### Research Areas in Mathematics

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

Geometric Invariant Theory:

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Faculty: S. Pattanayak

Structure theory of algebraic groups:

The main interest is to understand the structure of certain classes of algebraic groups either arising as automorphism groups or those with additional conditions or more generally to the class of group schemes over arbitrary bases. Such study naturally leads to understanding various other structures more systematically, like that of homogeneous spaces or homogeneous bundles etc.
Faculty: P. Samuel

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

Commutative Algebra:

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

Faculty : A. K. Maloo

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: A. Mandal

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.

Faculty : Prawal Sinha, B. V. Rathish Kumar

### 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

### 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:

Problems (not necessarily PDEs, can be purely variational in nature) set on cylindrical domains whose length tends to infinity, is analysed.
Faculty : P. Roy
Other Faculty : K. Bal

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:P. Shunmugaraj Other Faculty: M. Gupta

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

Non-Commutative Geometry

The main emphasis is on the metric aspect of noncommutative geometry.
Faculty: S. Guin

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

(iii) 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: S. Patnaik

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

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: P. Mohanty R. Rawat

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.

Modular Forms, Artihmetic Geometry

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Faculty: S. Jha, S. Shekhar

Iwasawa theory:

Work in this area is in Iwasawa Theory, Hida Theory and Galois representations. The basic objects of study are "Elliptic Curves" and "Modular Forms".
Faculty: S. Jha

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.

Faculty : P. Dutt, B. V. Rathish Kumar

### Representation theory of lie algebras and algebraic groups

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Faculty : S. Pattanayak

### Representations of algebraic groups and associated Weyl groups

Finding combinatorially interesting bases for certain invariant theoretic spaces associated to classical algebraic groups. Such bases lead to various modular representation theoretic consequences as well.

Classification of nilpotent and unipotent orbits of semisimple algebraic groups and its consequences.

Faculty : P. Samuel

### Representations of lie algebras and Kac-Moody algebras

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Faculty : S. Sharma

### Category theory and Model Theory

Category theory and logic, especially model theory, provide languages to talk about almost all areas of pure mathematics and to study their interconnections. In categorical logic one studies interpretations/models of theories (written in various fragments of logic) in different categories. For example, a model of the theory of groups in the category of topological spaces is a topological group. My main research interest in this area lies in the category-theoretic syntax-semantics dualities.

On the model theory side, computation and study of algebraic invariants, namely the the K-groups, associated with structures that classify the collections of sets definable in the structure is also my area of interest. In the particular case of model theory of modules/representation theory, I am currently studying the Ziegler spectrum--a geometric invariant--associated with a particular type of finite dimensional algebra, namely a string algebra.

Faculty :Amit Kuber

### Rough Set Theory and Modal Logic

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

### 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,

### 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

### 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 : A. 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 & 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

### Research Areas in Statistics and Probability Theory

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

Detection of different features (in terms of shape) of non-parametric regression functions are studied; asymptotic distributions of the proposed estimators (along with their robustness properties) of the shape-restricted regression function are also investigated. Apart from this, work on the test of independence for more than two random variables is pursued. Statistical Signal Processing and Statistical Pattern Recognition are the other areas of interest.

Faculty: Subhra Sankar Dhar

The manner in which a component (or system) improves or deteriorates with time can be described by concepts of aging. Various aging notions have been proposed in the literature. Similarly lifetimes of two different systems can be compared using the concepts of stochastic orders between the probability distributions of corresponding (random) lifetimes. Various stochastic orders between probability distributions have been defined in the literature. We study the concepts of aging and stochastic orders for various coherent systems. In many situations, the performance of a system can be improved by introducing some kind of redundancy into the system. The problem of allocating redundant components to the components of a coherent system, in order to optimize its reliability or some other system performance characteristic, is of considerable interest in reliability engineering. These problems often lead to interesting theoretical results in Probability Theory. We study the problem of optimally allocating spares to the components of various coherent systems, in order to optimize their reliability or some other system performance characteristic. Performances of systems arising out of different allocations are studied using concepts of aging and stochastic orders.

Faculty: Neeraj Mishra

Estimation of entropies of molecules is an important problem in molecular sciences. A commonly used method by molecular scientist is based on the assumption of a multivariate normal distribution for the internal molecular coordinates. For the multivariate normal distribution, we have proposed various estimators of entropy and established their optimum properties. The assumption of a multivariate normal distribution for the internal coordinates of molecules is adequate when the temperature at which the molecule is studied is low, and thus the fluctuations in internal coordinates are small. However, at higher temperatures, the multivariate normal distribution is inadequate as the dihedral angles at higher temperatures exhibit multimodes and skewness in their distribution. Moreover the internal coordinates of molecules are circular variables and thus the assumption of multivariate normality is inappropriate. Therefore a nonparametric and circular statistic approach to the problem of estimation of entropy is desirable. We have adopted a circular nonparametric approach for estimating entropy of a molecule. This approach is getting a lot of attention among molecular scientists.

