Geometric Invariant Theory:
Faculty: Santosha Pattanayak

The geometry of algebraic varieties:
The geometry of algebraic varieties with reductive group actions, including flag varieties, toric varieties, torus actions on algebraic varieties, and spherical varieties. I am also interested in studying the structure of algebraic groups and GIT (Geometric Invariant Theory) quotients of algebraic varieties and exploring toric degenerations of algebraic varieties. Moreover, my interest extends to areas such as the Seshadri constants of algebraic varieties and the symplectic invariants of smooth projective algebraic varieties, including the Gromov width.
Faculty: Narasimha Chary Bonala

The research areas are Derivations, Higher Derivations, Differential Ideals, Multiplication Modules and the Radical Formula.
Faculty: A. K. Maloo

I mainly consider various analytic function spaces defined on the unit disk or on some half plane of the complex plane and various operators on these spaces such as multiplication operators, composition operators, Cesaro operators. Also, I work on similar operators on some discrete function spaces defined on an infinite rooted tree (graph), in particular, on the discrete analogue of Hardy spaces. I deal with number of other problems which connects geometric function theory with function spaces and operator theory.
Faculty: P. Muthukumar

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, B. V. Rathish Kumar

Development of Numerical Schemes for Incompressible Newtonian and Non-Newtonian Fluid Flows based on FDM, FEM, FVM, Wavelets, SEM, BEM etc. Development of Parallel Numerical Methods for Heat & Fluid Flow Analysis on Large Scale Parallel Computing systems based on MPI-OpenMP-Cuda programming concepts, ANN/ML methods for Flow Analysis. Global Climate Modelling on Very Large Scale Parallel Systems.
Faculty: B. V. Rathish Kumar, Saktipada Ghorai

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 equation
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, B.V. Rathish Kumar

Functional inequalities on Sobolev space
Sobolev spaces are the natural spaces where one looks for solutions of Partial differential equations (PDEs). Functional inequalities on this spaces ( for example Moser-Trudinger Inequality, Poincare Inequality, Hardy- Sobolev Inequality and many other) plays a very significant role in establishing existence of solutions for various PDEs. Existence of extremal function for such inequalities is another key aspect that is investigated

Asymptotic analysis on changing domains
Study of differential equations on long cylinders appears naturally in various branches of Physics, Engineering applications and real life problems.
Problems (not necessarily PDEs, can be purely variational in nature) set on cylindrical domains whose length tends to infinity, is analysed.
Faculty: Prosenjit Roy, Kaushik Bal, Indranil Chowdhury

Analysis of Nonlinear PDEs involving fractional/nonlocal operator:
Fully nonlinear elliptic and parabolic equation involving nonlocal operators. Equations motivated from stochastic control/ Game problems including Hamilton Jacobi Bellman Equations, Isaacs Equations, Mean Field Games problems -

1. Viscosity Solution theory, Comparison Principle, Wellposedness theory, Stability, Continuous Dependence.
2. Numerical Analysis –Wellposedness, Convergence, Error estimates of Finite difference method, Semi-lagrangian Method.

Faculty: Indranil Chowdhury

Control Theory and its applications:
We study several aspects of controllability, say exact controllability, null controllability, approximate controllability, controllability to the trajectories of a given system of ordinary and partial differential equations (both linear and nonlinear). We study stabilizability (exponential, asymptotic) of a system of differential equations and construct feedback control for that system. Currently we are studying controllability of reaction diffusion systems of partial equations using Carleman inequalities and fixed point technique. Multiplier techniques are also used to show controllability of system of hyperbolic partial differential equations. Several mixed systems (hyperbolic and parabolic) are also been studied.
Faculty: Mrinmay Biswas

Banach space theory
Geometric and proximinality aspects in Banach spaces.
Faculty: P. Shunmugaraj

Function-theoretic and graph-theoretic operator theory
The primary goal is to implement methods from the complex function theory and the graph theory into the multivariable operator theory. The topics of interests include de Branges-Rovnyak spaces and weighted shifts on directed graphs.
Faculty: Sameer Chavan

Non-commutative geometry
The main emphasis is on the metric aspect of noncommutative geometry.
Faculty: Satyajit Guin

Bounded linear operators
A central theme in operator theory is the study of B(H), the algebra of bounded linear operators on a separable complex Hilbert space. We focus on operator ideals, subideals and commutators of compact operators in B(H). There is also a continuing interest in semigroups of operators in B(H) from different perspectives. We work in operator semigroups involve characterization of special classes of semigroups which relate to solving certain operator equations.
Faculty: Sasmita Patnaik

Operator spaces
The main emphasis is on operator space techniques in abstract Harmonic Analysis.

