

SUMMARY OF RESEARCH ACCOMPLISHMENTS
Though originally developed as a natural outgrowth
of multivariable calculus, algebraic topology and differential forms
have become an essential tool used to formulate many basic laws of
physics. Through my research this area of mathematics has found a
natural application to many areas of electrical engineering and
computational electromagnetics. A strong theme in my research is the
identification of geometric and topological aspects, which shed light on
dimensional dependence in the complexity of engineering problems and
their algorithmic solution. Specifically,
(numbers refer to publications list, see also “other significant
conference papers”)
·
Cohomology theory and 3D boundary value problems
[29],[30],[21],[25]Techniques from algebraic and differential topology
are used in the formulation of variational boundary value problems and
in the development of algorithms for handling topological constraints in
numerical solutions. These techniques are essential to the efficient
solution of magnetostatic and eddy current problems in threedimensions.
In this area Hodge theory for manifolds with boundary gives the right
framework for investigating the global aspects of gauge transformations
and conservation laws for both primal and dual variational principles
associated with a given boundary value problem.
·
Data Structures for Finite Element Analysis
[6],[17], Results on data structures for the finite element analysis of
three dimensional electromagnetic fields have come about as a byproduct
of applying topological tools to what I thought were more challenging
problems. First, Whitney forms were mentioned in my Ph.D. thesis as the
obvious way of reconciling Maxwell’s equations with simplicial data
structures. It was only through the heroic efforts of Alain Bossavit and
the work at EDF that this point of view has become widely accepted.
Second, in the process of freeing the computation of homology groups of
floatingpoint arithmetic, my algorithms for the computation of cuts for
scalar potentials were rooted in the notion of a simplicial complex.
This data structure is quickly becoming a basic tool in the field and
the implicit homotopical information is leading to some very surprising
new developments. Third, the Whitney form dicretization of helicity
functionals has lead to unexpected ties to basic three dimensional
manifold invariants through the formalism of quantum field theory. The
irony is that in all of these cases, the data structures have gained
wider acceptance than the concrete problems where I introduced them as
essential tools.
·
Cuts for magnetic scalar potentials
[5],[7],[9],[21],[23],[25],[27],[28]Cohomology theory is exploited in
the scalar potential formulation of quasistatic magnetic field problems
involving multiply connected current free regions. Specifically, torsion
information in integral cohomology groups yields the information
required for coupling magnetic scalar potentials to stream functions
describing surface currents. Level sets of harmonic maps into the circle
(the EilenbergMacLane space K(Z,1)) play a key role in representing
“cuts” for scalar potentials in terms of embedded manifolds. Cohomology
operations (Bockstein homomorphism and Massey products) yield many
intuitive insights into the topological structure of 3D boundary value
problems. All of the above aspects can be treated systematically and
algorithmically in the context of the finite element method by appealing
to simplicial cohomology theory. Our work shows that beyond the work
required to solve Poisson's equation, an O(n^{4/3}) time and O(n^{2})
storage algorithm can always be implemented. When heuristic assumptions
are made about the region, (the exact consequences of which are being
explored), we have implemented an O(n) storage algorithm which requires
the solution of Poisson's equation and an additional O(n^{2})
time. This work enables a speed up of a factor of 79 for solving time
varying problems (for example: eddy current nondestructive testing) and
nonlinear problems such as magnet and motor design. Recently, this work
has opened the door to examining the role of knotted current paths in
device design.
·
Helicity functionals in compuational electromagnetics
[10],[19], [24], [26] The metric independence of helicity
invariants has been identified as being very useful in the finite
element formulation of certain 3D boundary value problems of
technological importance. The use of a helicity functional leads to
smaller condition numbers for resulting finite element matrices as well
as a simpler matrix assembly and numerical methods. These techniques
are presently being developed in the context of nonlinear and inverse
problems.
Here, a mathematical
remark is in order. Early in the investigation of Whitney form based
finite element analysis of helicity functionals, I developed a discrete
version of metric invariance at about the same time as Ed Witten
developed a quantum field theoretic view of the Jones polynomial. The
latter has created a lot of interest in “Abelian ChernSimmons Theory”
which, being related to the helicity functional, is enriched by my work
on Whitney form discretizations. Specifically, both Werner Muller's
proof of the RaySinger conjecture (equating analytic torsion with
Reidemeister torsion,) and Albert Schwartz's reinterpretation of the
conjecture in terms of quantum field theoretic partition functions, can
be made very concrete by appealing to my Whitney form discretizations of
helicity functionals. On manifolds with boundary, the partition
function is no longer Reidemeister torsion and hence no longer a
topological invariant. Instead, it is a product of a topological term
and the determinant of a Dirichlet to Neumann map. This and other
aspects of Whitney form discretizations of helicity functionals yield
geometric insights into threedimensional impedance tomography.
·
Micromagnetics [8], [14], [15], [18], [20], [22] In
the area of micromagnetics, the investigation of Bloch points and their
effect on vertical Bloch line memories requires a 3D formulation which
does not make use of the usual ``thin film'' assumptions. Here the use
of obstruction theory from algebraic topology naturally points to the
important role played by the degree of a mapping between compact
twodimensional manifolds. The expression of this winding number as an
integral offers insight into Bloch points both on the conceptual level
and in the context of numerical simulations. These techniques are
similar to those encountered in condensed matter physics and the
classification of defects in ordered media. By pursuing the use of
topological methods, a complete understanding of the "topological
accessibility" problem has been obtained by appealing theory underlying
the Hopf invariant of algebraic topology. Here one wants to know whether
one micromagnetic configuration can be deformed into another in a
continuous fashion. These topological conservation laws developed for
micromagnetics have been considered in the context of reversible
computation with micromagnetic structures.
·
Arithmetical Topology for the Analysis of Algorithms
[11], [12]
Simple considerations of combinatorial topology enable one to estimate
the number of degrees of freedom and the number of nonzero entries in a
finite element stiffness matrix arising from an unstructured mesh. Such
an analysis enables one to derive exact formulas for the number of
FLOPS/CG iteration and evaluate the relative merits of different finite
element formulations of three dimensional boundary value problems for
vector fields (e.g. tetrahedral vs. hexahedral element; node vs. edge
based elements). These techniques are used to articulate why direct
solvers such as Gaussian elimination are often used for accurate two
dimensional analyses, but semidirect algorithms such as conjugate
gradients are indispensible for threedimensional problems. These
analyses are also applicable to solution strategies involving h
and p refinement on both serial and massively parallel computers
