IPDF Talk
Speaker: Supriyo Jana, IITMadras
Host: G.K. Srinivasan
Title: Dynamics and integrability of polynomial vector fields on certain compact manifolds
Time, day and date: 11:00:00 AM - 12:00:00 PM, Tuesday, May 19
Venue: Ramanujan Hall
Abstract: We present a study of polynomial vector fields on algebraic manifolds, including the n-dimensional sphere and product of spheres. We provide a complete algebraic characterization of polynomial vector fields on $S^n$ of arbitrary degree, and investigate their first integrals and invariant algebraic sets. Sharp bounds on the number of invariant meridian and parallel hyperplanes are established. We also determine when a Hamiltonian vector field on Euclidean space induces a vector field on these manifolds. Darboux theory of integrability is employed to show that cubic Kolmogorov vector fields on the product of 1-sphere and 2-sphere, while not Hamiltonian, admit rational first integrals.

We conclude by motivating the study of polynomial vector fields on real projective spaces. We outline several open questions concerning the characterization of such vector fields, the classification of global phase portraits on $RP^2$, and the extension of Darboux integrability theory to the projective setting.

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.

PDE Seminar
Speaker: Pardeep Sharma
Host: Harsha Hutridurga
Title: Impulsive inverse problem for the Navier-Stokes equations.
Time, day and date: 10:00:00 AM – 11:00:00 AM, Friday, May 22
Venue: Online (https://zoom.us/j/95223583387?pwd=i49UhXlwzHiLr6aLK6tqnlesWKIc8C.1)
Abstract: Attached (https://drive.google.com/file/d/1R6eEC2IhOQldZlImzdyH2z3OfWwTYANF/view?usp=sharing)

PDE Seminar
Speaker: Dr. Rakib Mondal
Host: Sivaji Ganesh Sista
Title: Rarefaction Wave Interaction and Existence of Global Solutions for a Hyperbolic System of PDEs
Time, day and date: 11:30:00 AM – 12:30:00 PM, Friday, May 22
Venue: Online (https://meet.google.com/ika-sdnw-yuf)
Abstract: In this talk, we discuss the first-order hyperbolic systems of partial differential equations and their mathematical challenges due to lack of regularity. In particular, we introduce a simplified 1-D system of balance laws governing blood flow with time-dependent body force. We discuss the interaction of two centered rarefaction waves and show that the interaction gives rise to a Goursat boundary value problem (GBVP). By transforming the system into Riemann-invariant coordinates, we discuss the existence and uniqueness of a global $C^1$ solution to the GBVP.

We further extend this system of balance laws considering both time-space dependent body force and the viscosity resistance, and prove the existence and stability of global weak entropy solutions. We establish a priori uniform $L^\infty$ bounds for a sequence of solutions to the regularized two-parameter viscosity-flux approximated systems. Using the vanishing-diffusion limit and the compensated compactness framework, we show the strong convergence and establish the global weak entropy solutions to the original Cauchy problem with large initial data.

IPDF Talk
Speaker: Rahul Bhardwaj, IIT Ropar
Host: Suman Kumar Sahoo
Title: Reconstruction Results for Inverse Problems in Integral Geometry and Hyperbolic PDEs
Time, day and date: 4:00:00 PM – 5:00:00 PM, Friday, May 22
Venue: Online (https://zoom.us/launch/jc/93166883629) (Meeting ID: 931 6688 3629 Passcode: 506739)
Abstract: In this talk, we address two inverse problems arising in integral geometry and hyperbolic partial differential equations (PDEs). The first problem concerns the reconstruction of a vector field in the plane from a set of generalised V-line transforms, namely longitudinal and transverse V-line transforms. Specifically, we focus on recovering a vector field which is supported inside a disk of radius R centered at the origin. This recovery process is accomplished through a combination of the aforementioned generalized V-line transforms, using two different techniques for different sets of data. The second problem concerns an inverse problem for a semi-linear wave equation posed on a bounded subset of Euclidean space. The objective is to reconstruct the coefficients from the associated Dirichlet-to-Neumann map. The analysis is based on the method of higher-order linearization.