Georgia Institute of Technology College of Sciences

In this talk, I will talk about recent developments on the point mass problem on the real line. Starting from the point mass formula for orthogonal polynomials on the real line, I will present new methods employed to compute the asymptotic formulae for the orthogonal polynomials and how these formulae can be applied to solve the point mass problem when the recurrence coefficients are asymptotically identical. The technical difficulties involved in the computation will also be discussed.

We present results on the asymptotic behavior of a family of polynomials which are orthogonal with respect to an exponential weight on certain contours of the complex plane. Our motivation comes from the fact that the zeros of these polynomials are the nodes for complex Gaussian quadrature of an oscillatory integral defined on the real axis and having a high order stationary point. The limit distribution of these zeros is also analyzed, and we show that they accumulate along a contour in the complex plane that has the S-property in the presence of an external field. Additionally, the strong asymptotics of the orthogonal polynomials is obtained by applying the nonlinear Deift--Zhou steepest descent method to the corresponding Riemann--Hilbert problem. This is joint work with D. Huybrechs and A. Kuijlaars, Katholieke Universiteit Leuven (Belgium).

The tangent cone of a set X in R^n at a point p of X is the limit of all rays which emanate from p and pass through sequences of points p_i of X as p_i converges to p. In this talk we discuss how C^1 regular hypersurfaces of R^n may be characterized in terms of their tangent cones. Further using the real nullstellensatz we prove that convex real analytic hypersurfaces are C^1, and will also discuss some applications to real algebraic geometry.