### Research Areas in Mathematics

Here are the areas of Mathematics in which research is being done currently.

1. Banach Space Theory:

In Banach space theory the key areas of research are the following: (i) Approximation theory in infinite dimensional spaces with special emphasis on classical spaces. (ii) Isomorphic theory of separable Banach spaces, saturation and decomposition Faculty: S.Dutta ,  P. Shunmugaraj

2. Operator Spaces The main emphasis is on operator space techniques in Abstract Harmonic Analysis. Faculty: S. DuttaP. Mohanty

3. Operator Theory The interest in this area, as represented by our department, is along the following two directions:

(i) Unbounded Subnormals

The most outstanding example of an unbounded subnormal is the Creation Operator of the Quantum Mechanics. Our analysis of these operators is essentially based on the theory of Sectorial forms, a sophisticated tool from PDEs. In particular, one may combine the theory of sectorial forms with the spectral theory of unbounded subnormals to derive polynomial approximation results on certain unbounded regions.

(ii) Operators Close to Isometries

This is huge subclass of left-invertible operators which behave like isometries of Hilbert spaces. One may develop an axiomatic approach to these operators. Via this axiomatization, one may obtain the Beurling-type theorems for Bergman shift and Dirichlet shift in one stroke. Important examples of these operators include 2-hyperexpansive operators and Bergman-type operators. There is a transform which sends 2-hyperexpansive operators to Bergman-type operators. For instance, one may use this transform to obtain the Berger-Shaw theory for 2-hyperexpansive operators from the classical Berger-Shaw theory. Faculty: S. Chavan Other Faculty: M. Gupta

### Real and Complex Analysis, Nonlinear Analysis

The current research interest is on Computational Complex Analysis and Complex Analytic Dynamics and Fractals which primarily deal with analysis of Discrete Nonlinear Dynamical Systems and development of algorithm-friendly results. Broadly, this direction of research involves Generation and Applications of Fractals; Fractal Interpolation and Approximation; Simulation, Compression and Analysis of Images of Natural Objects and Investigation of Bifurcation and Chaos.

Another major area of interest is nonlinear analysis with a stress on semi linear elliptic equations. The techniques are variational methods or monotone method. Compact embedding theorems of Sobolev spaces play a crucial role in the existence theorems while degree theory is handy for establishing multiple solutions. Semi linear equations with “jumping nonlinearities” are interesting from bifurcation point of view also.

### Harmonic Analysis

Harmonic analysis on Euclidean spaces, Lie groups and abstract harmonic analysis are represented in the department.

In the Euclidean set up the theory of multipliers, in particular, bilinear multipliers, and combinatorial harmonic analysis are major thrust areas in the department.In Lie groups, the focus is on convolution operators and Kunze-Stein Phenomenon for semi-simple Lie groups. Problems related to integral geometry on semi-simple and nilpotent Lie groups are also being studied.

In abstract harmonic analysis, the emphasis is on Banach algebra techniques and operator space.,

Faculty: S. Madan, P. Mohanty, R. Rawat, S. K. Ray

The research in this area is mostly related to the study of Laplacian Matrices of Trees and Distinguishing Chromatic Number of Graphs.

Faculty :  A. K. Lal

The research areas are Derivations, Higher Derivations, Differential Ideals, Multiplication Modules and the Radical Formula.

Faculty :  A. K. Maloo

The study of interaction of electromagnetic fields with physical objects and the environment constitutes the main subject matter of Computational Electromagnetics. One of the major challenges in this area of research is in the development of efficient, accurate and rapidly-convergent algorithms for the simulation of propagation and scattering of acoustic and electromagnetic fields within and around structures that possess complex geometrical characteristics. These problems are of fundamental importance in diverse fields, with applications ranging from space exploration, medical imaging and oil exploration on the civilian side to aircraft design and decoy detection on the military side - just to name a few.

Computational modeling of electromagnetic scattering problems has typically been attempted on the basis of classical, low-order Finite-Difference-Time-Domain (FDTD) or Finite-Element-Method (FEM) approaches. An important computational alternative to these approaches is provided by boundary integral-equation formulations that we have adopted owing to a number of excellent properties that they enjoy. Listed below are some of the key areas of interest in related research:

1. Design of high-order integrators for boundary integral equations arising from surface and volumetric scattering of acoustic and electromagnetic waves from complex engineering structures including from open surfaces and from geometries with singular features like edges and corners.

2. Accurate representation of complex surfaces in three dimensions with applications to enhancement of low quality CAD models and in the development of direct CAD-to-EM tools.

3. High frequency scattering methods in three dimensions with frequency independent cost in the context of multiple scattering configurations. A related field of interest in this regard includes high-order geometrical optics simulator for inverse ray tracing.

4. High performance computing.  Faculty :Akash Anand

Active work has been going on in the area of "Tribology". Tribology deals with the issues related to lubrication, friction and wear in moving machine parts. Work is going in the direction of hydrodynamic and elastohydrodynamic lubrication, including thermal, roughness and non-newtonian effects. The work is purely theoretical in nature leading to a system on non-linear partial differential equations, which are solved using high speed computers.

### Semigroups of Linear Operators and Their Applications, Functional Differential Equations, Galerkin Approximations

Many unsteady state physical problems are governed by partial differential equations of parabolic or hyperbolic types. These problems are mostly prototypes since they represent as members of large classes of such similar problems. So, to make a useful study of these problems we concentrate on their invariant properties which are satisfied by each member of the class. We reformulate these problems as evolution equations in abstract spaces such as Hilbert or more generally Banach spaces. The operators appearing in these equations have the property that they are the generators of semigroups. The theory of semigroups then plays an important role of establishing the well-posedness of these evolution equations. The analysis of functional differential equations enhances the applicability of evolution equations as these include the equations involving finite as well as infinite delays. Equations involving integrals can also be tackled using the techniques of functional differential equations. The Galerkin method and its nonlinear variants are fundamental tools to obtain the approximate solutions of the evolution and functional differential equations.

