Georgia Institute of Technology College of Sciences

In this work, a novel approach is used to study geometric properties of the indicatrix bundle and the natural foliations on the tangent bundle of a Finsler manifold. By using this approach, one can find the necessary and sufficient conditions on the Finsler manifold (M; F) in order that its indicatrix bundle has the Sasakian structure.

In this talk, we consider the dynamical properties of solutions to the fractional nonlinear Schrodinger equation (FNLS, for short) arising from pseudorelativistic Boson stars. First, by establishing the profile decomposition of bounded sequences in H^s, we find the best constant of a Gagliardo-Nirenberg type inequality. Then, we obtain the stability and instability of standing waves for (FNLS) by the profile decomposition. Finally, we investigate the dynamical properties of blow-up solutions for (FNLS), including sharp threshold mass, concentration and limiting profile. (Joint joint with Jian Zhang)

The purpose of this talk is to introduce some recent works on the field of Sobolev orthogonal polynomials. I will mainly focus on our two last works on this topic. The first has to do with orthogonal polynomials on product domains. The main result shows how an orthogonal basis for such an inner product can be constructed for certain weight functions, in particular, for product Laguerre and product Gegenbauer weight functions. The second one analyzes a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which involves the outward normal derivatives on the sphere. Using the representation of these polynomials in terms of spherical harmonics, algebraic and analytic properties will be deduced. First, we will get connection formulas relating classical multivariate orthogonal polynomials on the ball with our family of Sobolev orthogonal polynomials. Then explicit expressions for the norms will be obtained, among other properties.