Arithmetic Applications of Definable and Hyperbolic Geometry

在数学中，忘记某些结构是很有用的，这是一个经常出现的现象。例如，当一个人在研究一个多项式函数时，使用代数结构可能会适得其反，相反，一个人应该记住，他正在处理一个连续函数。这使人们可以自由地执行在代数世界中不可能执行的操作（例如剪切和粘贴），但代价是某些良好的特性（例如具有有限多个方程的解）。更一般地说，数学家们非常努力地寻找合适的工作环境：足够普遍，以便在“允许”做的事情上灵活，但又足够具体，以便有许多令人愉快的性质。一个常见的例子是代数函数的世界，和全纯函数的世界。我的建议的很大一部分涉及开发一个中间类别，可以被粗略地描述为“0 -极小全纯函数”。事实证明，我们感兴趣的许多函数，比如指数函数，以及在代数变量研究中出现的自同构函数，并不是完全代数的，但比一般的全纯函数表现得更好。这个理论是由Peterzil和Starchenko提出的，已经在泛函超越和数论中得到了广泛的应用。与我的同事一起，我们正在进一步推动这一理论，以允许研究更细微的代数现象（幂零增厚，变形理论，相干束等…）这已经在霍奇理论中有了巨大的应用——霍奇理论是通过上同调来理解代数变异的一个特别强大的工具。我们已经证明了hodge理论的自然设置实际上是“0 -极小全纯函数”，并使用这个在该领域证明了长期存在的猜想。更重要的是，许多现有的结果变得更加精简，使整个主题更容易理解。霍奇理论的圣杯（也是代数几何的中心问题之一）是霍奇猜想。这使得人们可以从代数子变量（多项式方程的解）的混合结构（更简单的线性代数信息）中推导出代数子变量的信息。本提案的目标之一是试图利用该技术在霍奇猜想上取得进展。具体地说，我们希望用这些方法解决一个叫做“绝对霍奇猜想”的重要问题。

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.