Electrons in atoms, molecules and solids are described by the
fermionic (antisymmetric) wavefunctions obtained as solutions of
Schrodinger equation. The antisymmetry means that the wavefunction
changes the sign whenever two electrons exchange their positions.
The place where the fermionic wavefunction goes through zero is called
the fermion node. The fermion node condition Psi(r1,r2,...,rN)=0,
where arguments are positions of N fermions, specifies a (3N-1)-
dimensional hypersurface in the 3N-dimensional space. I will talk
about known properties of fermion nodes (very few) and mention some
of the unsolved problems (many). I will point out the key importance
of fermion nodes for stochastic methods of solving the Schrodinger
equation (quantum Monte Carlo) and give two examples for which
the exact nodes are known. I will try to explain how one can construct
approximate wavefunctions with "reasonable" nodes and mention
about possible projects such as investigations of fermion nodes for
a few-electron system(s). Typical questions to ask: is there a topological
change in the node when going from one approximate solution to another,
are there any general features which could be explored for obtaining
more accurate solutions, etc.
Here are some illustrations of
fermion nodes and some
quantum Monte Carlo work. Please visit the
homepage of the NCSA Quantum Simulations of
Condensed Matter Systems Group.
11 AM Tuesday 1 July 1997
Topology of Riemannian manifolds with boundary having
cut locus of low degree
Previously, Stephanie Alexander and I have shown that if a Riemannian
manifold with boundary has
sufficiently small curvature-normalized inradius, then its cut
locus has at most a given degree. For 3-manifolds with a 2-sphere
as boundary, we classify those with cut locus of degree at most 3
in terms of a ``representing graph''. Filling in the boundary with a
disk gives a classification of Riemannian 3-manifolds without
boundary having a point with cut locus of degree 3. The homology
of each of these manifolds is described in terms of its
representing graph.
11 AM Tuesday 8 July 1997
Ropelength of Knots
Suppose we want to tie knots and links in one-inch rope. What length of
rope is required to construct each different knot type? To answer this
question we define a new notion of thickness for space curves, based
on the total turning angle of arcs. We will relate this thickness
to earlier measures based on curvature bounds and on distortion of arcs.
For the new notion, we can prove that in each knot type there is
a shortest curve of thickness one, and we have some further understanding
of the geometry of such curves. We will also discuss new results about
the asymptotic growth of ropelength for different families of knots
with increasing crossing number.
11 AM Tuesday 15 July 1997
First Summer Projects Potpourri
Birgit Bluemer, John Estabrook, Ulises Cervantes
In lieu of a single speaker, several of us will present summaries of
projects under way this summer. Birgit
will report progress in (1) inserting the graviLens into Marcus
Thiebaux's Virtual Director, and (2) factoring the rtica into a
calculator/viewer pair as prototyped by illiConnect. John and
Ulises will report on the current status of the Vosaic/MBone project.
We will see brief excerpts of three prior seminar talks by Lubos
Mitas, Richard Bishop, and John Sullivan, as they will appear in
a Java-enabled Netscape session, once we put these files on illiWeb.
Other projects will be presented at future seminars. The summaries
will be available on Vosaic.
11 AM Tuesday 22 July 1997
Second Summer Projects Potpourri
John Estabrook, Chris Hartman, Matthew Stiak, George Francis
We continue the presentation of brief summaries of summer projects.
John will demonstrate graviConnect and review the status
of the illiConnect project. Chris will demonstrate
equiVert , which is the current, OpenGL version of our
Minimax Eversion viewer. It connects with a possibly
remote and parallelized Brakke Evolver via illiConnect
sockets. Chris will also demonstrate his implementation of
homotopies in philEver. Stik will demonstrate the gui
for philEver. George will, if there is time, describe
Bryce DeWitt's proposed sphere eversion and how the gui/philever
system will realize it.
11 AM Tuesday 29 July 1997
Virginia Tech Department of Material Science and Engineering
Eigenvalue-Eigenvector Glyphs: Visualizing Zeroth, Second,
Fourth and Higher Order Tensors in a Continuum
In engineering and the sciences many examples now exist where visual
tools
have been effectively used by researchers to study massive-complex data
sets
generated by supercomputer simulations. With well designed visual tools
our
data-rich but information-poor world has been transformed into an
information
rich experience from which new insights are possible. Greater access to
computer resources in all disciplines has prompted users to go beyond
the
"number crunching" paradigm and establish visual methodologies that are
discipline independent. In mechanics visualizing gradients in scalar
properties (zeroth order tensors) and glyphs of stresses (second order
tensors)
have become the most common examples. Closer examination reveals that
the
common eigenvalue problem, with eigenvalues of zeroth order and
eigenvectors of
first order, can be used to characterize physical properties that are
tensors
of second, fourth, and higher order. Higher order tensors can be used
to
characterize material anisotropy. For some anisotropic materials new
geometries have been discovered that can now be used to subclassify
within
existing material class symmetries. The need to study the distribution
of
tensorial properties in a continuum has lead to the development of a
visual
method that can be used to study tensor equation invariance. This same
visual
method can be used to extract simple functions from "raw data": massive
data
sets resulting from numerical simulations, computer controlled
experiments or
both. Portions of this presentation can be accessed
.
here.
