### Research Areas in Mathematics

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

Geometric Invariant Theory:
Faculty: Santosha Pattanayak

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

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 noncommutative 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.

Cardiac electrophysiology

Theory, Modeling & Simulation of Cardiac Electrical Activity (CEA) in Human Cardiac Tissue based on PDEODE 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

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:)

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

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

Differential geometry

Geometric Analysis and Geometric PDEs. Interested in geometry of the eigenvalues of Laplace operator, Geometry of geodesics.

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 : 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

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

### Research Areas in Statistics and Probability Theory

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

Exponential growth in computing power in the past few decades has made Bayesian methods for infinitedimensional models possible, which is termed as the Bayesian nonparametric (BN) methods. BN is a vast area dealing with modelling and making inference in various fields of Statistics, including, and not restricted to density estimation, regression, variable selection, classification, clustering. Irrespective of the field of execution, a BN method deals with prior construction on an infinite-dimensional parameter space, posterior computation and thereby making posterior predictive inference. Finally, the method is validated by supportive asymptotic properties to show the closeness of the proposed method to the true underlying data generating process.

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

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

Faculty: Shalabh, Sharmishtha Mitra

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 Misra

The main goal of environmental statistics is to build sophisticated modelling techniques that are necessary for analysing temperature, precipitation, ozone concentration in air, salinity in seawater, fire weather index, etc. There are multiple sources of such observations, like weather stations, satellites, ships, and buoys, as well as climate models. While station-based data are generally available for long time periods, the geographical coverage of such stations is mostly sparse. On the other hand, satellite-derived data are available only for the last few decades, but they are generally of much higher spatial resolution. While the current statistical literature has already explored various techniques for station-based data, methods available for modelling high-resolution satellite-based datasets are relatively scarce and there is ample opportunity for building statistical methods to handle such datasets. Here, the data are not only huge in volume, but they are also spatially dependent. Modelling such complex dependencies is challenging also due to the high nonstationary often present in the data. The sophisticated methods also need suitable computational tools and thus provide scopes for novel research directions in computational statistics. Apart from real datasets, statistical modelling of climate model outputs is a new area of research, particularly keeping in mind the issue of climate change. Under different representative concentration pathways (RCPs) of the Intergovernmental Panel for Climate Change (IPCC), different carbon emission

Faculty: Arnab Hazra

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 Misra

The mathematical discipline of Game theory models and analyses interactions between competing and cooperative players. Some research areas in game theory are choice theory, mechanism design, differential games, stochastic games, graphon games, combinatorial games, evolutionary games, cooperative games, Bayesian games, algorithmic games - and this list is certainly not exhaustive. Gametheoretic models are used in many real-life problems such as decision making, voting, matching, auctioning, bargaining/negotiating, queuing, distributing/dividing wealth, dealing with cheap talks, the evolution of living organisms, disease propagation, cancer treatment, and many more. Game Theory is also a popular research area in computer science where equilibrium structures are explored using computer algorithms. Mathematical topics such as combinatorics, graph theory, probability (discrete and measure-theoretic), analysis (real and functional), algebra (linear and abstract), etc., are used in solving game-theoretic problems.

Build machine learning algorithms based on statistical modeling of data. With a statistical model in hand, we apply probability theory to get a sound understanding of the algorithms.

Faculty: Subhajit Dutta

Markov chain Monte Carlo (MCMC) algorithms produce correlated samples from a desired target distribution, using an ergodic Markov chain. Due to the lack of independence of the samples, and the challenges of working with Markov chains, many theoretical and practical questions arise. Much of the research in this area can be divided into three broad topics: (1) development of new sampling algorithms for complicated target distributions, (2) studying rates of convergence of the Markov chains employed in various applications like variable selection, regression, survival analysis etc, and (3) measuring the quality of MCMC samples in an effort to quantify the variability in the final estimators of the features of the target.

Faculty: Dootika Vats

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 shaperestricted 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 area of optimal experimental design has been an integral part of many scientific investigation including agriculture and animal husbandry, biology, medicine, physical and chemical sciences, and industrial research. A well-designed experiment utilizes the limited recourse (cost, time, experimental units, etc) optimally to answer the underlying scientific question. For example, optimal cluster/crossover designs may be applied to cluster/cross randomized trials to efficiently estimates the treatment effects. Optimal standard ANOVA designs can be utilized to test the equality of several experimental groups. Most popular categories of optimal designs include Bayesian designs, longitudinal designs, designs for ordered experiments and factorial designs to name a few.

Faculty: Satya Prakash Singh

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.

Faculty: Neeraj Misra

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 auto correlated, 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

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 Mestimators 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

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

The branch of statistics that focuses on the methods for analysing data observed across some spatial locations in 2-D or 3-D (most common), is called spatial statistics. The spatial datasets can be broadly divided into three types: point-referenced data, areal data, and point patterns. Temperature data collected by a few monitoring stations spread across a city on some specific day is an example of the first type. When data are obtained as summaries of some geographical regions, they are of the second type, crime rate dataset from the different states of India on a specific year is an example. An example of the third type is the IED attack locations in Afghanistan during a year, where the geographical coordinates are themselves the data. Because of the natural dependence among the observations obtained from two close locations, the data cannot be assumed to be independent. When the study domain is large, often we have a large number of observational sites and at the same time, those sites are possibly distributed across a nonhomogeneous area. This leads to the necessity of models that can handle a large number of sites as well as the nonstationary dependence structure and this is a very active area of research. Apart from common geostatistical models, a very active area of research is focused on spatial extreme value theory where max-stable stochastic processes are the natural models to explain the tail-dependence. While the available methods for such spatial extremes are highly scarce, specifically for moderately highdimensional problems, different future research directions are being explored currently in the literature. For better uncertainty quantification and computational flexibility using hierarchically defined models, the Bayesian paradigm is often a natural choice.

Faculty: Arnab Hazra

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

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 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 Misra