Oxford

I will give a survey-type introduction to manifolds equipped with $G_2$ structures, emphasizing the similarities and differences with Riemannian manifolds equipped with almost complex structures, and with oriented Riemannian 3-manifolds. Along the way I may discuss the Berger classification of Riemannian holonomy, the Calabi-Yau theorem, exceptional geometric structures arising from the algebra of the Octonions, and calibrated submanifolds. This talk is the second of two parts.

I will give a survey-type introduction to manifolds equipped with $G_2$ structures, emphasizing the similarities and differences with Riemannian manifolds equipped with almost complex structures, and with oriented Riemannian 3-manifolds. Along the way I may discuss the Berger classification of Riemannian holonomy, the Calabi-Yau theorem, exceptional geometric structures arising from the algebra of the Octonions, and calibrated submanifolds. This talk will be in two parts.

I will present a universal definition of the integers in the field of rational numbers, building on work discussed by Bjorn Poonen in his seminar last term. I will also give, via model theory, a geometric criterion for the non-diophantineness of Z in Q.