Methods for arithmetic distance, distribution and complexity of rational points

有理点算术距离、分布和复杂度的计算方法

• 批准号: RGPIN-2021-03821
• 负责人: Grieve, Nathan
• 金额: $ 1.31万
• 依托单位: Carleton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=742736
• 关键词: Methods; arithmetic; distance; distribution; complexity

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.

目前的提案要求资助 PI 的研究项目，该项目与代数、算术和微分几何的最重要领域有着密切的联系，它涉及有理点、丢番图近似和高维双有理几何。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。因此，PI 的学员将获得可转移的技能。 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 Vojta 对 Nevanlinna 价值分布理论的数论解释的影响。维双有理几何，包括 K-稳定性，以及卡勒几何的思想，PI 还对阿贝尔簇、向量丛和等方向有科学兴趣。 PI 和他的团队将在代数、数论和表示论的组合和计算方面研究围绕 Vojta 的主要猜想、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。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的有效计算方法。几何、数论和交换代数。 PI 将确保历史上代表性不足的群体的成员享有平等机会。 PI 和他的学员将为 STEM 领域的学生规划推广活动。目标是与数学研究中心（蒙特利尔）协调，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。其实验室以及菲尔兹研究所（多伦多）将成为 PI 计划的首要组成部分，以长期持续培养高素质人才。 PI 提案的主要科学内容侧重于以下更专业的领域。几何、数论和抽象代数 (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、 (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) 计算计算机代数仍然处于交叉领域的前沿。代数、几何和数论。