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.
Also, more details on these topics are contained in the following survey articles written by Bjorn Poonen:
http://math.mit.edu/~poonen/papers/h10_notices.pdf
http://math.mit.edu/~poonen/papers/sampler.pdf