**Introduction**** **

We live in the era of mathematics. It permeates all aspects of economic and social activity, and its influence and impact are profound.Technology and mathematics are closely connected in several ways. Modern technology would be unthinkable without mathematics. It has both an inherent logic and beauty while also providing the structure and models from which physicists, chemists, biologists, medics, engineers, economists, and social scientists build an understanding of our world and construct the tools to improve our lives.

Mathematics is the abstract study of topics such as numbers, structures, shapes, space, quantity, and change. Mathematics may be studied in its own right (pure mathematics) or as it is applied to other disciplines such as natural sciences, engineering, medicine, finance, computer science, and social sciences (applied mathematics).

**Research in Mathematics at IITs**

The Department of Mathematics at various IITs offer doctoral research program for motivated students interested in pursuing their career in mathematics – either academic or industry and postgraduate programs, namely MSc and MTech. They provide excellent opportunities in recent and emerging research areas in pure and applied mathematics to prepare motivated researchers who can work in frontline areas. The departments pursue collaborative programs with highly reputed national and international institutions. The departments encourage interdisciplinary trends. The vibrant academic environment of the departments is nurtured by strongly motivated faculty who work in almost all broad areas of Mathematics. The faculty members are involved in interdisciplinary research collaborations with various departments and research centers both within and beyond the institute.

Ph.D. Program in Mathematics offers an exciting and unique opportunity to students for pursuing research in the following areas:

*Algebra and Number Theory*: Algebraic Geometry and Noncommutative Geometry, Representation Theory, Number Theory, Algebraic Groups, and Related Structures, Commutative Algebra

*Analysis*: Multivariable Operator Theory, Numerical Functional Analysis, Approximate Solutions of Operator Equations and Eigenvalue Problems, Spline Theory, Noncommutative Probability, Several Complex Variables, Operator Algebras, and Operator Spaces

*Combinatorics and Theoretical Computer Science*: Graph Theory, Graph Ideas, Algebraic and enumerative combinatorics, Probabilistic Combinatorics, Logic, Computing, Approximation Algorithms, Cryptology, Computational Number Theory

*Differential Equations and Applied Mathematics*: Linear and Nonlinear Partial Differential Equations, Nonlinear Functional Analysis, Systems, and Control Theory, Game Theory, Fluid, and Solid Mechanics, Mathematical Modelling & Numerical Simulation, Integral Equations, Wave-Structure Interaction, Computational Fluid Dynamics, Convective Heat And Mass Transfer

*Geometry and Topology*: **Algebraic topology and Homotopy theory, Computational Geometry**, **Differential Geometry**, **Geometric group theory and Hyperbolic geometry**, **Moduli spaces of hyperbolic surfaces, Topological graph theory**

*Probability and Statistics*: Probability theory and Applications, Stochastic Processes, Time Series Analysis, Mathematical Finance, Econometric modeling, Regression modeling, Systems, and Control Theory, Game Theory, Statistical

After completing the program, the scholars get excellent opportunities to pursue research and teaching careers in education and research institutions worldwide. And for those who are willing to apply mathematics to practical problems, there has been a dramatic change in the job scene over the past few years. The scholars also get jobs in industries. They can even join software industries. Many financial services companies, research labs of multinational companies, and others aggressively recruit mathematics researchers.

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