Title: Diophantine tuples

Abstract: A tuple of numbers $(a,b,c)$ is a Diophantine tuple if the product of two distinct elements added by $1$ is a perfect square. This was introduced by Diophantus and he gave an example of such a quadruple of rational numbers. Fermat came up with such a triple with integer entries. Many generalizations of this notion came up over the years.

The aim is to talk about these tuples and their generalizations. In the way I will touch upon some of my works in this direction.

Title: Understanding of the Solute Dispersion in Blood Flow in Microvessels.

Abstract: This lecture is intended to present the developments on the solute dispersion in circular tubes. The study on solute dispersion was initiated by G. I. Taylor for the Newtonian fluid flow. A plethora of literature is available on the solute dispersion in Newtonian and non-Newtonian fluid flow in tubes of various cross-sections. Current research shows the study is lead up to exploring unsteady solute dispersion in the pulsatile flow of non-Newtonian fluids under the periodic body acceleration/deceleration. Several variants of the non-Newtonian fluid models are considered in this lecture.

Understanding solute dispersion involves finding the time-dependent velocity profile analytically for smaller values of the Womersley frequency parameter for yield stress fluid such as the K-L fluid or the non-yield stress model such as the Ellis fluid while for larger values of this parameter, a numerical solution is computed using an explicit finite difference method. The Aris’ method of moment is widely used by researchers to examine the solute dispersion; Estimates for the exchange coefficient due to the irreversible first order boundary reaction, the convection and the dispersion coefficients, the skewness and the kurtosis are examined and their response to the system governing parameters is analysed. The discussion on the solute dispersion is made in the three flow and dispersion regimes: (i) viscous flow with diffusive dispersion regime, (ii) viscous flow with unsteady dispersion regime and (iii) the unsteady flow with unsteady dispersion regime. These regimes are characterized by the interplay between the values of the Péclet number, the Womersley frequency parameter, which is associated with the pressure pulsation, and the oscillatory Péclet number which has inherently the Schmidt number. 

For a specific non-Newtonian fluid such as the Ellis fluid, the impact of the body acceleration parameter, the wall absorption parameter, the degree of shear thinning behaviour index, shear stress parameter, the Womersley frequency parameter, and the fluctuating pressure parameter, on the axial mean solute concentration is discussed here. It has been noticed that the value of the dispersion coefficient decreased monotonically in the viscous flow with the diffusive dispersion region, while the skewness and kurtosis both have shown significant variations in the unsteady dispersion regime, which lead to the significant variation in the axial mean concentration. Graphical analysis shows that a leftward shift and also a reduction in the peak of the mean concentration leading to the non-Gaussianity of the solute dispersion in the non-Newtonian fluid flow under the body acceleration/deceleration conditions. The contrast in the axial mean concentration results with and without the higher order moments is tabulated along with the presence and absence of the wall absorption parameter. This analysis can be used in understanding the solute dispersion in blood flow, with specific applications such as nutrient transport and directed drug delivery.

Title: Cholesky factorization of almost all non-positive matrices

Abstract: The Cholesky decomposition has been a fundamental and important result in matrix theory, since it appeared 100 years ago. It applies only to the positive definite matrices, aka the PD-cone. A parallel factorization is the LDU procedure, which applies to the larger open dense cone of LPM matrices - these are real symmetric, with all Leading Principal Minors nonzero.

Motivated by LDU, we extend the Cholesky factorization to one for every LPM matrix, which is distinct from the LDU factorization. Moreover, the LPM cone splits into 2^n LPM sub-cones of "equal" measure, one of which is the PD-cone. We show that each of these sub-cones admits uncountably many factorizations, each of which generalizes the one by Cholesky, and is algorithmic and a smooth diffeomorphism of the LPM sub-cone with lower triangular matrices. These diffeomorphisms equip each sub-cone with a rich structure: it is an abelian Lie group with a bi-invariant Riemannian metric; as well as isometrically isomorphic to a Euclidean space; and moreover, it is amenable to tools from random matrix theory. This is based on the joint work arXiv:2508.02715 with Prateek Kumar Vishwakarma.

Title: Flows and solitons on some Riemannian manifolds

Abstract: This talk consists of some new results on Ricci flow, Ricci soliton, Ricci-Yamabe flow and soliton, conformal Ricci flow and soliton. Also new types of Bach flow and soliton have been introduced and studied their applications in various product manifolds and relativity. These flows have great importance in Mathematical Physics and in Engineering.

Title: Discretization of Climate Forecasting Model: Nonlinearity and Applications

Abstract: Climate systems are influenced by a variety of interconnected factors, including external forcing, and greenhouse gas emissions. They are also naturally complex and nonlinear. Although continuous climate models offer valuable insights, they often experience problems with numerical simulation and analytical precision. An effective approach for reducing complex systems while retaining fundamental dynamics is discretization. We investigate stability, explore tipping points, and analyze long-term behavior through the transformation of continuous equations into discrete-time frameworks. This talk emphasizes how discrete models are used to represent the forcing of CO₂ and CH₄ emissions and how they affect temperature anomalies. Applications include comparison with observational data, scenario analysis, and forecasting. Discrete frameworks provide an efficient and informative tool to enhance our understanding of global warming mechanisms.

Title: Refining boundary value problems in non-local elasticity theory

Abstract: This study introduces novel refined boundary conditions for traction-free surfaces in nonlocal elasticity, representing a significant advancement in nonlocal elasticity theory. Revisiting Eringen’s differential model, we show that its classical equivalence with the integral formulation breaks down near boundaries and can be restored only through additional boundary constraints. Using asymptotic boundary-layer analysis, the refined conditions lead to corrected dispersion relations for Rayleigh waves in a nonlocal elastic half-space. The framework extends naturally to multilayered and semi-infinite media, enhancing its applicability to complex material systems. Phase-velocity and sensitivity analyses demonstrate how nonlocal parameters and Poisson’s ratio affect Rayleigh wave dispersion, and numerical results from finite element analysis (FEA) further illustrate the influence of nonlocality on phase velocity, offering valuable insights for material design and wave-control strategies.