Why computers can't replace mathematicians (yet)

In 1900, David Hilbert presented his list of 23 mathematical challenges for the 20th century. Hilbert’s 10th problem asked for what we would now call a computer algorithm to decide whether a given Diophantine equation has integer solutions. Diophantine equations are among the oldest problems in mathematics. In spite of their innocent appearance, and substantial effort over the centuries, these equations remain quite thorny.

Surprisingly, Yuri Matiyasevich proved in 1970 that the algorithm that Hilbert was asking for does not exist. This is a manifestation of the fundamental limits of formal mathematical systems that were discovered by Kurt Gödel in 1931 and further developed by the mathematician and computer pioneer Alan Turing.

Nevertheless, enormous progress had been made in our understanding of Diophantine equations over the last decades. One of the most profound insights of 20th century mathematics are the intimate connections between number theory and geometry: geometric properties of Diophantine equations (the “shapes” defined by these equations) determine their arithmetic properties.

Johannes Nicaise is a Professor of Mathematics at Imperial College London. In his inaugural lecture he will illustrate these connections between number theory and geometry, present the results by Gödel, Turing and Matiyasevich, and outline some important open problems to demonstrate that different branches of mathematics are often intertwined in fascinating ways.

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