I am a sophomore currently enrolled in an integrated Masters course in Mathematics and Computing at IIT Kharagpur. What should be my course of action from this point on if I want to pursue research in Mathematics after the completion of my course?

Okay, all most all of the stuff I can think of have already been addressed. Here are some points, mainly mistakes I think I made in your position four years ago and some general funda about the skill sets you need to develop. 1. Studying mathematics properly will result in very little tangible gains in your kgp life, understanding pure maths or knowing the proofs of theorems are unlikely to help your grades . Don't get bogged down or frustrated by this, keep on learning, 2. Kgp doesn't have much a of a culture for pure mathematics. Do not feel bitter about the environment you are in.  Feeling bitter will not get you anywhere, make the best of the situation. Here are some advantages of being in kgp: Many, (not all) courses in the department are taken by professor's not concerned about student attendance. If some of these courses do not interest you, bunk them, use the spare time to read. In the department never ever be impolite to professors. For such courses ideally the professor should  have no  idea who you are by the end of the course. Most exams constitute of questions being repeated from previous years, you can still make these courses useful. For example the modern Algebra mid sem, constitutes entirely of problem and theorems directly from IN Herstein's chapter on groups (hell some problems from sections not covered in class were included). Most of these are problems or counterexamples which you must know  before hand, even if you have proved a result before, you should remember the method to be able to reproduce it during the exam. Don't get bogged down by this extremely uncreative course structure, if you can look past this you have an excellent opportunity of thoroughly reading a very important area of pure mathematics from what is a classic book. Many of the problems in the exercises are very important results by themselves, and some of the methods are so widely used that they are worth remembering. I was reading a book once and the author commented, " a method is worth a thousand theorems"! This is extremely true, remembering a method just for the sake of an exam is bad, being familiar with a method to the extent you remember it, is actually very useful and gives you a line of attack to tackle other problems. 3. Read books, reading good books will really help you a lot. Most books have exercises appended to the chapters, almost all of such problems these days have hints(a place to start thinking) etc with them, work through them, they will help you immensely in your understanding of the matter. Almost all areas near about pure mathematics will require you to be familiar with basic Abstract Algebra, Linear Algebra, Real Analysis (includes Measure Theory which is an optional elective in the final year!!), Complex Analysis. If you use Herstein for Abstract Algebra, you can find a solution manual for it on DC, this if not abused is an useful companion. 4. Getting into a summer research program or into a grad school will require recos. Since you will find very few pple from a pure mathematics background in kgp, these reco's will be tough to procure. One such person is Prof. V.K Jain. I did my Msc thesis under him, his research interests were different to mine, and also his topic of research has become obsolete.  However working with him is still worth-while, since firstly you will be reading papers published from many respected journals of pure mathematics(Prof Jain himself has published in the Proceedings of the American Mathematical Society and the Bulletin Of The ams), further the area he works on is far simpler than most areas of pure maths. An undergraduate can read the initial papers with very basic understanding of complex analysis, of course as you go through all the literature over the years, you will see many lemma's and methods repeatedly used , and you will have to specialize yourself in identifying spots to apply these techniques. But the Msc  thesis comes in the final year, I would advice you to go regularly to any course Prof Jain is taking. His exams will mostly test your ability to write mathematics well. This itself is an extremely important skill and should not be underestimated, something in which I used to be very bad and still am quite bad (though better than before). One way to improve yourself in this regard is to pay more attention to the mathematical literature you read. Try to frame arguments yourself and then see how the author states the same thing. Any mathematical literature will leave out a lot of details for the reader to figure out. A good exposition strikes the perfect balance between what is included and excluded (taking into account the audience). Among the two sins of writing too much and too little, the later in my opinion is the lighter sin. Writing less details will make it easier for you to make your proofs clear. A clear proof  with some steps skipped (within a reasonable limit ) will be understood by a professor. http://erickson.sites.truman.edu/files/2012/04/guide1.pdf may help you in this regard. Good Luck

I agree completely with Anurag's points. I would also recommend reading widely many introductory books on various disciplines of mathematics. The IIT KGP library used to have a decent if old fashioned collection of books on various parts of mathematics. There I encountered many interesting books. Even if you read the first 50 pages of these books you learn a lot. I would also recommend reading Halmos's autobiography "I want to be a Mathematician" as an inspirational read (along with similar works.) Another thing that I would like to add is to work on interesting problems. There are many excellent collections of problems available nowadays, and one can learn much more by doing problems than by merely reading stuff. Finally, while the courses in the CS dept are very good, they actually cover a rather narrow part mathematics. For anybody with an interest in say Differential Geometry, Harmonic Analysis etc., there are no options in KGP except self study.

It's an appreciable thing that you are interested in research in Mathematics. Few things that you should know are: Maintain a good CGPA. No professor believes you or encourages you (unless he/she knows you personally) to take up a project under him, if you do not have a good CGPA. I hope you have a good CGPA. Try to figure out your area of interest. One good way is to take up research projects during your summer and if you are interested, even in winter. There are various summer fellowship programs offered by very good institutes such as IAS, JNCASR, IMSc, IISR, CMI, TIFR etc. Some offer research projects in winter also. Keep track of them. If you cannot get any of those, do not get disappointed. Stay back in summers at KGP and take up a project in your area of interest (from the courses that you have taken as of now). Generally from my experience, any professor outside the institute, will look at your CGPA and the projects that you have taken, before giving you a project. So, do not waste your summers. The most important thing is know your seniors who are also interested in research and planning to go for research after graduation. They can guide you the best. All the best.