Faculty: Neeraj Mishra

About fifty years ago statistical inference problems were first formulated in the now-familiar "Ranking and Selection" framework. Ranking and selection problems broadly deal with the goal of ordering of different populations in terms of unknown parameters associated with them. We deal with the following aspects of Ranking and Selection Problems:1. Obtaining optimal ranking and selection procedures using decision theoretic approach;2. Obtaining optimal ranking and selection procedures under heteroscedasticity;3. Simultaneous confidence intervals for all distances from the best and/or worst populations, where the best (worst) population is the one corresponding to the largest (smallest) value of the parameter;4. Estimation of ranked parameters when the ranking between parameters is not known apriori;5. Estimation of (random) parameters of the populations selected using a given decision rule for ranking and selection problems. Neeraj Mishra

In many practical situations, it is natural to restrict the parameter space. This additional information of restricted parameter space can be intelligently used to derive estimators that improve upon the standard (natural) estimators, meant for the case of unrestricted parameter space. We deal with the problems of estimation parameters of one or more populations when it is known apriori that some or all of them satisfy certain restrictions, leading to the consideration of restricted parameter space. The goal is to find estimators that improve upon the standard (natural) estimators, meant for the case of unrestricted parameter space. We also deal with the decision theoretic aspects of this problem.

Faculty: Neeraj Mishra

The outcome of any experiment depends on several variables and such dependence involves some randomness which can be characterized by a statistical model. The statistical tools in regression analysis help in determining such relationships based on the sample experimental data. This helps further in describing the behaviour of the process involved in experiment. The tools in regression analysis can be applied in social sciences, basic sciences, engineering sciences, medical sciences etc. The unknown and unspecified form of relationship among the variables can be linear as well as nonlinear which is to be determined on the basis of a sample of experimental data only. The tools in regression analysis help in the determination of such relationships under some standard statistical assumptions. In many experimental situations, the data do not satisfy the standard assumptions of statistical tools, e.g. the input variables may be linearly related leading to the problem of multicollinearity, the output data may not have constant variance giving rise to the hetroskedasticity problem, parameters of the model may have some restrictions, the output data may be autocorrelated, some data on input and/or output variables may be missing, the data on input and output variables may not be correctly observable but contaminated with measurement errors etc. Different types of models including the econometric models, e.g., multiple regression models, restricted regression models, missing data models, panel data models, time series models, measurement error models, simultaneous equation models, seemingly unrelated regression equation models etc. are employed in such situations. So the need of development of new statistical tools arises for the detection of problem, analysis of such non-standard data in different models and to find the relationship among different variables under nonstandard statistical conditions. The development of such tools and the study of their theoretical statistical properties using finite sample theory and asymptotic theory supplemented with numerical studies based on simulation and real data are the objectives of the research work in this area.

Faculty: Shalabh

Signal processing may broadly be considered to involve the recovery of information from physical observations. The received signals are usually disturbed by thermal, electrical, atmospheric or intentional interferences. Due to the random nature of the signal, statistical techniques play an important role in signal processing. Statistics is used in the formulation of appropriate models to describe the behaviour of the system, the development of appropriate techniques for estimation of model parameters, and the assessment of model performances. Statistical Signal Processing basically refers to the analysis of random signals using appropriate statistical techniques. Different one and multidimensional models have been used in analyzing various one and multidimensional signals. For example ECG and EEG signals, or different grey and white or colour textures can be modelled quite effectively, using different non-linear models. Effective modelling are very important for compression as well as for prediction purposes. The important issues are to develop efficient estimation procedures and to study their properties. Due to non-linearity, finite sample properties of the estimators cannot be derived; most of the results are asymptotic in nature. Extensive Monte Carlo simulations are generally used to study the finite sample behaviour of the different estimators.

Faculty: Debasis Kundu,Amit Mitra

Efficient estimation of parameters of nonlinear regression models is a fundamental problem in applied statistics. Isolated large values in the random noise associated with model, which is referred to as an outliers or an atypical observation, while of interest, should ideally not influence estimation of the regular pattern exhibited by the model and the statistical method of estimation should be robust against outliers. The nonlinear least squares estimators are sensitive to presence of outliers in the data and other departures from the underlying distributional assumptions. The natural choice of estimation technique in such a scenario is the robust M-estimation approach. Study of the asymptotic theoretical properties of M-estimators under different possibilities of the M-estimation function and noise distribution assumptions is an interesting problem. It is further observed that a number of important nonlinear models used to model real life phenomena have a nested superimposed structure. It is thus desirable also to have robust order estimation techniques and study the corresponding theoretical asymptotic properties. Theoretical asymptotic properties of robust model selection techniques for linear regression models are well established in the literature, it is an important and challenging problem to design robust order estimation techniques for nonlinear nested models and establish their asymptotic optimality properties. Furthermore, study of the asymptotic properties of robust M-estimators as the number of nested superimposing terms increase is also an important problem. Huber and Portnoy established asymptotic behavior of the M-estimators when the number of components in a linear regression model is large and established conditions under which consistency and asymptotic normality results are valid. It is possible to derive conditions under which similar results hold for different nested nonlinear models.

