Arithmetic of Homogeneous Spaces under Linear Algebraic Groups

该奖项涉及算术几何方面的工作，算术几何是代数几何和数论交叉的一个重要数学领域。同质空间研究的最新进展提出了新的问题和猜想，这些问题和猜想的研究是本提案的一部分。研究生可以接触到相关的问题，他们将通过专题研究生课程、研讨会和讲习班参与其中。PI和Co-PI计划在相关领域组织讲习班，并通过讲座和研讨会向数学界传播研究成果。鼓励和吸引女研究生将是学院使命的一部分。PI和Co-PI计划研究与连通线性代数群下齐次空间研究相关的问题，特别涉及局部和全局场上曲线的函数场。本研究参照经典群，与场的Brauer群和周期指数界的研究以及二次型和u不变式的研究密切相关。PI和Co-PI计划通过Brauer-Manin类障碍物以及已经构建的更高互易障碍物来研究齐次空间中对Hasse原理的阻碍。他们将研究曲线上全虚数域上扭束模空间的Brauer-Manin障碍，以得到这类曲线函数场的u不变量，这类曲线的有限性是一个悬而未决的大问题。他们还将研究Serre/Totaro问题，该问题是关于承认d度零循环的绝对简单单连通群的主同质空间上d度闭合点的存在性。该奖项反映了美国国家科学基金会的法定使命，并通过基金会的智力价值和更广泛的影响审查标准进行了评估，认为值得支持。

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.