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、算术和微分几何等最重要的领域有着深厚的联系。它涉及有理点、丢番图近似和高维双有理几何。有数学物理学的影响。 所申请的资金将为博士后研究员、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、工业和高等教育领域获得研究职位。 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 的程序受到 Vojta 对 Nevanlinna 价值分布理论的数论解释的影响。它与更高维的双有理几何相交，包括 K-稳定性以及卡勒几何的思想。 PI 的科学兴趣还包括阿贝尔簇、向量丛以及代数、数论和表示论的组合和计算方面。 PI 和他的团队将研究围绕 Vojta 主要猜想、线丛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。在并行方向上，他们将研究阿贝尔簇、向量丛、代数曲线和李代数之间丰富的相互作用。最后，还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 将确保历史上代表性不足的群体成员在公平、多元化和包容性方面享有平等机会。 PI 和他的学员将为 STEM 领域的学生规划外展活动。 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性科学互动。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）的合作将是 PI 长期持续发展高素质人才计划的首要组成部分。 PI提案的主要科学内容侧重于几何、数论和抽象代数等更专业的领域。 (i) 线性级数、其正性度量和牛顿-奥孔科夫体。 (ii) 高维双有理代数几何（包括K-稳定性、最小模型程序和几何不变量理论）。 (iii) 射影簇和模空间的丢番图算术方面。 (iv) 阿贝尔簇、卡拉比-丘流形和代数曲线。 (v) 计算计算机代数。这些主题仍然处于代数、几何和数论交叉点的研究前沿。

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.