Title:  Stability of Ground States for cubic NLS in 1-D

Abstract:  In this talk, we discuss orbital stability of ground state (equilibrium) solutions for a 1D cubic nonlinear focussing Schrödinger equation under perturbations of H1(R) initial data for cNLS. Stability of equilibrium solutions is an important condition for being able to use theoretical models in physical applications. The method of proof is a generalization of Lyapunov stability theory for nite dimensional systems. We rely on global time existence of a unique H1(R) solution to the cNLS (Ginibre-Velo, 1977).

Title:  Homogenization of Elliptic Operators with Random Coefficients

Abstract:  In this talk, we mainly consider the homogenization problems of elliptic equations with rapidly-oscillating random coefficients. The homogenization theorem is proved for this case. We  also prove the general theorem of individual homogenization when the bounded coefficients are ergodic.

Title:  On Bounded Point Derivations and Analytic Capacity

Abstract:  Let X be a compact subset of the complex plane and let R(X) denote the uniform closure of the space of rational functions whose poles lie off X. We say that there is a bounded point derivation on R(X) at x if and only if there exists a constant k such that |f t(x)| ≤ k||f ||X for all f ∈ H(X), where H(X) is the space of all functions that are holomorphic in some neighborhood of X. In this talk we will give necessary and sufficient conditions for the existence of a bounded point derivation on R(X) at x.
 

Title:  Non-Vanishing Homology of the Matching Complex

Abstract: A matching on a graph G is any subgraph where the maximum vertex degree is 1.  Since edge-deletion preserves the property of being a matching, the set of all matchings on G forms a simplicial complex M(G).  We will survey results on the lowest non-vanishing homology group for M(K_n) and discuss the extension of these results to more general graphs, specifically the r-stable ones.  Prior familiarity with simplicial complexes and homology is assumed.

Title:  TBA

Abstract:  TBA