Title: Ovoids in PG(3,q) and Algebraic codes
Speaker; Prof. N. S. Narasimha Sastry, Formerly from ISI Bangalore and IIT Dharwad
Day, Date and Time: Tuesday, 20th December 2022 at 2 PM
Venue: Ramanujan Hall, Dept of Mathematics
Abstract: Ovoids in PG(3,q) are the Incidence geometric analogues of spheres in Euclidean 3-space. If q is odd,
Baralotti and Panella showed independently that elliptic quadrics are the only ovoids in PG(3,q). Further, an ovoid
and its set of tangent lines determine each other. However, if q is an odd power of two, then apart from elliptic
ovoids (which exist for all q), PG(3,q) admits one more projective class of ovoids which are not projectively equivalent
to elliptic ovoids. They were discovered by Tits, using the graph - field outer automorphism of PSp(4,2^{2n+1!}), and its
stabilizer in PSp(4,q) (called the Suzuki simple group, the same as ^2 B_2(q) in Lie notation) was discovered earlier
by Suzuki as the final piece in the long series of works on the classification of finite Zassenhaus groups by Zassenhaus,
Ito, Feit and Suzuki. Further, the set of tangent lines of two ovoids can coincide even if they are projectively nonequivalent.
These are the only families of ovoids in PG(3,q) known and classification of ovoids in PG(3,q) is a major problem in Incidence
Geometry. Because of their connections to many other combinatorial structures ( like inversive planes, generalized quadrangles,
group divisible designs, ...) and the very exceptional behavior of the Suzuki simple group and the Tits ovoid, understanding the properties of ovoids in general, and their distribution in PG(3,q) in particular, are of considerable significance.
In this talk, I will present some facts known about ovoids in general, their distribution and the role of algebraic codes in
understanding them. An effort will be made to clarify all the basic notions involved.