In the Euclidean setting
Analysis, boundedness and weighted boundedness of singular integral operators are major thrust areas in the department. In abstract Harmonic analysis we do work in studying Lacunary sets in the noncommutative Lp spaces.
Faculty: Parasar Mohanty

On Lie groups
Problems related to integral geometry on Lie groups are being studied.
Faculty: Rama Rawat 

Cohomology and Deformation theory of algebraic structures
Research work in this area encompasses cohomology and deformation theory of algebraic structures, mainly focusing on Lie and Leibniz algebras arising out of topology and geometry. In particular, one is interested in the cohomology and Versal deformation for Lie and Leibniz brackets on the space of sections of vector bundles e.g. Lie algebroids and Courant algebroids. This study naturally relate questions about other algebraic structures which include Lie-Rinehart algebras, hom-Lie-Rinehart algebras, Hom-Gerstenhaber algebras, homotopy algebras associated to Courant algebras, higher categories and related fields.
Faculty: Ashis Mandal

TPDE based Image processing for Denoising, Inpainting, Classification, Compression, Registration, Optical flow analysis etc. Bio-Medical Image Analysis based on CT/MRI/US clinical data, ANN/ML methods in Image Analysis, Wavelet methods for Image processing.
Faculty: B. V. Rathish 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.
Faculty: Malay Banerjee, Saktipada Ghorai, B.V. Rathish Kumar

Cardiac electrophysiology
Theory, Modeling & Simulation of Cardiac Electrical Activity (CEA) in Human Cardiac Tissue based on PDEODE models such as Monodomain Model, Biodomain model, Cardiac Arrhythmia, pace makers etc
Faculty: B.V. Rathish Kumar

Algebraic number theory and Arithmetic geometry
Iwasawa Theory of elliptic curves and modular forms, Galois representations, Congruences between special values of L-functions.
Faculty: Sudhanshu Shekhar

Analytic number theory
L-functions, sub-convexity problems, Sieve method
Faculty: Saurabh Kumar Singh

Number theory and Arithmetic geometry
Iwasawa Theory of elliptic curves and modular forms, Selmer groups
Faculty: Somnath Jha

Number theory, Dynamical systems, Random walks on groups
During the last four decades, it has been realized that some problems in number theory and, in particular, in Diophantine approximation, can be solved using techniques from the theory of homogeneous dynamics, random walks on homogeneous spaces etc. Indeed, one translates such problems into a problem on the behavior of certain trajectories in homogeneous spaces of Lie groups under flows or random walks; and subsequently resolves using very powerful techniques from the theory of dynamics on homogeneous spaces, random walk etc. I undertake this theme
 Faculty: Arijit Ganguly

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 are not limited to): Singular Perturbation problems, Multiscale Phenomena, Hyperbolic Conservation Laws, Elliptic and Parabolic PDEs, Integral Equations, Computational Acoustics and Electromagnetics, 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, Nyström Method, 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.
Faculty: B. V. Rathish Kumar, Akash Anand, Indranil Chowdhury

Broadly speaking, I work with topics in C*-algebras and von Neumann algebras. More precisely, my work involves Jones theory of subfactors and planar algebras.
Faculty: Keshab Chandra Bakshi

Representation of Lie and linear algebraic groups over local fields, Representation-theoretic methods, automorphic representations over local and global fields, Linear algebraic groups and related topics MSC classification (22E50, 11F70, 20Gxx:)
Faculty: Santosh Nadimpalli

Representations of finite and arithmetic groups
Current research interests: Representations of Linear groups over local rings, Projective representations of finite and arithmetic groups, Applications of representation theory.
Faculty: Pooja Singla

