Arithmetic of Homogeneous Spaces under Linear Algebraic Groups

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This award concerns work in Arithmetic Geometry, an important area of mathematics at the intersection of algebraic geometry and number theory. Recent progress on the topic of the study of homogeneous spaces has set forth new questions and conjectures whose study is part of the proposal. There are related problems accessible to graduate students and they will be engaged via topical graduate courses, seminars and workshops. The PI and Co-PI plan to organize workshops in related areas and disseminate the outcome of research to mathematical community through lectures and seminars. Encouraging and engaging women graduate students will be part of the PI's mission.The PI and Co-PI plan to investigate questions related to the study of homogeneous spaces under connected linear algebraic groups with special reference to function fields of curves over local and global fields. This study, with reference to classical groups, is closely related to the study of the Brauer group of the field and the period-index bounds as well as the study of quadratic forms and the u-invariant. The PI and Co-PI plan to investigate the obstruction to the Hasse principle for homogeneous spaces via the Brauer-Manin like obstructions as well as higher reciprocity obstructions which have already been constructed. They will investigate the Brauer-Manin obstruction for moduli spaces of twisted sheaves over curves over totally imaginary number fields to get a hold on the u-invariant of function fields of such curves whose finiteness is a big open question. They will also study Serre/Totaro question on the existence of closed points of degree d on principal homogeneous spaces under absolutely simple simply connected groups admitting a zero cycle of degree d.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.