Methods for arithmetic distance, distribution and complexity of rational points

目前的提案要求为PI的研究项目提供资金支持，该项目与代数、算术和微分几何中最重要的领域有着密切的联系。它处理有理点，丢番图近似和高维双分几何。有数学物理的影响。申请的资金将为博士后、研究生和本科生研究助理提供安全的资金来源。作为影响，PI的学员将获得可转移的技能，使他们能够在政府、工业和高等教育中获得研究职位。PI和他的团队将在学术会议和研讨会上传播他们的研究成果。PI的计划受到了Vojta对Nevanlinna的值分布理论的数论解释的影响。它与更高维度的双几何相交，包括K-稳定性，以及来自Kahler几何的思想。PI也有在阿贝尔变异方向的科学兴趣，向量束，代数，数论和表示理论的组合和计算方面。PI和他的团队将研究围绕Vojta的主猜想，线束的稳定性和正性以及施密特子空间定理的多种形式的问题。在平行方向上，他们将研究阿贝尔变量、向量束、代数曲线和李代数之间的丰富相互作用。最后，有一个方面涉及代数几何、数论和交换代数的有效计算方法。PI将确保在公平、多样性和包容性方面为历史上代表性不足的群体成员提供平等机会。PI和他的学员将为STEM领域的学生计划外展活动。PI的另一个目标是促进蒙特利尔-渥太华-金斯顿-多伦多走廊内研究人员之间的协作科学互动。与数学研究中心（蒙特利尔）及其实验室以及菲尔兹研究所（多伦多）进行协调将是PI制定长期持续培养高素质人才计划的一个主要组成部分。PI提案的主要科学内容强调以下更专业的领域：几何、数论和抽象代数。(i)线性级数及其正性度量和牛顿-奥昆科夫体。（ii）高维双空间代数几何（包括K—稳定性、最小模型规划和几何不变量理论）。(3)射影变分和模空间的丢番图算术方面。（iv） Abelian变种，Calabi—Yau流形和代数曲线。计算计算机代数。这些主题仍然是代数、几何和数论交叉研究的前沿。

The current proposal requests funding in support of the PI's research program, which has deep ties to the most important areas of Algebraic, Arithmetic and Differential Geometry. It deals with rational points, Diophantine approximation and higher dimensional birational geometry. There is an influence from Mathematical Physics. The requested funds will provide a secure source of funding for postdoctoral fellows, graduate students and undergraduate research assistants. As impact, the PI's trainees will obtain transferable skills that will enable them to obtain research positions within government, industry and higher education. The PI and his team will disseminate their research findings at academic conferences and workshops.   The PI's program is impacted by Vojta's number theoretic interpretation of Nevanlinna's value distribution theory. It intersects with higher dimensional birational geometry, including K--stability, and ideas from Kahler geometry. The PI also has scientific interests that are in the direction of Abelian varieties, vector bundles, and combinatorial and computational aspects of Algebra, Number Theory and Representation Theory. The PI and his team will work on questions that surround Vojta's Main Conjecture, stability and positivity for line bundles and the many forms of Schmidt's Subspace Theorem. In a parallel direction, they will study the rich interactions amongst Abelian varieties, vector bundles, algebraic curves and Lie algebras. Finally, there is an aspect that deals with effective computational methods for Algebraic Geometry, Number Theory and Commutative Algebra. The PI will ensure equal opportunity, in terms of Equity, Diversity and Inclusion, for members of historically underrepresented groups. The PI and his trainees will plan outreach activities for students in STEM fields. Another of the PI's objectives is to foster collaborative scientific interactions amongst researchers within the Montreal--Ottawa--Kingston--Toronto corridor. Coordinating with the Centre de Recherches Mathematiques (Montreal) and its laboratories, together with the Fields Institute (Toronto) will be an overarching component of the PI's plan to develop a long term continued development of highly qualified personnel. The main scientific content of the PI's proposal places an emphasis on the following more specialized areas of Geometry, Number Theory and Abstract Algebra. (i) Linear series, measures of positivity thereof and Newton--Okounkov bodies. (ii) Higher dimensional birational algebraic geometry (including K--stability, the Minimal Model Program and Geometric Invariant Theory). (iii) Diophantine arithmetic aspects of projective varieties and moduli spaces. (iv) Abelian varieties, Calabi--Yau manifolds and algebraic curves. (v) Computational computer algebra. These topics continue to be at the forefront of research that is at the intersection of Algebra, Geometry and Number Theory.