Representation theory of Lie algebras and algebraic groups
Faculty: Santosha Pattanayak

Representation theory of infinite dimensional Lie algebras
Current research interest: Representation theory of Kac-Moody algebras; Toroidal Lie algebras and extended affine Lie algebras.
Faculty: Sachin S. Sharma

Representation theory and Invariant theory
Current research interest: Representation and structure theory of algebraic groups, Classical invariant theory of reductive algebraic groups and associated Weyl groups.
Faculty: Preena Samuel

Combinatorial representation theory
String algebras form a class of tame representation type algebras that are presented combinatorially using quivers and relations. Currently I am interested in studying the combinatorics of strings to understand the Auslander-Reiten quiver that encodes the generators for the category of finite length R-modules as well as the Ziegler spectrum associated with string algebras whose topology is described model-theoretically
Faculty: Amit Kuber

Representation Theory of Algebraic groups: Representation theory of Algebraic groups and Lie algebras, and its applications to Invariant theory and Algebraic geometry.
Faculty: Narasimha Chary Bonala

Set theory (MSC Classification 03Exx)
We apply tools from set theory to problems from other areas of mathematics like measure theory and topology. Most of these applications involve the use of forcing to establish independence results. For examples of such results see https://home.iitk.ac.in/~krashu/
Faculty: Ashutosh Kumar

Rough set theory and Modal logic
Algebraic studies of structures and corresponding logics that have arisen in the course of investigations in Rough Set Theory (RST) constitute a primary part of my research. Currently, we are working on algebras and logics stemming from a combination of formal concept analysis and RST, and also from different approaches to paraconsistency.
Faculty: Mohua Banerjee

Broadly speaking, my work lies in the theory of functions of several complex variables. Two major themes of my work till now are related to _Pick-Nevanlinna interpolation problem_ and on the _Kobayashi geometry of bounded domains_. I am also interested in complex potential theory and complex dynamics in one variable setting.
Faculty: Vikramjeet Singh Chandel

Algebraic topology and Homotopy theory
The primary interest is in studying equivariant algebraic topology and homotopy theory with emphasis on unstable homotopy. Specific topics include higher operations such as Toda bracket, pi-algebras, Bredon cohomology, simplicial/ cosimplicial methods, homotopical algebra.
Faculty: Debasis Sen

Algebraic topology, Combinatorial topology
I apply tools from algebraic topology and combinatorics to address problems in topology and graph theory.
Faculty: Nandini Nilakantan

Low dimensional topology
The main interest is in Knot Theory and its Applications. This includes the study of amphicheirality, the study of closed braids, and the knot polynomials, specially the Jones polynomial.
Faculty: Aparna Dar

Geometric group theory and Hyperbolic geometry
Work in this area involves relatively hyperbolic groups and Cannon-Thurston maps between relatively hyperbolic boundaries. Mapping Class Groups are also explored.
Faculty: Abhijit Pal

Manifolds and Characteristic classes
We are interested in the construction of new examples of non-Kahler complex manifolds. We aim also at answering the question of existence of almost-complex structures on certain even dimension real manifolds. Characteristic classes of vector bundles over certain spaces are also studied.
Faculty: Ajay Singh Thakur

Moduli spaces of hyperbolic surfaces
The central question we study here to find combinatorial descriptions of moduli spaces of closed and oriented hyperbolic surfaces. Also, we study isometric embedding of metric graphs on surfaces of following types: (a) quasi-essential on closed and oriented hyperbolic surfaces (b) non-compact surfaces, where complementary regions are punctured discs, (c) on half-translation surfaces etc.
Faculty: Bidyut Sanki

Systolic topology and Geometry
We are interested to study the configuration of systolic geodesics (i.e., shortest closed geodesics) on oriented hyperbolic surfaces. Also, we are interested in studying the maximal surfaces and deformations on hyperbolic surfaces of finite type to increase systolic lengths.
Faculty: Bidyut Sanki

Topological graph theory
We study configuration of graphs, curves, arcs on surfaces, fillings, action of mapping class groups on graphs on surfaces, minimal graphs of higher genera.
Faculty: Bidyut Sanki

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.
Faculty: B. V. Rathish Kumar