Professor S. G. Dani


Mathematics

7886 Views Equations,inequalities and their solutions

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Mankind,across cultures,has grappled with various kinds of equations since ancient times,in a variety of contexts. The Pythagorean equation x2+y2=z2 played an important role in the works of Greek mathematicians and the geometric constructions in Sulvasutras of the Vedic period. It further inspired what is called "Fermat''s last theorem",that the analogous equation with higher exponents in place of 2 has no nontrivial solutions,stated by Fermat,whose proof involved works of numerous eminent mathematicians and was completed more than three centuries after its formulation. Linear Diophantine equations,and what is commonly known as "Pell''s equation" held sway in the works Indian mathematicians from the end of the 5th century until the 12th century,starting with Aryabhata,and culminating in the work of Bhaskaracharya,in which one finds a complete exposition of the approach to the solutions,through Kuttaka and Chakravala methods.During the modern era,apart from various equations generalizing the scenario founded on the special equations as above,there has also been interest in approximate solutions,solutions involving a small error - it being expressed in the form of inequalities. One of the well-known problems in this context,known as the Oppenheim conjecture,going back to 1930 was settled only in 1986,by the Fields medalist G.A. Margulis,and has subsequently led to a lot of further work,including some contributions of this speaker. The techniques towards the solutions have ranged from simple arithmetic,algebra,geometric methods,and most recently,study of dynamics of certain flows. In this talk we shall discuss the developments around these problems and conclude with some unresolved questions in the area.

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