## Abraham A. Ungar

0000-0003-2882-1663Abraham A. Ungar was born in Haifa, Israel. He attended the Hebrew University in Jerusalem as both an undergraduate and a graduate student, earning his B.S. and M.S. in 1965 and 1967. He completed his Doctorate in mathematical geophysics at Tel-Aviv University in 1973 under the supervision of Zepora Alterman. After being a Postdoctoral Fellow at the University of Toronto, Department of Physics (1974–1975), he took faculty positions at the National Research Institute for Mathematical Sciences (Pretoria, South Africa, 1975–1977), Rhodes University (Grahamstown, South Africa, 1978–1983), Simon Fraser University (Vancouver, Canada, 1983–1984), and North Dakota State University (Fargo, ND, USA, 1984–present), where he currently teaches. His interest in understanding why Einstein velocity addition law of relativistically admissible velocities is seemingly structureless, being neither commutative nor associative, led him to the discovery of the gyrocommutative gyrogroup structure that Einstein velocity addition and Möbius addition encode. Both Einstein addition and Möbius addition thus turn out to be gyrocommutative gyrogroup operations, just as the common vector addition is a commutative group operation. Furthermore, both Einstein gyrogroups and Möbius gyrogroups admit scalar multiplication, giving rise to Einstein gyrovector spaces and Möbius gyrovector spaces. The latter form, respectively, the algebraic setting for the Cartesian-Beltrami-Klein ball model of hyperbolic geometry and for the Cartesian-Poincaré ball model of hyperbolic geometry, just as vector spaces form the algebraic setting for the standard Cartesian model of Euclidean geometry. Since 2001, he authored six books on analytic hyperbolic geometry and its applications in Einstein's special theory of relativity.

Biography Updated on 1 March 2011