Title: Local-to-global principles for solving Diophantine equations

Abstract: The modern approach to the question of whether a polynomial equation admits rational solutions is to first check whether local solutions exist at every completion of the rationals (a finite computation), and then check whether the Hasse principle holds. If the Hasse principle holds, then the existence of local solutions everywhere guarantees the existence of a rational solution.

In this talk, I will introduce the Hasse principle and give examples of some nice families of equations for which the principle holds. I will then discuss this local-to-global approach for a natural class of equations coming from norms of number fields (the basic objects of algebraic number theory). I will provide an overview of what is known on this topic, the main techniques one uses to study it and present some recent developments and lines of research.

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*Abstract*: A finitely-presented group *G* is hyperbolic if there is a linear bound on the number of relators required to prove that a word of length *n* is equal to the identity in *G*. The word problem in a group that is known to be hyperbolic is solvable in linear time. However, it is undecidable in general whether a group is, in fact, hyperbolic. This talk will present some efficient, low-degree polynomial-time procedures which seek to prove that a given finitely-presented group is hyperbolic. If successful, they can also often construct, in low-degree polynomial time, a linear time word problem solver and a quadratic time conjugacy problem solver. This is joint work with Derek Holt, Steve Linton, Max Neunhoffer, Richard Parker and Markus Pfeiffer.