PUBLIC VIVA-VOCE NOTIFICATION
Subject : Mathematics
Wednesday, 19 April 2023, 03:30 PM.
Venue: Seminar Hall
Homological Invariants of Hibi Rings and Polyominoes
Dharm Veer
Chennai Mathematical Institute.
19-04-23

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Abstract

In this thesis, we study the minimal graded free resolution of Hibi rings and the h-polynomial of polyomino algebras.

Green and Lazarsfeld defined property Np to study the graded minimal free resolution of S/I, where S is a polynomial ring over a field and I is an ideal generated by quadratics. The ring S/I satisfies property Np if S/I is normal and the graded minimal free resolution of S/I over S is linear up to p-th position. We prove necessary conditions for Hibi rings to satisfy Green-Lazarsfeld property Np for p = 2 and 3. We also show that a Hibi ring satisfies property N4 if and only if either it is a polynomial ring or it has a linear resolution. In particular, it satisfies property Np for all p.

Let P be a polyomino. Qureshi associated a finitely generated graded algebra K[P] over a field K to P. Rinaldo and Romeo showed that if P is a simple thin polyomino, then the h-polynomial of K[P] is the rook polynomial of the polyomino P and they conjectured that this property characterises thin polyominoes.

In this thesis, we verify the conjecture of Rinaldo and Romeo when P is a non-thin convex polyomino such that its vertex set is a sublattice of N2 . We also show that the Gorenstein rings associated with simple thin polyominoes satisfy the Charney-Davis conjecture.

All are invited to attend the viva-voce examination.