Research Interests:

I consider myself a combinatorialist, specializing in finite geometries. Most of my work has dealt with constructing various geometrical structures or configurations, and then analyzing or characterizing those structures. The general geometric setting is typically some projective or affine space (or perhaps a circle geometry of some type) over a finite field. Some of my work on caps , point sets in some finite projective space with no three points collinear, has application to coding theory. Techniques used are typically a blend of finite field theory, elementary number theory, and finite group theory together with the usual counting arguments. In recent years I have been using various software packages, such as Magma, to help construct relatively ‘large’ examples and to analyze those examples in the hope of recognizing patterns that can be generalized and eventually proven as theorems. Thus I use the computer in the discovery process as opposed to using the machine as a ‘number-cruncher.’

Ph.D. Students

• Bryan Petrak (jointly advised with Felix Lazebnik, 2012)
• Craig Culbert (2009), Kutztown University, Kutztown, PA
• Marina Capursi (2006), University of Central Florida, Orlando, FL
• Keith Mellinger (2001), chair of Mathematics, University of Mary Washington, Fredericksburg, VA
• Jeremy Dover (1996), Dover Networks LLC, Edgewater, MD
• Kenny Wantz (1995), Regent University, Virginia Beach, VA
• Richard Weida (1987), chair of Mathematics, Lycoming College, Williamsport, PA

Grants

• National Security Agency, H98230-09-1-0074, 2008-2010
  “Semifield Spreads, Nearfield Spreads, and Related Combinatorial Objects”
• National Security Agency, H98230-06-1-0071, 2005-2007
  “Ovoids and Spreads in Polar and Projective Spaces”
• National Security Agency, MDA904-03-1-0099, 2002-2004
  “Problems in Fnite Geometry”
• National Security Agency, MDA904-00-1-0029, 1999-2001
  “Partitioning Problems in Finite Geometry and their Applications”
• North Atlantic Treaty Organization, SA.5-2-05(CRG.970185), 1997-1999
  “Geometrical Configurations and their Symmetries”
  joint with G. Korchmaros (Potenza, Italy) and J.W.P. Hirschfeld (Brighton, England)
• National Security Agency, MDA904-94-H-2033, 1993-1995
  “Conics and Unitals in Finite Desarguesian Planes”
• National Science Foundation, DMS-8804569, 1988-1989
  “Mathematical Sciences: Research Equipment”
  joint with R.D. Baker, R.P. Gilbert, J. Hemmeter, and P.B. Monk
• Alfred P. Sloan Foundation, 1983
  “Delaware Revision in Undergraduate Mathematics”
  joint with R.D. Baker and I. Stakgold
• University of Delaware Research Foundation, 1978-1979
  “Blocking Sets and Partial Spreads in Projective Spaces”