31 Mart 2017,
Gebze Teknik Üniversitesi Matematik Bölümü Seminerleri
The geometry of finite non-associate division algebras.
Sabancı University, Türkiye
Finite non-associative division algebras (also called semifields) were first studied
by L. E. Dickson in 1905 as axiomatically defined algebraic structures satisfying
almost all of the axioms in the definition of a finite field. A study which naturally
arose in the aftermath of the well-known theorem: A finite skew-field is a field.
This theorem essentially says that the axiom of commutativity of multiplication
is implied by the other axioms of a finite field.
Dickson showed that this does not hold true for the axiom of associativity of multiplication,
by constructing explicit examples of finite non-associative division
algebras. Later, when the coordinatisation of projective planes was established
(1940’s), it turned out that Dickson’s examples also implied the existence of projective
planes in which Desargues configuration does not hold, a configuration
of points and lines, whose importance emerged from Hilbert’s axiomatisation of
Once this connection between the theory of semifields and the theory of projective
planes was established, the topic received a considerable amount of attention
from both geometers and algebraists. A good survey of the state of the
art at that time, can be found in the book Projective planes (1970) by Hughes
and Piper, the book Finite Geometries (1968) by Dembowski, or in Knuth’s
dissertation Finite semifields and finite projective planes (1963).
In the last decade a second wave of interest in the theory of semifields arrived,
partly due to applications, partly due to new connections between the algebra
and the geometry of semifields, e.g. , . For a survey, see . In this
talk we will elaborate on the geometry of finite semifields, and explain how
the interplay between algebra and geometry has allowed us to obtain many
new results, e.g. [4, 5], including the classification of 8-dimensional rank two
Gebze Teknik Üniversitesi, Matematik Bölümü Seminer Odası