University of Colorado Denver College of Liberal Arts and Sciences

Bio

My work has generally been focused on the interplay between algebra and geometry in the construction and study of new infinite families of finite generalized quadrangles, non-classical ovals in finite desarguesian projective planes of even order, and in some cases new families of finite projective planes. This work has afforded me the opportunity to serve as a visiting research faculty member on several occasions at universities in Belgium, Italy and Australia. Recently I have begun to indulge my interests in algebraic number theory and its use in finite geometry, and hope to expand my research expertise to include algebraic number theory in my work. At the present time, Dr. Laurel Rogers (University of Colorado at Colorado Springs, Professor Emerita) and I are writing a major monograph titled TOPICS IN FINITE GEOMETRY: OVALS, OVOIDS AND GENERALIZED QUADRANGLES.

Select Publications

Books:
S. E. Payne and J. A. Thas, Finite Generalized Quadrangles, a research monograph first published by Pitman in 1984 with a second edition published by the European Mathematics Society in 2009.

I. Cardinali and S. E. Payne, q-Clan Geometries in Characteristic 2, a research monograph in the series Frontiers in Mathematics published by Birkhauser in 2007.

Journal Articles:
S. E. Payne, "Nonisomorphic generalized quadrangles," Journal of Algebra, 18 (1971), 201 – 212.

S. E. Payne, "A complete determination of translation ovoids in finite desarguesian planes," Rend. Accad. Naz. Lincei LI (1971), 226 – 229.

S. E. Payne, "A new infinite family of generalized quadrangles," Congressus Numerantium, 49 (1985), 115 – 128.

S. E. Payne, "An essay on skew translatioin generalized quadrangles," Geometriae Dedicata, 32 (1989), 93 – 118.

J. A. Thas and S. E. Payne, "Spreads and ovoids in finite generalized quadrangles," Geometriae Dedicata, 52 (1994), 227 – 253.

S. E. Payne, "The fundamental theorem of q-clan geometry," Designs, Codes and Cryptography, 8 (1996), 181 – 202.

N. L. Johnson and S E. Payne, "Flocks of Laguerre planes and associated geometries," in Mostly Finite Geometries, (ed. N. L. Johnson), Marcel Dekker, 1997, 51 – 122.

M. Brown, C. M. O’Keefe, S. E. Payne, T. Penttila, and G. Royle, "Spreads of T_2(O), alpha – flocks and ovals," Designs, Codes and Cryptography, 31 (2004), 252 – 282.

Courses Taught

Math 3140, 7413, 7414: abstract algebra

Math 3191, 5718: linear algebra, both undergraduate and graduate level

Math 3195: introduction to ordinary differential equations with linear algebra

Math 4110, 5110: elementary number theory and introductory algebraic number theory

Math 4310, 4320, 5070: introduction to real analysis

Math 7409: applied combinatorics

Math 7421, 7023: projective geometry, with emphasis on finite geometries