Symmetry Methods, Conservation Laws and Exact Solutions for Differential Equations: LUMS; MATHEMATICS LABS

FINANCIAL MATHEMATICS, STOCHASTIC PORTFOLIO

Stochastic Portfolio Theory is a branch of Mathematical Finance. Robert Merton studied the portfolio allocation problem as a Stochastic Control Problem. In his model, an investor allocates his/her wealth between a risky asset and a riskless asset, then chooses the consumption rate to maximize the total expected utility

ALGEBRAIC GEOMETRY

Algebraic Geometry is the study solutions of multivariate polynomial equations. The solutions of these polynomial equations which are the geometric objects of study in Algebraic Geometry are called algebraic varieties. Dr. Qureshi is interested in the classification of families of algebraic varieties. In particular, he has interested the classification Fano varieties and Calabi-Yau varieties. In dimension one and two, the classification is almost complete. Dr. Qureshi mainly works on the classification of 3-folds and 4-folds by using the bi-regular constructions of the graded rings of such algebraic varieties

OPERATOR THEORY

C* Algebras were first studied in connection with modeling observables in quantum mechanics. They have subsequently generated a lot of interest as an area of research. Fixed point theory is the investigation of existence, uniqueness, and approximation of fixed points of mappings. From economics (Nash’s theorem) to physics (phase transitions) there is a wide variety of applications of this exciting area of research. Dr. Shah’s research focuses on these two areas.

SPECTRAL THEORY

Spectral theory can be seen as a generalization of the ideas and concepts of eigenvalues and eigenvectors of a square matrix to a broader class of operators acting in different spaces. Dr. Usman’s research interest lies in the spectral theory of Quantum graphs. 

SCIENTIFIC COMPUTATION

Real world problems can be formulated in mathematical terms (modeling), as the models have grown complex, a great deal of computing power is required to study these models. The Scientific computation involves investigation of robust and efficient methods for studying problems that require heavy computations. Dr. Sial’s has been working on using finite element methods, on problems form material science. One current project looks at optimization of functionals arising in physics, control theory, and differential equations. 

ALGEBRAIC TOPOLOGY

Using ideas from algebra, how can one classify topological spaces?  This is the fundamental question which investigators in Algebraic Topology try to address. Dr. Haniya is interested in the cohomology of n-pointed configuration spaces of complex projective varieties and rational models for the cohomology of such spaces. There is a natural action of the symmetric group on these spaces as well as an induced action on the model which she studies to facilitate computations for cohomology. In particular, her interest is in the cohomology groups of configurations of Rieman surfaces with fewer points and the algebra structure for the cohomology of the unordered configuration spaces. 

MATHEMATICAL ANALYSIS

Modeling of complex biological phenomena has become fashionable over the past two decades, as the computing power required to analyze such models has become available. On the other side, non-local calculus has gained significance owing to its applications in various biological and physical phenomena. The theory of non-local calculus has to be developed to better analyze and understand these physical and biological phenomena

EVOLUTION EQUATIONS

Evolution equations can be interpreted as differential laws describing the development of a system.  Beginning with the study of differential equations investigators now look at these in a more abstract setting. Dr. Imran Naeem studies exact solutions of evolution equations using Lie Symmetry methods.  Lie symmetry analysis of differential equations was initiated by the Norwegian mathematician Sophus Lie (1842 – 1899). Today, this area of research is being actively pursued. The Lie approach is a systematic way of unraveling exact solutions of ordinary and partial differential equations. It works for linear as well as nonlinear differential equations.