PUBLIC VIVAVOCE 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. 190423 Abstract In this thesis, we study the minimal graded free resolution of Hibi rings and the hpolynomial 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 pth position. We prove necessary conditions for Hibi rings to satisfy GreenLazarsfeld 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 hpolynomial 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 nonthin 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 CharneyDavis conjecture. All are invited to attend the vivavoce examination.
