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的研究计划，该计划与代数学，算术和微分几何的最重要领域有着深刻的联系。它涉及有理点，丢番图逼近和高维双有理几何。这是数学物理学的影响。 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可靠的资金来源。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。首席研究员及其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的值分布理论的数论解释的影响。它与高维双有理几何相交，包括K-稳定性和Kahler几何的思想。PI也有科学的兴趣是在阿贝尔品种，向量束，代数，数论和表示论的组合和计算方面的方向。PI和他的团队将围绕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开展工作。在一个平行的方向，他们将研究阿贝尔簇，向量丛，代数曲线和李代数之间的丰富的相互作用。最后，有一个方面，处理有效的计算方法，代数几何，数论和交换代数。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。该方案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Centre de Recherches Mathematiques（蒙特利尔）及其实验室以及Fields Institute（多伦多）的协调将是PI计划的首要组成部分，以开发高素质人员的长期持续发展。PI提案的主要科学内容强调以下更专业的几何学，数论和抽象代数领域。(i)线性级数，其正性测度和牛顿-奥昆科夫体。(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理论）。(iii)投射簇与模空间的丢番图算术。(iv)阿贝尔簇，卡拉比-丘流形和代数曲线. (v)计算计算机代数。这些主题继续处于研究的前沿，处于代数，几何和数论的交叉点。