IIT

An AIR of 1410 in IIT JAM Mathematics does not get you through premier IITs such as Bomb, Kanpur, Madras, and Delhi because they have a very low closing rank. However, it gives a good opportunity for newer IITs like IIT Indore, IIT Hyderabad, IIT Gandhinagar, and maybe even IIT Bhilai or IIT Palakkad, which tend to have higher cutoffs but increased opportunities in research in Pure Mathematics. If you want to pursue a research-oriented career in the strong research area, then explore IISERs, CMI, or TIFR as an option along with considering the above ones for further studies or a Ph.D. in mathematics.

If an applicant wants to write the GATE Statistics (ST) paper and subsequently apply for Data Science courses in IITs, he/she must be either pursuing his/her final year studies or holding a bachelor's degree in Statistics, Mathematics, Engineering, Technology, or any appropriate Science stream. Many of these IITs accept GATE ST scores for the M.Tech or M.Sc. degrees in Data Science, but eligibility varies according to institutes, making it wise to consult the specific IIT's admission brochure to determine which GATE papers are accepted for their Data Science programs.

The closing rank for general category in M.Sc Chemistry was around 787.

You rank is actually very close to the cutoff.

You may receive a seat in 2nd round or 3rd round as it might change across the rounds when seats freed.

Congratulations on your excellent rank and all the best.

Trying to get into NITs or IITs for B.Arch with NATA? Hate to break it to you—but NATA scores don't work there. NITs only accept JEE Main Paper 2, and IITs want you to clear JEE Advanced + the AAT (Architecture Aptitude Test). Save your NATA scores for private architecture schools or other institutes that actually accept it. For NIT/IIT, it's JEE route only—no shortcuts.

Yes, it is possible to get admission to IIT Kharagpur for MSc+PhD in Mathematics with a rank of 250 in IIT JAM for candidates belonging to the General AI category. Round 4 of the cutoff list has been released and the cutoff rank for admission to MSc+PhD in Mathematics was closed at 254 for candidates belonging to the General AI category. For candidates belonging to the OBC AI category, the cutoff rank was closed at 391. For candidates belonging to the SC AI category, the cutoff was closed at 1014. For more categories such as ST AI and EWS AI category, getting admission to the college with this rank is possible. 

There are only 2 IITs that offer Nanotechnology courses in India at present. You can pursue those courses at the PG level. You can study Nanotechnology courses in IITs, like IIT Roorkee and IIT Madras, which offer M.Tech in Nanotechnology and M.Tech in Functional Materials and Nanotechnology, respectively. The Nanotechnology course fees at the IITs is around INR 20,000. 

It will require a very high JEE Advanced rank to get admissions in an IIT. In the top IITs (e.g Bombay, Delhi) popular branches such as Computer Science Engineering (CSE) may require an All India Rank (AIR) of below 100-200 in the General category. In other faculty members and newer IITs ranks can go up to 10,000 or more depending upon the institute, course or degree. These cut offs vary every year depending on the difficulty level of the exam, applicants and seats.

To get into B.Sc Chemistry at an IIT, you must qualify JEE Advanced. For General category, a rank under 6000 is generally needed. Top IITs may require ranks within 1500–3000. For OBC, ranks up to 8000 may work, and for SC/ST, ranks up to 25,000–35,000 might be accepted.

Class 12 marks do not impact  into IIT admissions. But if you get 70 and above then there are more chances to be highlighted after GATE.  

Gate is given for Masters admission especially M.Tech . 

Gate score between 400 and 800 is required to get admission in IIT or NITs.