Faculty : D. Bahuguna

### Homogenization and Variational Methods for Partial Differential Equations

The main interest is on Aysmptotic Analysis of partial differential equations. This is a technique to understand the macroscopic behaviour of a composite medium through its microscopic properties. The technique is commonly used for PDE with highly oscillating coefficients. The idea is to replace a given heterogeneous medium by a fictitious homogeneous one (the homogenized' material) for numerical computations. The technique is also known as `Multi scale analysis''. The known and unknown quantities in the study of physical or mechanical processes in a medium with micro structure depend on a small parameter $\varepsilon$. The study of the limit as $\varepsilon \rightarrow0$, is the aim of the mathematical theory of homogenization. The notion of $G$-convergence, $H$-convergence, two-scale convergence are some examples of the techniques employed for specific cases. The variational characterization of the technique for problems in calculus of variations is given by $\Gamma$-convergence.

Faculty : T. Muthukumar
Other Faculty : B.V. Ratish Kumar

There is an active group working in the area of Mathematical Biology. The research is carried out in the following directions.

### Mathematical Ecology

1. Research in this area is focused on the local and global stability analysis, detection of possible bifurcation scenario and derivation of normal form, chaotic dynamics for the ordinary as well as delay differential equation models, stochastic stability analysis for stochastic differential equation model systems and analysis of noise induced phenomena. Also the possible spatio-temporal pattern formation is studied for the models of interacting populations dispersing over two dimensional landscape.

2. Mathematical Modeling of the survival of species in polluted water bodies; depletion of dissolved oxygen in water bodies due to organic pollutants.

### Mathematical Epidemiology

1. Mathematical Modeling of epidemics using stability analysis; effects of environmental, demographic and ecological factors.

2. Mathematical Modeling of HIV Dynamics in vivo

### Bioconvection

Bioconvection is the process of spontaneous pattern formation in a suspension of swimming micro-organisms. These patterns are associated with up- and down-welling of the fluid. Bioconvection is due to the individual and collective behaviours of the micro-organisms suspended in a fluid. The physical and biological mechanisms of bioconvection are investigated by developing mathematical models and analysing them using a variety of linear, nonlinear and computational techniques.

### Bio-fluid dynamics

Mathematical Models for blood flow in cardiovascular system; renal flows; Peristaltic transport; mucus transport; synovial joint lubrication.

The faculty group in the area of Numerical Analysis & Scientific Computing are very actively engaged in high quality research in the areas that include (but not limited to): Singular Perturbation problems, Multiscale Phenomena, Hyperbolic Conservation Laws, Elliptic and Parabolic PDEs, Computational Fluid Dynamics, Computer Aided Tomography and Parallel Computing. The faculty group is involved in the development, analysis and application of efficient and robust algorithms for solving challenging problems arising in several applied areas. There is expertise in several discretization methods that include: Finite Difference Methods, Finite Element Methods, Spectral Element Methods, Boundary Element Methods, Spline and Wavelet approximations etc. This encompasses a very high level of computation that requires software skills of the highest order and parallel computing as well.

Rough Set Theory (RST) addresses imprecision that arises from a difficulty in describing reality. In everyday discourse, we place a grid over reality, the grid being typically induced by attributes. Then pieces of data having the same values for a set of attributes, cannot be distinguished. As a result, our concepts, generally, are not definable in terms of the grid. RST prescribes approximations to describe such concepts, and there may be several concepts with the same approximations describing them. RST thus serves as a means for reasoning with objects and concepts that are rendered indiscernible, due to incomplete information about the domain of discourse.

A major concern here is to look for appropriate formal logical frameworks to represent reasoning in RST. Inherent modalities point to the domain of modal logics. It is thus that we have some modal systems capturing ‘rough truth’, and different versions of ‘rough modus ponens’. Modal logics also come in while studying dynamic aspects of RST. Data is presented in RST with the help of an information system, which may be complete, incomplete or non-deterministic. One then investigates sequences of information systems that evolve with time, or which arise from multiple sources (agents), and notions of information updates in the context. New temporal and quantified modal logics, and logics for information systems along with their ‘dynamic’ versions, have surfaced during this study.

Algebraic studies of structures that have arisen in the course of RST investigations constitute an important part of the research. Of special interest is a category-theoretic study of rough sets, and in fact, of concepts in a general framework of ‘granulations’. Other applications of RST are also being studied, e.g. in dialogues between participants of a discourse, in communicative approximations, or in representing ‘open universes’. Techniques for computation of ‘minimal’ sets of attributes required for classification (reducts) of objects also hold interest.

On another side, there is interest in the use of modal systems for reasoning with beliefs revealed by agents.

Faculty :  Mohua Banerjee

### Topology

The main interest is in Knot Theory and its Applications, in particular Low Dimensional Topology.   Faculty :  A. Dar

### Computational Geometry

The interest is in studying Abelian Polyhedral Maps and Polyhedral Manifolds, in particular the aim is to minimize the total number of faces and flag numbers in all polyhedral manifolds of the same p.l. type.
Faculty :   N. Nilakantan

### Differential Geometry

Faculty :  G. Santhanam