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: Optimal Control and Differential Game Approaches for Dynamic Supply Chains

Abstract: Optimal control theory focuses on determining optimal decisions for dynamic systems by optimizing an objective function subject to system dynamics, control variables, state variables, and constraints. Two major approaches exist for solving optimal control problems: Pontryagin’s Maximum Principle, which relies on maximizing the Hamiltonian function, and the dynamic programming method, which uses the Hamilton-Jacobi-Bellman (HJB) equation.

A significant extension of optimal control theory is differential game theory, which addresses situations involving multiple decision-makers (players), each with their own objective function and strategy set. In differential games, players interact strategically while attempting to optimize their individual objectives. The solutions to differential game problems are known as equilibrium solutions, typically classified as Nash equilibrium or Stackelberg equilibrium.

Differential game theory has been extensively applied in diverse domains, including supply chain management (SCM). A supply chain consists of coordinated organizations and stakeholders involved in product creation and delivery. Supply chain management refers to the planning, coordination, and control of activities such as production, logistics, and marketing with the goal of delivering value to customers while minimizing economic, environmental, and social costs.

Dynamic decision-making offers significant advantages over static policies because dynamic strategies evolve over time and help reduce business risks and uncertainties. Profit maximization problems in supply chains involving dynamic strategies can therefore be modeled as differential game problems, where supply chain members act as players, profit functions represent objectives, and business policies serve as strategies. Consequently, optimal control theory, specifically differential game theory, provides a rigorous framework for determining optimal policies in supply chain management problems.

Keywords: Supply chain; Optimal control theory; Differential game; Dynamic decisions; Equilibrium solutions

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.

Title: Modeling Heat Transport and Temperature Dynamics in Biological Tissues and Organs

Abstract: This talk presents advanced methodologies for modeling heat transport and temperature dynamics in biological tissues and organs, with special emphasis on skin thermomechanics under rapid heating conditions. The presentation introduces two bioheat models formulated within the nonlocal, memory-dependent Moore–Gibson–Thompson (MGT) framework. These models integrate key physiological processes—such as sweating-driven evaporative cooling, nonlocal elastic interactions, and memory-based thermal responses—to more accurately capture the coupled thermal and mechanical behavior of living tissues.

Analytical solutions are obtained using Laplace transform techniques, while numerical simulations are carried out through Riemann-sum approximation methods. The findings demonstrate that both nonlocality and memory effects significantly affect temperature distribution, displacement fields, and stress generation within biological tissues. The MGT-based formulations further exhibit enhanced predictive accuracy compared with classical bioheat models, particularly for high-frequency and fast transient thermal scenarios.

The talk aims to provide insights relevant to thermal injury prediction, hyperthermia treatment planning, laser-assisted medical procedures, and a broad range of biomedical heat-transfer applications.