The interplay between finding solutions to sets of equations and questions in geometry has been very fruitful in Mathematics. The PI's research fits into this way of thinking about both topics. The study of linear algebraic groups and homogeneous spaces provides a unified plank to understanding distinct interesting objects in algebra, geometry and number theory. Extending the classical study over number fields, for example the rational numbers, to function fields, which includes the class of simple functions such as polynomials, which is part of the PI's proposal,is useful from the geometric perspective. Engaging graduate students on topics related to algebraic groups and homogeneous spaces, an area ripe with questions accessible to students, via seminars and workshops, will be part of the activities of the PI during this project execution.
The study of quadratic forms and their zeros over function fields over number fields and p-adic fields is an example of the objects studied during this proposal period. The PI plans to investigate questions related to the study of homogeneous spaces with special reference to quadratic forms and Brauer groups. The PI shall study the period-index questions for the Brauer group of function fields of curves over number fields with a view to bounding the u-invariant of such fields. It is an open question whether quadratic forms in sufficiently many variables over function fields of curves over totally imaginary number fields have a nontrivial zero with conditional results dependent on the Hasse principle for twisted moduli spaces over curves over number fields; the PI will investigate the obstruction to the Hasse principle for such spaces. Higher reciprocity obstructions using the Bloch-Ogus theory will be used to study the existence of rational points on homogeneous spaces over function fields. The PI also proposes to study G-trace forms, via construction of invariants, towards answering realisability questions.
