Symplectic Reflection Algebras and their Generalizations

这个研究项目是朝着更好地从数学上理解镜像对称的某些方面迈出的一步。镜面对称一方面预测了奇异Calabi-Yau变体的变形模量与镜面对偶Calabi-Yau变体的（Kahler结构）分辨率的模量之间的自然对应关系。然而，有很多奇异的Calabi-Yau品种的例子，要么没有变形，要么根本没有分辨率，要么两者兼而有之。在这种情况下，镜像对称预测了某种“非对易”替代不存在的对易变形的存在。分辨率，它恢复了上述变形和分辨率之间的对偶性。建议的目标可简述如下：将P. Etingof和PI近5年来提出的辛反射代数的概念推广到其他奇异辛变量。发展辛分辨非交换变形的一般机制，类似于早先由PI和D. kaldin发展的泊松变形理论。开始发展Calabi-Yau代数理论。在不一定是辛的Calabi-Yau轨道的情况下，这些代数应该取代辛反射代数。非交换几何是20世纪80年代初在经典几何和量子理论的交汇处兴起的一个相对较新的数学领域。19世纪以现代形式出现的古典几何是一种数学理论，旨在精确地描述空间和时间的各种现象。它在控制电力和磁力的方程中起着基本的作用，对爱因斯坦的相对论来说更是基础。著名的“测不准原理”的发现清楚地表明，经典几何不能充分描述量子理论的所有复杂性。后者是20世纪理论物理学最重要的成就之一，它奠定了本世纪包括激光技术和计算机在内的技术革命的基础。它是一种物理理论，允许描述非常小的物体，如原子和电子的行为。非交换几何是一种新的几何，也可以被称为量子几何，因为它被设计为在非常小的距离上提供足够的几何。因此，非交换几何的发展是量子物理学的绝对基础。本提案旨在进一步发展这一理论，并探索其与其他数学和物理领域的相互作用，包括与概率论的意想不到的联系。

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.