Research

My research in mathematics is in the general area of differential geometry.  In particular, I am interested in certain kinds of geometric singularities including the classical notion of an umbilic.  Umbilics are singular points of a symmetric 2-form, namely the second fundamental form.  However, there are higher order notions of geometric singularities which are of interest.  These geometric singularities are related to conjectures of Carathéodory and Loewner.  The Carathéodory conjecture states that a sphere immersed in \mathbb{R}^3 has at least 2 umbilics.  Loewner’s conjecture states that if f:D\to\mathbb{R} is C^{n+1} and \partial_{\bar z}^n f\neq0 for all z\neq0, then the index of the vector field \partial_{\bar z}^n f in a neighborhood of 0 is at most n.

My Ph.D. dissertation is entitled “Index formulas for higher order Loewner vector fields.”  It establishes a formula to compute the index of a Loewner vector field by counting the radial eigenvectors of a specific matrix valued function and it provides a defect term for Loewner’s conjecture.

The study of Loewner’s conjecture is also deeply connected to ideas in dynamical systems.

Steven Broad’s Research Statement

Dissertation Abstract

Dissertation

1 Comment

  • Brendan Guilfoyle
    July 1, 2009 at 10:49 am

    Nice dissertation – perhaps could have included the reference:

    V. V. Ivanov, The analytic Carathéodory conjecture, Sib. Math. J. 43, No. 2, 251—322 (2002).

    Reply

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