Yes there are such research institutions at the UG level. In basic sciences :



NISER-Bhubaneshwar (National Institute for Science Education and Research)



UM-DAE-CBS, Mumbai (Centre for Excellence in Basic Sciences)



Both the institutions are DAE funded autonomous UG research institutions awarding the degree of Int.M.Sc. of 5 years. They have a common entrance test ie National Entrance Screening Test -NEST. NISER has 100 and CBS has 45 seats. For further information on NEST and both of these institutes you may refer to the respective websites.



For pure Mathematics, the following two are undoubtedly the best:



ISI - Indian Stastical Institute : Kolkata (for B.Stat.) & Bengaluru (for B.Math.) (http://isical.ac.in)



CMI -Chennai Mathematical Institute



ISI and CMI have their own entrance tests which is one of the toughest mathematics test at the 12th standard level.



ISI has two courses for which the entrance test (written exam) is common followed by separate interview for the two courses of the selected candidates :



B.Stat. at Kolkata center



B.Math. at Bengaluru Center



CMI too has two courses but in its Chennai campus itself, the entrance to which is common :



B.Sc. in Mathematics and Computer Science



B.Sc. in Mathematics and Physics

IISC Bangalore is top among all other universities, which stands 1st in NIRF university rankings 2017.

You can enter into IISC based on JEE MAINS rank.



NISER is also one of the top institute  located at bhuvaneshwar, which u can get through it through NEST examination.

IISc can be a option you can try .it's application form  will available in February