UTA Department of Mathematics

Metroplex Algebraic Geometry, Algebra and Number Theory (AGANT) Seminar,
a joint venture of   UNT,   UTA   and   TCU

Date/Time/Room: Friday (10/24/2008) at 4:00 pm in 304 Pickard Hall

Speaker: Dr. Norman Johnson, Professor,
Department of Mathematics, University of Iowa.

"Combinatorics of Spreads and Parallelisms"

Abstract: 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 http://www.uta.edu/maps/, 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.