In my opinion the following Institutes are extremely good and involved in very good quality Mathematical research.

1) Chennai Mathematical Institute

2) Tata Institute of Fundamental Research

3) Harish Chandra Research Institute

4) Indian Institute of Science

5) IIT KHARAGPUR, IIT ROORKEE and IIT Bombay.

P.S : Politics and Corruption has become part of our daily lives, but we have to keep up doing the good work under these excruciatingly painful circumstances , i understand how you feel because i myself has gone and perhaps still going through such circumstances.

Politics is everywhere. If you are seeking for a particular place, there may not be vacancies. There are thousands of available places and vacancies. To be aware get in touch with bodies like researchgate.