In this area, real analysis, harmonic analysis, complex variables, ordinary differential equations, partial differential equations, integral equations, operator theory and all analytical problems originating from applied science are studied. Applications of the research results are employed to solve concrete problems that arise in natural science, engineering, and financial mathematics. Computerized tomography(CT) using the Radon transform and image processing using the wavelets are conspicuous applications of analysis.
Here, the structures and the properties of manifolds are studied using algebraic, geometric, and combinatorial methods. Active research areas include (i) knots, links, braids, and 3-manifolds (ii) the geometric structures on low-dimensional manifolds including hyperbolic and discrete group theory (iii) 4-manifolds through Seiberg-Witten theory, symplectic and contact structures, and (iv) symmetries of manifolds in terms of group actions on differential manifolds, algebraic varieties, and semi-algebraic sets. In addition applications are effectively being made to computer graphics and non-commutative cryptography, in which braid groups are used.
Algebra and Number Theory
Work in these areas often involves theoretical problems in algebraic number theory and algebraic geometry, class field theory, modular forms, and representations. Applicable problems in cryptography, coding theory and game theory are also studied using methods in algebraic geometry, number theory and linear algebra.
Using differential manifold theory and Riemannian manifolds, those working in geometry study such topics as curvature pinching problems, curvature and group actions, closed geodesics, finiteness theorems, comparison theorems, geometric structure and isometric immersions, harmonic maps and non-linear problems.
Computational Mathematics and Scientific Computing
?Computational mathematics involves the study of methods of expressing complex phenomena as mathematical models and discovering techniques of numerically solving the models. Research is also directed towards theoretical studies based on the analysis and developments of new techniques applicable to science and engineering.
Combinatorics is an area of mathematics that studies mathematical objects having discrete or combinatorial structures. It involves combinatorial problems from various fields of mathematics and allows for the development of theories about diverse combinatorial objects. Emphasis is put on enumerative combinatorics, graph theory and algebraic combinatorics.
Topics studied in this field include Shannon’s information theory, computation theory, complexity theory, Hoffman code, entropy, data compression, error correcting codes, cryptography, and information security.
The area of financial mathematics involves the study and design of mathematical models of financial derivatives and markets using stochastic integral equations or stochastic differential equations. Real data from the markets are used to test mathematical models and the techniques to predict the market movements are studied.
Probability and Statistics
In probability, random phenomena in nature and society are studied rigorously in terms of measure theory. Research emphasis is on stochastic process, martingale, Markov chain, stochastic differential equations, queueing theory for the analysis of telecommunication systems, stochastic control theory and optimization. In statistics, emphasis is on multivariate statistical analysis, data analysis, learning theory, neural network models, graphic models, time series analysis, Bayesian analysis, parameter estimation, hypothesis verification, regression analysis, etc.