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Geometry Seminar

Date: December 1, 2017

Time: 4:00PM - 5:00PM

Location: BLOC 628

Speaker: Jose Rodriguez, University of Chicago


Title: Numerical computation of Galois groups and braid groups

Abstract: Galois groups are an important part of number theory and algebraic geometry. To a parameterized system of polynomial equations one can associate a Galois group whenever the system has k (finitely many) nonsingular solutions generically. This Galois group is a subgroup of the symmetric group on k symbols. Using random monodromy loops it has already been shown how to compute Galois groups that are the full symmetric group. In the first part of this talk, we show how to compute Galois groups that are proper subgroups of the full symmetric group. We give examples from formation shape control and algebraic statistics. In the second part, we discuss the generalization to braid groups. Braid groups were first introduced by Emil Artin in 1925 as a generalization of the symmetric group and have more refined information than the Galois group. We develop algorithms to compute a set of generators for these groups using homotopy continuation. We conclude with an implementation using Bertini.m2, an interface to the numerical algebraic geometry software Bertini through Macaulay2. This is joint work with Jonathan Hauenstein and Frank Sottile and with Botong Wang.