Math Links

Flocks of Cones

Hyperoval Page


Bill Cherowitzo's
Home Page


[Fano's Plane with Fractal Background] Fano's Plane, a finite geometry

Research Interests | Ref. Pubs. | In Prep | Postscript Files | Talks | Web Pages

Research Interests

My primary research is in the area of Finite Geometry, in particular the combinatorial structures known as ovals (or hyperovals) which can be found in these geometries. The Math Reviews section which covers this material is 51E21. Related to this area and also of interest to me is Design Theory (05Bxx), Coding Theory (94Bxx) and Cryptology (94A60) in so far as it relates to these areas. My work tends to be heavily algebraic, in particular dealing with finite fields and their structure (11Txx).

My most recent activity involves Flocks of Cones in PG(3,q). This work can be directly viewed at the Flocks of Cones web site.

Refereed Publications

2010 The Classification of Inherited Hyperconics in Hall Planes of Even Order, Europ. J. Comb, 31(2010), pp. 81-86. (available on-line 12 May 2009)

2009 Constructing the Tits Ovoid from an Elliptic Quadric, Innovations in Incidence Geometry, 8-9(2009), pp. 73-79.

2008 Groups of Hyperovals in Desarguesian Planes (with L. Bayens and T. Penttila), Innovations in Incidence Geometry, 6-7(2008), pp. 37-51.

2005 Hyperfocused Arcs (with L. Holder), Bull. Belg. Math. Soc. (Simon Stevin), 12(2005), pp. 685 - 696.

2003 The cyclic q-clans with q = 2e (with S.E. Payne), Adv. Geom., Special Issue(2003), S158-S185.

2003 A Unified Construction of Finite Geometries Related to q-clans in Characteristic Two (with C.M. O'Keefe and T. Penttila), Adv. Geom., 3(2003), 1-21.

2000 The Lunelli-Sce Hyperoval in PG(2,16) (with J.M.N. Brown), J. Geom., 69, pp. 15-36.

1998 Alpha-Flocks and Hyperovals, Geom. Ded., 72, pp. 221-246.

1998 Monomial Flocks of Monomial Cones in Even Characteristic, J. Belg. Math. Soc. - Simon Stevin, 5, pp. 241-253.

1998 Alpha-Flocks with Oval Herds and Monomial Hyperovals (with L. Storme), Finite Fields and their Applications, 4, pp. 185-199.

1996 Hyperovals in Desarguesian planes: An update, Discrete Mathematics, 155, pp. 31-38.

1996 Flocks and Ovals (with I.Pinneri, G.Royle and T.Penttila), Geom. Dedicata,60, pp. 17-37.

1991 Hyperovals in the Translation Planes of Order 16, J. of Combinatorial Mathematics and Combinatorial Computing, 9 (1991), pp 39-55. (.pdf file - 5.8 MB)

1990 Ovals in the Figueroa Planes, Journal of Geometry, 37 , pp. 84-86. (.pdf file - 1.9 MB)

1988 Hyperovals in Desarguesian Planes of Even Order, Annals of Discrete Mathematics, 37 , pp. 87-94.

1988 Closed Arcs in Finite Projective Planes, Congressus Numerantium, 62, (1988), pp. 69-77. (.pdf file - 3.8 MB)

1986 Ovals in the known planes of order nine, International Conference on Finite Geometry and Combinatorics, pp. 33-38.

1986 On the Projectivity of B-Ovals, Journal of Geometry, 27, pp 119-139. ( .pdf file - 10.1MB)

1986 Ovals and Other Configurations in the known Planes of Order 9 (with R. Killgrove and D. Kiel), Congressus Numerantium, 55, pp. 167-179. ( .pdf file 9MB )

1985 Harmonic Ovals of Even Order in Finite Geometries, edited by C.A. Baker and L.M. Batten, Marcel Dekker, pp. 65-81.

1985 Translation B-Ovals, Congressus Numerantium, 47, pp. 161-172.

Publications In Preparation

Accepted (2009) Parallelisms of Quadric Sets (with N.L. Johnson), to appear in Innovations in Incidence Geometry.

Submitted (2009) Net Replacements in the Hughes-Kleinfeld Semifield Planes (with N.L. Johnson).

Submitted (2009) (alpha)-flokki, partial (alpha)-flokki and Baer groups , (with N.L. Johnson).

Submitted (2009) Flocks of Cones: Herds and Herd Spaces (see web page below)

Submitted (2009) Flocks of Cones: Star Flocks (see web page below)

In Prep Flocks of Cones: Semifield Type Flocks (see web page below)

In Prep 1-Factorizations and Near 1-Factorizations Associated with Conics (with L. Holder)

In Prep Ovals and Hyperovals in Hall Planes

Postscript Files of Some Papers

Recent Presentations

This link will go to an index of recent presentations that I have given where you can view the slides of the talks. In some cases there is more material presented in these slides than was actually given in the talk.

Index of Recent Presentations

Research Oriented Web Pages