Metroplex Algebraic Geometry, Algebra and
Number Theory (AGANT) Seminar,
a joint venture of
Friday (10/24/2008) at 4:00 pm in 304 Pickard Hall
Dr. Norman Johnson, Professor,
Department of Mathematics,
University of Iowa.
"Combinatorics of Spreads and Parallelisms"
This seminar will give an introduction and overview of spreads of vector
spaces and parallelisms of projective spaces (a projective space is the
lattice of vector subspaces of a vector space). A k-spread of a vector
space of dimension kt over a field K is a partition of the
non-zero vectors by a set of mutually disjoint (as subspaces)
k-dimensional subspaces. Hence, a k-spread is a set of lines
such that every point (non-zero vector) is incident with exactly one line.
When t = 2, every k-spread constructs an affine plane whose
points are the vectors and whose lines are translates of the
k-subspaces of the k-spread. This affine plane admits a
translation group of the vector space as a group of automorphisms
(collineations) and hence is called a ``translation plane''. A
``k-parallelism'' is a set of k-spreads such every
k-dimensional subspace is contained in exactly one k-spread of
the set. Considering that a spread of a point-line geometry is a set of
lines such that every point is incident with exactly one line of the set, a
parallelism is a set of spreads such that every line of the geometry is
contained in exactly one spread of the set. When t = 2, finite
k-spreads and finite k-parallelisms form the building blocks
of the geometry of finite projective and affine planes.
The only background required for this seminar is a basic understanding of
linear algebra and group theory.
A map of UTA may be found at
and directions to UTA are on that website.
Note that PKH is the triangular building in the central-campus region of
the map. Select Pickard Hall from the building selector. PKH is next to
the ``garage'' & is on Nedderman Dr between Planatarium Place (aka College
St) & West St. West St is one-way northbound between W Mitchell St and
3rd St. Parking is available in the nearby (East and SE of PKH) parking lots
with a UTA permit (but many people park there late on Fridays with no permit
and are not caught). There are a few metered parking spots on the East side
of Pickard Hall on West St. Visitor (free), & metered, parking is available
by Davis Hall (marked DH in SW part of the submap). A parking garage
(expensive) is nextdoor to PKH (use West St to get to it). Note that if
heading south on Cooper, then no left turn is allowed to Nedderman Dr.
Note that PKH 304 is on the 3rd floor of PKH; turn right out of elevator
on 3rd floor.