Srinivasa Ramanujan was a self-trained mathematical genius and self-styled mystic from Madras, India who died in 1920 at the age of thirty-two. Contrary to the Western scientific tradition, he didn't reveal his methods, other than to say that mathematical observations came to him in dreams provided by a Hindu god, didn't provide proofs, and didn't show his work.
But, observations about number theory and abstract algebra that he made from his death bed in 1920, one of a veritable horde of conjections neither proved nor disproved that he advanced. Some of these have just been proved using modern mathematical methods developed in the last decade.
A televised documentary of his life and accomplishments is coming soon in honor of the 125th anniversary of his birth.
Modern mathematicians are still struggling to figure out how he saw relationships that three generations of mathematicians since him, informed by far more research upon which they build their own work, have not managed to see. What fruitful component was there to his work that has eluded the entire profession in the West for so long?
Few mathematicians believe that Srinivasa Ramanujan was genuinely divinely inspired, but this isn't to say that they don't think he was on to some amazing unstated principal or approach to their trade that they lack, and which makes the relationships more obvious. In the same way, while it was an accomplishment for Wiley to finally prove Fermat's Last Theorem, using methods that were clearly not available centuries earlier when it was formulated, mathematicians still daydream over and ponder what simpler approach (even if not fully rigorous) Fermat could have used to reach his conclusion.
Then again, mathematics is a mature discpline. With a handful of notable exceptions (e.g. fractals, and the simplex method of solving linear equations), particularly in applied mathematics, almost all of the material studied by mathematics students in undergraduate and early level graduate level courses had been worked out by the deaths of Swiss mathematician Leonhard Euler in 1783 and French mathematician Jean Baptiste Joseph Fourier in 1830, several generations before Srinivasa Ramanujan was born.
There is very little being taught in graduate school mathematics classes today that Srinivasa Ramanujan would have either been immediately familiar with and able to grasp, or had the foundational knowledge to figure out in a matter of a few days or weeks. The state of mathematics in 1920 was not so very behind what it is today in a great many of its subfields, including number theory, where he was most renowned.
Number theory remains a subfield of mathematics where many easy to understand problems remain unsolved and where each new advance seems like some sort of miracle not easily inferred by just anyone from the knowledge that came before it in the field. It has progressed not in some logical and orderly fashion, but with a crazy quilt of zen-like observations whose connection to a larger context and structure of the theorems of the field is obscure.