Indian Institute of Science is one of the Institutions which has Research facilities in Department of Mathematics. The department is active in research in most areas of pure and applied mathematics, including algebra and number theory, analysis, discrete mathematics, geometry/topology, mathematical finance, numerical analysis, partial differential equations, probability, representation theory, and time-series analysis. For admission process, you can reach them at http://math.iisc.ac.in/contact-us.html. 

For mathematics the Indian statistical Institute is considered to be the most premier institute for mathematics you can pursue Masters or PhD program at Indian statistical Institute and fulfill your dream of research in mathematics.The Indian statistical Institute software programs on mathematics and statistics so based on your interest you can choose your fields.