부산대학교  기하학, 위상수학  세미나
(PNU Geometry and Topology seminar)

Title: Dynamics of polynomial vector fields on a sphere. 

Abstract: David Hilbert proposed a problem at the beginning of the 1900s. It is known as Hilbert's 16th problem, which asks for a bound for the number of invariant algebraic curves in terms of the degree of polynomial vector fields in the plane. This problem is solved for a few particular cases. One can ask a similar question for the polynomial vector fields on the sphere. In this talk, I'll characterize and study dynamical properties of cubic vector fields on the sphere. Then, I'll show that there exist completely integrable cubic vector fields on the sphere and also study the maximum number of various types of invariant circles for homogeneous cubic vector fields on the sphere. Finally, I'll discuss phase portraits of certain cubic Kolmogorov vector fields on the sphere.

Titile: Equidistribution of expanding translates of smooth curves in homogeneous spaces under the action of a product of SO(n,1)'s

Abstract: Let G be a semisimple Lie group, Gamma be a lattice in G, and let A = {a_t} be a one-parameter diagonal subgroup. Given a point x in G/Gamma and a curve phi: I -> U^+ in the expanding horospherical subgroup of A, we consider the family of expanding translates a_t phi(I)x as t -> infinity. The equidistribution problem asks for conditions under which these translates eventually spread uniformly throughout G/Gamma with respect to the Haar probability measure.

Motivated by applications to Diophantine approximation and hyperbolic geometry, Shah initiated the study of this problem and obtained results in several important special cases involving SL_n(R) and a single copy of SO(n,1). Lei Yang later established equidistribution for analytic curves satisfying an algebraic condition when G = SO(n,1)^k. In this talk, I will discuss an extension of Lei Yang's result from analytic curves to smooth curves when G = SO(n_1,1) x ... x SO(n_k,1).