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.

Commutative algebra seminar

Date and time: Tuesday, 20 December 2022, 3pm

Venue: Ramanujan Hall

Speaker: Shravan Patankar, University of Illinois, Chicago, IL, USA

Title: Vanishing of Tors of absolute integral closure in equi-characteristic zero

Abstract: We show that the vanishing of Tors of the absolute integral closure forces regularity assuming further that the ring under consideration is IN-graded of dimension 2. This answers a question of Bhargav Bhatt, Srikanth Iyengar, and Linchuan Ma. We use almost mathematics over R+ to deduce properties of the Noetherian ring R and the theory of rational surface singularities. In particular, in spite of being a question purely in commutative algebra our proof uses algebro-geometric methods.