Problems of Unlikely Intersections: The Zilber-Pink Conjecture for Shimura varieties

丢番图方程是数学中最古老的课题之一（以3世纪希腊化数学家亚历山大的丢番图命名）。丢芬图方程只是一个多项式方程，通常有两个或多个未知数，其系数是整数（整数）或分数（有理数）。例如，考虑5X+7Y=1或Y=3X^2。给定丢芬图方程，其目的是为未知数（上面的X和Y）找到整数或有理值，从而使方程成立。这样的值的组合被称为方程的积分解或有理解。一般来说，寻找丢芬图方程的积分或有理解的问题是极其困难的，只有最简单的情况才能明确地处理。采用更概念化的方法，可以将丢番图方程视为几何对象（例如，考虑抛物线Y=X^2）。在更现代的时代，很明显，这种观点发挥了关键作用，20世纪一些最伟大的数学进步导致了一个惊人的发现，即几何在诸如此类的算术问题中起着主导作用。事实上，这在1983年得到了深刻的证明，当时Faltings证明了莫德尔猜想，该猜想指出，满足一定几何条件的两个变量的丢芬图方程只有有限多个有理解。1986年，法尔廷斯因他的证明获得了菲尔兹奖。莫德尔的猜想引起了安德烈、朗、马宁、芒福德、奥尔特等人的许多其他有限猜想。然而，尽管这些猜想有明显的相似之处，但它们之间的关系尚不清楚。直到Zilber-Pink猜想出现；在一个巨大的新猜想中，它同时概括了上述所有猜想。它实现这一目标的部分原因是在称为（混合）志村变量的丰富数学对象中工作。Zilber-Pink猜想是一个不可能相交的问题，它的名字来源于一个简单的原理，即在d维空间中，如果n和m的总和小于d，则分别为n和m维的两个几何物体极不可能相遇。例如，考虑从实验室的对角发射的两束激光；我们希望它们彼此错过，因为激光是在三维空间中发射的线（因此是维度1），并且1+1=2小于3。近年来，“不可能相交问题”引发了一系列研究活动，很大程度上是由于来自数学逻辑的新工具。皮拉和赞尼尔首先将这些方法应用到所谓的曼宁-芒福德猜想中，他们的方法启发了一种总体策略，这种策略已经对该学科产生了深远的影响。拟议的研究旨在获得Shimura品种的新算法结果，由于作者和他的合作者以前的工作，已知通过扩展Pila-Zannier策略在Zilber-Pink猜想方面取得了重大进展。它还寻求在Zilber-Pink猜想的设置中获得有效的结果（换句话说，具有主要可计算的数值输出的结果）。它将使用微分方程几何中的新工具来实现后一个目标，这些工具已经在更简单的情况下产生了结果。

One of the oldest topics in mathematics is the study of Diophantine equations (named after the 3rd century Hellenistic mathematician Diophantus of Alexandria). A Diophantine equation is simply a polynomial equation, usually with two or more unknowns, whose coefficients are whole numbers (integers) or fractions (rationals). Consider, for example, 5X+7Y=1, or Y=3X^2.The aim, given a Diophantine equation, is to find integer or rational values for the unknowns (X and Y in the above) so that the equation holds. Such a combination of values is referred to as an integral or rational solution to the equation. In general, the problem of finding integral or rational solutions to a Diophantine equation is extremely difficult and only the simplest cases can be handled explicitly. To take a more conceptual approach, one can think of a Diophantine equation as a geometric object (consider, for example, the parabola Y=X^2). In more modern times, it has become clear that this perspective has a key role to play, and some of the greatest mathematical advances of the 20th century have led to the striking discovery that geometry plays a governing role in arithmetic problems such as these. Indeed, this was profoundly demonstrated in 1983, when Faltings proved Mordell's conjecture, which states that a Diophantine equation in two variables satisfying certain geometric conditions has only finitely many rational solutions. In 1986, Faltings was awarded a Fields Medal for his proof.Mordell's conjecture gave rise to a number of other finiteness conjectures, due to Andre, Lang, Manin, Mumford, Oort, and others. However, although these conjectures shared obvious similarities, it was unclear how they related to one another. This was until the Zilber-Pink Conjecture came along; in a vast new conjecture, it simultaneously generalised all of the aforementioned conjectures. It achieved this in part by working within the rich mathematical objects known as (mixed) Shimura varieties.The Zilber-Pink Conjecture is a problem of Unlikely Intersections, which is a name derived from the simple principal that, in a space of dimension d, two geometric objects of dimensions n and m, respectively, are highly unlikely to meet if the sum of n and m is less than d. Consider, for example, two lasers fired from opposite corners of a laboratory; we expect them to miss each other because the lasers are lines (and, hence, of dimension 1) being fired in 3-dimensional space, and 1+1=2 is less than 3. Problems of Unlikely Intersections have produced a flurry of activity in recent years, in large part due to new tools coming from mathematical logic. These were first applied by Pila and Zannier to the so-called Manin-Mumford Conjecture, and their approach has inspired a general strategy, which has already had profound effects in the subject.The proposed research seeks to obtain new arithmetic results for Shimura varieties that, due to previous work of the author and his collaborator, are known to yield significant progress towards the Zilber-Pink Conjecture via extensions of the Pila-Zannier strategy. It also seeks to obtain effective results (in other words, results with numeric outputs that are in principal computable) in the setting of the Zilber-Pink Conjecture. It will achieve this latter aim using new tools from the geometry of differential equations that have already produced results in simpler settings.