IPDF Talk
Speaker: Annot Kumar Yadav
Host: Tony J P
Title: Bounds on Hilbert coefficients and reduction number
Time, day and date: 3:30:00 PM – 4:30:00 PM, Wednesday, May 20
Venue: Online (meet.google.com/qwt-rmfj-hgm) 
Abstract: Let $(A, m)$ be a Cohen–Macaulay local ring and $I$ an $m$-primary ideal. In this talk, we study how Hilbert coefficients, reduction numbers, and the Ratliff–Rush filtration govern the depth and homological properties of associated graded rings and related blow-up algebras. Our work is motivated by a conjecture of Maria Evelina Rossi, which predicts a linear upper bound for the reduction number in terms of multiplicity and the first Hilbert coefficient.
Using higher Hilbert coefficients, particularly the third Hilbert coefficient, we obtain improved bounds on reduction numbers under suitable depth conditions. In dimension three, our results provide the best-known bound for a class of integrally closed ideals. We also establish inequalities for the third Hilbert coefficient and show that equality cases correspond to favourable structural properties of the Ratliff–Rush filtration. These results extend to $I$-admissible filtrations, where Rossi’s conjecture is verified in certain extremal cases.
Finally, we study the behaviour of the Ratliff–Rush filtration modulo superficial elements and obtain computable criteria for its compatibility with superficial elements. Overall, the talk highlights how numerical invariants detect structural properties of blow-up algebras.
This talk is based on joint work with Kumari Saloni.

Mathematics Colloquium
Speaker: Saurabh Kumar Singh
Host: Sumit Kumar
Title: Counting special points on quadratic surfaces.
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, May 20
Venue: Ramanujan Hall
Abstract: We show that the modern version of the circle method powered by the equidistribution of quadratic roots allows us to count special points on quadratic surfaces. For example, we obtain asymptotic for integer points on quadratic surfaces with prime coordinates and in short intervals.