Mathematical Sciences: Conjectures of Ramanujan in the      Unorganized Portions of his Notebooks
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In the years following the death of the greatest Indian                    mathematician, Srinivasa Ramanujan, three notebooks containing                  unfinished and unpublished work of his were discovered.  The                    contents of the notebooks reflected much of the mathematical                    style of the man. There are unsupported conjectures, claims and                 formulas written in an unorganized manner, covering vast areas of               mathematics.  No other recorded genius is known to have practiced               mathematics like Ramanujan.  He worked in isolation with only a                 few old books at his disposal.  He was not aware - or apparently                concerned - with structured mathematics as it had developed in                  the western world during the early part of the twentieth century.               Indeed, even today, none of those who have studied his remarkable               contributions and legacy, can comprehend how he came to have such               remarkable insight with so little formal training.  It is the                   search for this insight which motivates much of the work to be                  supported by this award.                                                             The means by which this effort will be carried out is simple               to state. One goes through the notebooks statement by statement                 (there are roughly 3000 results stated without proof), verifying                or negating each one, using all the powerful mathematical tools                 now available.  This task has been going on for many years.  The                first notebook is essentially understood and the second is also                 nearly complete. As a result of this, over 40 papers and two                    definitive books have appeared. Work is now progressing on the                  third which will involve higher order transformations of                        hypergeometric series and inversion formulas stated by Ramanujan.                    Some of the deepest work and potentially most difficult                    challenges will be in the area of asymptotic expansions of                      infinite series, products and integrals.  At some later time                    attention will have to be given to brief allusions to                           generalizations in the theory of elliptic functions made by                     Ramanujan in a 1914 paper.  Any progress here is expected to be                 of considerable interest to physicists.
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[{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}, {'Code': '1264', 'Text': 'ALGEBRA,NUMBER THEORY,AND COM'}]
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