Symplectic Reflection Algebras and their Generalizations

这个研究项目是为了更好地从数学上理解镜像对称的某些方面。镜像对称一方面预测了奇异Calabi-Yau簇的变形模与镜像对偶Calabi-Yau簇的(Kahler结构上的)分辨率模之间的自然对应。然而，有大量的奇异Calabi-Yau变体的例子要么没有变形，要么根本没有分辨率，或者两者兼而有之。在这种情况下，镜面对称性分别预测了不存在的交换变形的某种“非对易”替代品的存在。决议，这恢复了变形和决议之间的上述二元性。提案的目标可以简短地表述如下1.将P.Etingof和Pi在过去5年中引入的基于辛反射代数概念的技巧推广到其他类型的奇异辛族。发展非对易辛解的一般变形机制，类似于PI和D.Kaledin早期发展的Poisson变形理论。开始发展Calabi-Yau代数理论。在不一定是辛的Calabi-Yau代数的情况下，这些代数应该代替辛反射代数。非对易几何是一个相对较新的数学领域，它产生于1980年初-S在经典几何和量子理论的交界处。古典几何于19世纪以其现代形式被创造出来，是一种被设计来精确描述各种空间和时间现象的数学理论。它在控制电磁力的方程式中起着基础性的作用，对爱因斯坦的相对论来说更是基础性的。著名的“不确定原理”的发现清楚地表明，经典几何不能充分描述量子理论的全部复杂性。量子理论的复杂性是20世纪理论物理学最重要的成就之一，它是本世纪技术革命的基础，包括激光技术和计算机。这是一种物理理论，它允许描述非常小的物体的行为，比如原子和电子。非对易几何是一种新的几何，也可以称为量子几何，因为它的设计是为了在很小的距离上提供适当的几何。因此，非对易几何的发展是量子物理的绝对基础。目前的提议试图进一步发展这一理论，并探索它与数学和物理的其他领域的相互作用，包括与概率论相当意想不到的联系。

This research project is a step towards bettermathematical understanding of some aspects of Mirror Symmetry.Mirror Symmetry predicts a natural correspondencebetween the moduli of deformations of a singular Calabi-Yau variety, on onehand, and the moduli of (Kahler structures on) resolutionsof the mirror dual Calabi-Yau variety, on the other hand.However, there are plenty of examples of singular Calabi-Yau varietieswhich have either no deformations or no resolutions at all,or both. In such cases, Mirrror Symmetry predicts theexistence of some sort of `noncommutative' substitutes forthe nonexistent commutative deformations, resp. resolutions,that recovers the above mentioned duality between deformationsand resolutions.The goals of the proposal could be briefly formulated as follows.1. Generalize the techniques based on the notion ofSymplectic reflection algebra introduced by P. Etingof and the PI duringthe last 5 years to other classes of singular symplecticvarieties.2. Develop general machinery of noncommutative deformationsof symplectic resolutions analogous to the theory of Poissondeformations developed earlier by the PI and D. Kaledin.3. Begin development of the theory of Calabi-Yau algebras.These algebras should replace symplectic reflection algebrasin the case of not necessarily symplectic Calabi-Yau orbifolds.Noncommutative Geometry is a relatively new area ofmathematics which arose in early 1980-s at the junction ofclassical geometry and quantum theory. Classical geometrywhich was created in its modern form in the 19-th century, is amathematical theory which is designed to describe preciselyvarious phenomena of Space and Time. It plays a fundamental rolein the equations governing electric and magnetic forces,and is even more fundamental for Einstein's Relativity theory. The discovery of the famous `Uncertainty Principle' had clearlydemonstrated that Classical geometry cannot adequately describe all of the complexity of Quantum theory.The latter is one of the most important achievements of theoreticalphysics of the 20-th century, which lies at the foundation ofthis century's technological revolution, includinglaser technology and computers. It is a physical theorywhich allows to describe the behavior of very small objects, likeatoms and electrons. Noncommutative Geometry is a new kind ofgeometry, that may also be called Quantum Geometry since it isdesigned to provide the adequate geometry at very small distances.Development of Noncommutative Geometry is thus absolutelyfundamental for Quantum physics. The present proposalseeks to further develop this theory and to explore itsinteractions with other areas of mathematics and physics,including quite unexpected connections to Probability theory.