Arithmetic Applications of Definable and Hyperbolic Geometry

可定义几何和双曲几何的算术应用

• 批准号: RGPIN-2019-04178
• 负责人: tsimerman, jacob
• 金额: $ 2.33万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758702
• 关键词: Arithmetic; Applications; Definable; Hyperbolic; Geometry

It is a frequent phenomenon in mathematics that it can be useful to forget certain structure. For example, when one is studying a polynomial function, it can be counterproductive to use the algebraic structure, and instead one should merely remember that one is dealing with, say, a continuous function. This gives one the freedom to perform operations that are impossibly in the algebraic world (such as cutting-and-pasting) but comes at the expense of certain nice properties (such as having finitely many solutions to equations). More generally,  mathematicians work quite hard to find just the right setting to work in: sufficiently general so as to be flexible in what one is 'allowed' to do, but sufficiently  concrete so as to have many enjoyable properties. A frequent example of this is the world of algebraic functions, versus the world of holomorphic functions. A large part of my proposal deals with developing an intermediate category that can be loosely described as `o-minimal holomorphic functions'. It turns out that many of the functions we are interested in - such as the exponential function, and automorphic functions that come up in the study of algebraic varieties - are not quite algebraic, but are much more well-behaved then general holomorphic functions. This theory was initiated by Peterzil and Starchenko and has already found much use in functional transcendence and number theory. Together with my coworkers, we are pushing this theory further to allow for studying much more nuanced algebraic phenomena (nilpotent  thickenings, deformation theory, coherent sheaves, etc...) This has already had enormous applications to Hodge Theory - a particularly powerful tool for understanding algebraic varieties via their cohomology. We have shown that the natural setting for hodge theory is in fact 'o-minimal holomorphic functions', and using this proven long-standing conjectures in the field. More importantly, many of the existing results become much more streamlined, making the whole subject more accessible. The holy grail of hodge theory (and one of the central questions of algebraic geometry) is the hodge conjecture. This allows one to derive information about algebraic subvarieties (solutions to polynomial equations) from their hodge structures (much simpler linear algebraic information). It is one of the goals of this proposal to attempt to make progress on the hodge conjecture using this technology. Specifically, we hope that an important piece called the "absolute hodge conjecture" can be resolved using these methods.

数学中的一种常见现象是，忘记某些结构可能是有用的，例如，当一个人在研究多项式函数时，使用代数结构可能会适得其反，相反，人们应该只记住正在处理的，比如说，一个连续函数。这使得人们可以自由地执行代数世界中不可能的操作（例如剪切和粘贴），但代价是某些良好的属性（例如方程的有限多个解）。更一般地说，数学家非常努力地寻找合适的工作环境：足够普遍，以便在“允许”做的事情上保持灵活性，但又足够具体，以便拥有许多令人愉快的属性，这是一个常见的例子。是代数函数的世界，而不是全纯函数的世界。我的建议的很大一部分涉及开发一个可以宽松地描述为“o-最小全纯函数”的中间类别。事实证明，我们的许多函数都是。人们感兴趣的——例如代数簇研究中出现的指数函数和自同构函数——并不完全是代数函数，但比一般的全纯函数表现得更好。这个理论是由 Peterzil 和 Starchenko 发起的，并且已经得到了应用。在泛函超越和数论中发现了很多用途，我们正在与我的同事一起进一步推动这一理论，以研究更微妙的代数现象（幂零增厚、变形）。理论、相干滑轮等……）这已经在霍奇理论中得到了巨大的应用——霍奇理论是一种通过上同调来理解代数簇的特别强大的工具。我们已经证明，霍奇理论的自然环境实际上是“o-最小全纯”。更重要的是，许多现有的结果变得更加精简，使得整个主题更容易理解（也是霍奇理论的核心问题之一）。代数几何）是霍奇猜想，这使得人们可以从其霍奇结构（更简单的线性代数信息）中导出有关代数子类型（多项式方程的解）的信息，该提案的目标之一是尝试在这方面取得进展。具体来说，我们希望可以使用这些方法来解决称为“绝对霍奇猜想”的重要问题。