Colloquium: Tuesday, November 17, 2pm. Speaker: Ishai Dan-Cohen (Duisburg-Essen). Title: A fundamental group approach to the unit equation.
Over the course of the last 15 years or so, Minhyong Kim has developed a method
for making effective use of the fundamental group to bound sets of solutions to hyperbolic
equations; his method opens a new avenue in the quest for an effective version of the Mordell
conjecture. But although Kim’s approach has led to the construction of explicit bounds in
special cases, the problem of realizing the potential effectivity of his methods remains a diffcult
and beautiful open problem. In the case of the unit equation, this problem may be approached
via “motivic” methods. Using these methods we are able to describe an algorithm; its output
upon halting is provably the set of integral points, while its halting depends on conjectures.