In his address to the Paris meeting of the International Congress of Mathematics in 1900, David Hilbert delivered one of the most influential speeches ever given by a mathematician. In his speech titled "The future problems of mathematics," Hilbert presented (a subset of) his famous list of 23 unsolved problems to be studied in the coming century. Attempts to solve these problems have since resulted in interesting developments throughout mathematics.

This talk will briefly present some of Hilbert's problems, their current status, and broader mathematical developments resulting from them. Since these problems range greatly in the precision of their statement, it is hard to describe the current status of every single one of them. Furthermore, depending on how "resolved" is defined, there may be different answers as to whether some of these problems have been resolved. The talk will address this in more detail.

After explaining the relevant background (for example, consistency and decision problems), we will study in detail Hilbert's first and tenth problems, i.e. the continuum hypothesis and the existence of an algorithm to determine whether any diophantine equation has an integer solution. Through these problems, we will address the notions of independence and undecidability.