant meridian and parallel hyperplanes are established. We also determine when a Hamiltonian vector field on Euclidean space induces a vector field on these manifolds. Darboux theory of integrability is employed to show that cubic Kolmogorov vector fields on the product of 1-sphere and 2-sphere, while not Hamiltonian, admit rational first integrals.

We conclude by motivating the study of polynomial vector fields on real projective spaces. We outline several open questions concerning the characterization of such vector fields, the classification of global phase portraits on $RP^2$, and the extension of Darboux integrability theory to the projective setting.