Seminar: Degrees of Vector Fields

Time:

Venue/Location: Phòng B102, VIASM

Báo cáo viên: Professor B. Ulrich (Purdue University)

Tóm tắt: Motivated by the problem to find algebraic curves that are left invariant by a given vector field in the plane, Poincare asked whether the degree of such curves can be bounded above in terms of the degree of the vector field. Although the question has a negative answer in general, it has inspired a great deal of work for over a century. A broader goal of this research is to relate the degree of vector fields in projective n-space to properties of curves or even varieties that they leave invariant. We will survey some of the numerous earlier results and report on recent joint work with Marc Chardin, Hamid Hassanzadeh, Claudia Polini, and Aron Simis, where the question is approached from a more algebraic point of view. We provide lower bounds for the degree of vector fields in terms of local and global invariants of the curves they leave invariant. Higher dimensional varieties are considered as well, and the sharpness of the bounds will be discussed.