Hilbert’s tenth problem asked for an algorithm that, given a multivariable polynomial equation with integer coefficients, would decide whether there exists a solution in integers. Around 1970, Matiyasevich, building on earlier work of Davis, Putnam, and Robinson, showed that no such algorithm exists. But the answer to the analogous question with integers replaced by rational numbers is still unknown, and there is not even agreement among experts as to what the answer should be.
Even before the resolution of Hilbert’s tenth problem, some problems in group theory were proved undecidable. This lecture will discuss how undecidability in group theory led to undecidability in topology, and how undecidability in number theory led to undecidability in analysis.
Undecidable problems have arisen in nearly every major branch of mathematics. This lecture will present a sampling of these and will discuss a few problems whose undecidability status is not yet known, including one about chess.
(The three lectures will be mostly independent of each other.)