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

数学中最古老的主题之一是丢番图方程（以3世纪希腊数学家亚历山大丢番图的名字命名）的研究。丢番图方程是一个简单的多项式方程，通常有两个或更多的未知数，其系数是整数（整数）或分数（有理数）。例如，考虑5X+7 Y =1，或Y= 3X^2。给定一个丢番图方程，目标是找到未知数（上面的X和Y）的整数或有理值，使方程成立。这样的值的组合被称为方程的积分或有理解。在一般情况下，问题的整体或合理的解决方案，以丢番图方程是非常困难的，只有最简单的情况下，可以明确处理。为了采取更概念化的方法，我们可以将丢番图方程看作一个几何对象（例如，考虑抛物线Y=X^2）。在更近的时代，人们已经清楚地认识到这种观点起着关键作用，20世纪一些最伟大的数学进步已经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，这一点在1983年得到了深刻的证明，当时Faltings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200个有理解。在1986年，法尔明斯被授予菲尔兹奖，他的证明。莫德尔的猜想引起了一些其他的有限性假设，由于安德烈，朗，马宁，芒福德，奥尔特，和其他人。然而，尽管这些建筑有着明显的相似之处，但它们之间的关系却并不清楚。直到沿着出现了齐尔伯-平克猜想，这个新的猜想同时推广了前面提到的所有猜想。齐伯-平克猜想（Zilber-Pink Conjecture）是一个不可能相交的问题，这个名字来源于一个简单的原理，即在一个d维空间中，如果n和m的和小于d，那么两个分别为n和m维的几何对象极不可能相交。例如，考虑从实验室的相对角落发射的两束激光;我们期望它们彼此错过，因为激光是在三维空间中发射的线（因此，维度为1），并且1+1=2小于3。近年来，不可能相交问题产生了一系列的活动，这在很大程度上是由于来自数理逻辑的新工具。Pila和Zannier首先将这些应用于所谓的Manin-Mumford猜想，他们的方法启发了一种通用策略，该策略已经在该主题中产生了深远的影响。拟议的研究旨在获得Shimura品种的新算术结果，由于作者及其合作者以前的工作，已知通过Pila-Zannier策略的扩展，对Zilber-Pink猜想产生重大进展。它还寻求在Zilber-Pink猜想的设置中获得有效的结果（换句话说，具有原则上可计算的数值输出的结果）。它将实现后一个目标，使用新的工具，从几何的微分方程，已经产生的结果，在更简单的设置。

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.