Date:  Tuesday, June 28, 2022
Time:  9:00 a.m.
Location:       Weber 201

You may also participate virtually.

To Join visit: https://zoom.us/j/96875783998?pwd=d0NHcHBGTlAxRGErRmNMclU0NW02UT09
Meeting ID: 968 7578 3998
Passcode: 353726

Advisor:  Dr. Clayton Shonkwiler

Committee:  Dr. Wolfgang Bangerth, Dr. Henry Adams, Dr. Jacob Roberts

Title: Hodge and Gelfand Theory in Clifford Analysis and Tomography

Abstract: There is an interesting inverse boundary value problem for Riemannian manifolds called the Calderon problem which asks if it is possible to determine a manifold and metric from the Dirichlet-to-Neumann (DN) operator. Work on this problem has been dominated by complex analysis and Hodge theory and Clifford analysis is a natural synthesis of the two. Clifford analysis analyzes multivector fields, their even-graded (spinor) components, and the vector-valued Hodge–Dirac operator whose square is the Laplace–Beltrami operator. Elements in the kernel of the Hodge–Dirac operator are called monogenic and since multivectors are multi-graded, we are able to capture the harmonic fields of Hodge theory and copies of complex holomorphic functions inside the space of monogenic fields simultaneously. We show that space of multivector fields has a Hodge–Morrey-like decomposition into monogenic fields and the image of the Hodge–Dirac operator. Using the multivector formulation of electromagnetism, we generalize the electric and magnetic DN operators and find that they extract the absolute and relative cohomologies. Furthermore, those operators are the scalar components of the spinor DN operator whose kernel consists of the boundary traces of monogenic fields. We define a higher dimensional version of the Gelfand spectrum called the spinor spectrum which may be used in a higher dimensional version of the boundary control method. For compact regions of Euclidean space, the spinor spectrum is homeomorphic to the region itself. Lastly, we show that the monogenic fields form a sheaf that is locally homeomorphic to the underlying manifold which is a prime candidate for solving the Calderon problem using analytic continuation.