Noncommutative Geometry and Cherednik Algebras

这个项目是关于“非对易射影几何”或射影代数几何与非对易代数的相互作用的理论和应用。粗略地说，与交换情形相类比，二次、三次增长的非交换分次环上的模扭转分次模范畴应分别被认为是投射曲线的非交换类似曲面。这一直觉导致了许多非平凡的见解，并导致了非交换代数的产生。事实上，对非对易曲线(以及二次增长的非对易分次环)进行分类的问题可以被认为是确定的，而这一建议背后的许多激励主题将是理解非对易曲面。这类代数中的一大类可以根据与Keeler和Rogalski合作开发的“幼稚”爆破来分类。尽管这些爆炸的构造方式让人联想到交换爆炸，并依赖于几何数据，但它们的结构与经典物体非常不同。该项目的主要部分将是进一步了解这些对象并扩大它们的应用。该项目的另一个主要主题将是将这一一般理论应用于特定类别的代数。这里有一个特别有用的技巧，那就是把非交换代数上的模范畴“完备”到分次代数上的模范畴，然后应用非交换射影几何。例如，这已经被用来将A型有理Cherednik代数与Hilbert格式以及Haiman在n！猜想。该项目将继续这项研究，以更深入地了解这些重要的代数及其与其他数学领域的关系，例如可积系统和对称空间上不变特征分布的研究。代数几何是现代数学中最古老的领域之一，起源于多项式方程的研究；例如，平面曲线是两个变量的多项式方程的解的集合。这导致了几何对象和相关多项式的(交换)代数之间的丰富的相互作用。非对易代数是一个年轻得多的学科，它也起源于方程式理论，在这种情况下是矩阵方程式，近年来在数学(例如微分方程式理论)和物理的许多领域变得越来越重要(海森伯格的测不准原理是一个经典的例证，但更微妙的非对易，例如在弦理论中)。近年来，很明显，在这些非对易物体中隐藏着明确的、尽管往往相当微妙的几何结构，两者之间的相互作用本身就产生了丰富的理论，实际上是几种理论。这些统称为非对易几何。

This project concerns the theory and application of ``noncommutative projective geometry'' or the interaction of projective algebraic geometry with noncommutative algebra. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has lead to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled and the motivating theme behind much of this proposal will be to understand noncommutative surfaces. A large class of these algebras can be classified in terms of the ``naive'' blow-ups developed in collaboration with Keeler and Rogalski. Although these blow-ups are constructed in a manner reminiscent of commutative blowups, and depend upon geometric data, their structure is quite unlike the classical objects. A major portion of the project will be to further understand these objects and to extend their applications. The other major theme of the project will be in applications of this general theory to specific classes of algebras. A particularly useful technique, here, is to ``complete'' the category of modules over a noncommutative algebra to those over a graded algebra and then to apply noncommutative projective geometry. This has, for example, been used to relate rational Cherednik algebras in type A to Hilbert schemes, and to Haiman's work on the n! conjecture. The project will continue this research to gain a deeper understanding of these important algebras and their relation to other areas of mathematics, for example to integrable systems and to the study of invariant eigendistributions on symmetric spaces.Algebraic geometry, which is one of the oldest areas of modern mathematics, has its origins in the study of polynomial equations; for example a plane curve is the set of solutions of a polynomial equation in two variables. This leads to a rich interplay between that geometric object and the (commutative) algebra of the associated polynomials. Noncommutative algebra, which is a much younger subject, also has its origins in the theory of equations, in this case matrix equations, and in recent years has become increasingly important in many areas of mathematics (for example the theory of differential equations) and physics (Heisenberg's uncertainty principle is a classic illustration, but more subtle non-commutativity occurs, for example, in string theory). It has become apparent in recent years that there are definite, though often rather subtle, geometric structures hidden in these noncommutative objects and the interplay between the two has led to a rich theory, actually several theories, in their own right. These are collectively called noncommutative geometry.