This proceedings records the 31st International Colloquium on Group Theoretical Methods in Physics (¿Group 31¿). Plenary-invited articles propose new approaches to the moduli spaces in gauge theories (V. Pestun, 2016 Weyl Prize Awardee), the phenomenology of neutrinos in non-commutative space-time, the use of Hardy spaces in quantum physics, contradictions in the use of statistical methods on complex systems, and alternative models of supersymmetry. This volume¿s survey articles broaden the colloquiäs scope out into Majorana neutrino behavior, the dynamics of radiating charges, statistical pattern recognition of amino acids, and a variety of applications of gauge theory, among others. This year¿s proceedings further honors Bertram Kostant (2016 Wigner