Arithmetic Applications of Definable and Hyperbolic Geometry

忘记某些结构是很有用的，这是数学中经常出现的现象。例如，当学习多项式函数时，它可能会适得其反 使用代数结构，而不是只记住，我们正在处理的是，比如说，一个连续函数。这使人们可以自由地执行以下操作 在代数世界中是不可能的(例如剪切和粘贴)，但以某些良好的性质为代价(例如具有有限多个方程的解)。 更广泛地说，数学家们非常努力地寻找合适的工作环境：足够普遍，以便在允许做什么时保持灵活性，但也足够 混凝土，以便有许多令人愉快的特性。 一个常见的例子是代数函数的世界与全纯函数的世界。我的建议中有很大一部分涉及到开发一个中间类别 这可以松散地描述为‘O-极小全纯函数’。事实证明，我们感兴趣的许多函数--如指数函数和代数簇研究中出现的自同构函数--并不完全是代数的，但比一般的全纯函数表现得更好。这一理论是由彼得齐尔和斯塔琴科首创的，已经在功能超越和数论中得到了广泛的应用。与我的同事们一起，我们正在进一步推动这一理论，以使 用于研究更多微妙的代数现象(幂零加厚、变形理论、相干层等) 这对霍奇理论已经有了巨大的应用--霍奇理论是通过上同调来理解代数簇的一个特别强大的工具。我们已经证明了Hodge理论的自然背景实际上是‘o-极小全纯函数’，并利用这一领域长期存在的猜想证明了这一点。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。 霍奇猜想是霍奇理论的圣杯(也是代数几何的中心问题之一)。这使得人们可以从它们的Hodge结构(简单得多的线性代数信息)中获得关于代数子簇(多项式方程的解)的信息。这项提议的目标之一是试图利用这项技术在霍奇猜想上取得进展。具体地说，我们希望能够用这些方法解决一个重要的问题，即“绝对霍奇猜想”。

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.