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

数学中最古老的话题之一是丢番图方程的研究(以公元3世纪希腊数学家亚历山大·丢番图斯的名字命名)。丢番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，给出一个丢番图方程，其目的是为未知数(上面的X和Y)找到整数值或有理值，从而使方程成立。这种值的组合被称为方程的积分或有理解。一般来说，寻找丢番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显式处理。采取更概念化的方法，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，这一观点显然发挥着关键作用，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发挥着主导作用。事实上，这一点在1983年得到了深刻的证明，当时Falings证明了Mordell猜想，该猜想指出，满足某些几何条件的两个变量的丢番图方程只有有限多个有理解。1986年，法林斯因证明而被授予菲尔兹奖。莫德尔的猜想引发了许多其他的有限猜想，如安德烈、朗、马宁、芒福德、奥尔特等人。然而，尽管这些猜想有明显的相似之处，但不清楚它们之间是如何联系的。直到Zilber-Pink猜想出现；在一个庞大的新猜想中，它同时推广了上述所有猜想。Zilber-Pink猜想是一个不可能相交的问题，这个名字源于一个简单的原理，即在d维空间中，如果n和m之和小于d，则n和m维的两个几何物体极不可能相遇。例如，考虑从实验室的相反角落发射的两个激光；我们预计它们会相互错过，因为激光是在三维空间中发射的直线(因此是1维的)，而1+1=2小于3。近年来，不太可能相交的问题引发了一系列活动，很大程度上是因为数理逻辑带来了新的工具。这些最早是由Pila和Zannier应用于所谓的Manin-Mumford猜想，他们的方法启发了一个普遍的策略，已经在这个主题中产生了深远的影响。拟议的研究试图获得关于Shimura变种的新的算术结果，由于作者和他的合作者之前的工作，已知通过Pila-Zannier策略的扩展产生了对Zilber-Pink猜想的重大进展。它还试图在Zilber-Pink猜想的背景下获得有效的结果(换句话说，具有主要可计算的数字输出的结果)。它将使用已在更简单的环境中产生结果的微分方程式几何中的新工具来实现后一目标。

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.