Title: Effectivity in Faltings' Theorem.

Abstract: In joint work with Brian Lawrence we show that, assuming
standard motivic conjectures (Fontaine-Mazur, Grothendieck-Serre,
Hodge, Tate), there is a finite-time algorithm that, on input (K,C)
with K a number field and C/K a smooth projective hyperbolic curve,
outputs C(K). On the other hand, in certain cases (i.e. after
restricting the inputs (K,C) --- e.g. so that K/Q is totally real and
of odd degree) there is an unconditional finite-time algorithm to
compute (K,C)→ C(K), using potential modularity theorems. I will
discuss these two results, focusing in the latter case on how to
unconditionally compute the K-rational points on the curves C_{a} : x^{6}
+ 4y^{3} = a^{2} (i.e. a in K^{x} fixed) when K/Q is totally real of
odd degree.

(The talk will cover Chapters 7, 9, and 11 of my thesis, available on e.g. my website.)