Faculty: Debasis Kundu,Amit Mitra

Econometric modelling involves analytical study of complex economic phenomena with the help of sophisticated mathematical and statistical tools. The size of a model typically varies with the number of relationships and variables it is applying to replicate and simulate in a regional, national or international level economic system. On the other hand, the methodologies and techniques address the issues of its basic purpose – understanding the relationship, forecasting the future horizon and/or building "what-if" type scenarios. Econometric modelling techniques are not only confined to macro-economic theory, but also are widely applied to model building in micro-economics, finance and various other basic and social sciences. The successful estimation and validation part of the model-building relies heavily on the proper understanding of the asymptotic theory of statistical inference. A challenging area of econometric modelling has been the application of advanced mathematical concept of wavelets, which are ideally suited to study the chaotic behaviour of financial indicators, to name just one. A successful combination of econometrics with the non-parametric artificial intelligence techniques is another interesting aspect of the modelling exercise. So, whether the purpose is to validate or negate age-old theories in the contemporary world, or to propagate new ideas in the ever-growing complexities of physical phenomena, econometric modelling provides an ideal solution.

Faculty: Shalabh, Sharmishtha Mitra

Economic globalization and evolution of information technology has in recent times accounted for huge volume of financial data being generated and accumulated at an unprecedented pace. Effective and efficient utilization of massive amount of financial data using automated data driven analysis and modelling to help in strategic planning, investment, risk management and other decision-making goals is of critical importance. Data mining techniques have been used to extract hidden patterns and predict future trends and behaviours in financial markets. Data mining is an interdisciplinary field bringing together techniques from machine learning, pattern recognition, statistics, databases and visualization to address the issue of information extraction from such large databases. Advanced statistical, mathematical and artificial intelligence techniques are typically required for mining such data, especially the high frequency financial data. Solving complex financial problems using wavelets, neural networks, genetic algorithms and statistical computational techniques is thus an active area of research for researchers and practitioners.

Faculty: Amit Mitra, Sharmishtha Mitra

Traditionally, life-data analysis involves analysing the time-to-failure data obtained under normal operating conditions. However, such data are difficult to obtain due to long durability of modern days. products, lack of time-gap in designing, manufacturing and actually releasing such products in market, etc. Given these difficulties as well as the ever-increasing need to observe failures of products to better understand their failure modes and their life characteristics in today's competitive scenario, attempts have been made to devise methods to force these products to fail more quickly than they would under normal use conditions. Various methods have been developed to study this type of "accelerated life testing" (ALT) models. Step-stress modelling is a special case of ALT, where one or more stress factors are applied in a life-testing experiment, which are changed according to pre-decided design. The failure data observed as order statistics are used to estimate parameters of the distribution of failure times under normal operating conditions. The process requires a model relating the level of stress and the parameters of the failure distribution at that stress level. The difficulty level of estimation procedure depends on several factors like, the lifetime distribution and number of parameters thereof, the uncensored or various censoring (Type I, Type II, Hybrid, Progressive, etc.) schemes adopted, the application of non-Bayesian or Bayesian estimation procedures, etc.

Faculty: Debasis Kundu, Sharmishtha Mitra

The study of Stochastic calculus, more specifically, that of stochastic differential equations and stochastic partial differential equations, has a broad range of applications across various disciplines or branches of Mathematics, such as Partial Differential Equations, Evolution systems, Interacting particle systems, Finance, Mathematical Biology. Theoretical understanding for such equations was first obtained in finite dimensional Euclidean spaces. Later on, to describe various natural phenomena, models were constructed (and analyzed) with values in Banach spaces, Hilbert spaces and in the duals of nuclear spaces. Important topics/questions in this area of research include existence and uniqueness of solutions, Stability, Stationarity, Stochastic flows, Stochastic Filtering theory and Stochastic Control Theory, to name a few.

Faculty: Suprio Bhar

The seminal works of Terry Lyons on extensions of Young integration, the latter being an extension of Riemann integration, to functions with Holder regular paths (or those with finite p-variation for some 0 < p < 1) lead to the study of Rough Paths and Rough Differential Equations. Martin Hairer, Massimiliano Gubinelli and their collaborators developed fundamental results in this area of research. Extensions of these ideas to functions with negative regularity (read as "distributions") opened up the area of Regularity structures. Important applications of these topics include constructions of pathwise' solutions of stochastic differential equations and stochastic partial differential equations.

Faculty: Suprio Bhar