Colloquium: Prof. William Schwalm

Prof. William A. Schwalm
Department of Physics and Astrophysics, University of North Dakota

Friday Oct 24, 4:00–5:00pm, 211 Witmer Hall. Refreshments at 3:30pm.

Coupled Phenomena with Balls and Sticks

One is interested in surface phenomena, such as behavior of adsorbed molecules, or patterns of adsorbates, in response to various probes. To calculate local densities of states, conductance, vibrational modes and so on that arise from coupled electromagnetic, mechanical and quantum mechanical mechanisms, it is useful to develop as simple a model as possible. For the most part, the theories involved are in terms of coupled partial differential equations, often nonlinear, and the systems have extremely complex geometries. The plan here is to replace these formalisms by a far simpler one.

All the identities and integral theorems of vector calculus are contained in the calculus of differential forms. The analogy between the exterior calculus of differential forms and a discrete calculus based on boundary and coboundary operations on a cell complex yields discrete lattice models for an array of interesting physical phenomena. These include coupling between electrons and elastic distortions and a polarization field or magnetic interactions. The models are based on arbitrary of coupled scalar and polar or axial vector field quantities. It turns out that these can be manipulated in a unified way as conveniently as can the standard scalar tight binding models for independent electrons.

I will explain the basic ideas of this primarily topological formalism and hope to show some results for both complicated limited geometries and for an infinite hierarchy of uniform lattices.