Seminar Archive
Spring 2009
Fall 2008
F07-S08
F06-S07
Natural growth processes tend to result in bodies which possess no stress free configuration. We formulate a hyper-elastic theory for such bodies in which strain is measured with respect to a reference metric rather than a reference configuration. In this formulation, the residual stress arises from the geometrical frustration involved in the attempted isometric embedding of the non-Euclidean 3D metric in Euclidean space. Applying this formalism to thin sheets, we derive a reduced 2D elastic theory enabling us to treat thin bodies which are neither plates nor shells, which we term non-Euclidean plates.
In this talk I will present some of the phenomena exhibited by non-Euclidean plates such as, spontaneous buckling and the convergence to the Willmore energy minimizing isometry in the limit of vanishing thickness. I will also discuss the existence of such minimizers of the Willmore energy and their relation to minimal surfaces.
I will explain in a few examples how the integrable structure of certain non-linear PDEs arising in mathematical physics (Yang-Mills, harmonic maps, KdV etc.) can be used to describe their solution space in terms of moduli of algebraic curves.
In general, Einstein's gravitational field equations of the theory of general relativity cannot be solved exactly. One case in which exact solutions are possible is the Friedmann-Lema\^itre-Robertson-Walker (FLRW) model, in which one assumes that on large scales our current universe is homogeneous and isotropic. We will show that the resulting equations have solutions in terms of elliptic functions, which are the simplest of non-elementary functions and are known to appear in many branches of physics. In particular we will write these solutions in terms of Jacobi or Weierstrass elliptic functions, and in some cases also show an equivalent expression in terms of theta functions.
We find linearly independent solutions of the Goncharov-Firsova wave equation (solutions these author were unable to find) for a massive complex scalar field on a charged Kerr black hole.These solutions generalize the classical monopole spherical harmonic solutions known in the massless case.This is joint work with one of my former students,Dr.Shabnam Beheshti.