Duality theories of topological groups and topological algebras

对偶的概念可以追溯到法国数学家埃瓦里斯特·伽罗瓦（Evariste Galois, 1811-1832）的时代，他通过根号（n次根）阐述了多项式方程在单一变量中的溶解度标准。伽罗瓦解的核心在于包含给定多项式的一些解的代数结构（称为“中间域扩展”）与伽罗瓦发明的典型多项式代数结构的子结构（多项式的“伽罗瓦群”）之间的对应关系。这种对应关系的一个重要性质是它颠倒了包含关系：中间场扩展越大，对应的子结构越小，反之亦然。虽然对偶现象经常出现在数学中，但它们很少被认识到。例如，仿射代数几何的起点是若干变量多项式的零集（“代数集”）和具有特殊性质的多项式集（“根式理想”）之间的对偶性。所有有意义对偶共有的主要特征是对偶性质对的集合，即对偶（P，Q），使得一个对象满足P当且仅当其对偶具有Q。本文提出的研究的长期目标是发现具有足够多的具有上述类型的有趣性质对（P，Q）的拓扑群的对偶。我不排除这种对偶不存在的可能性，在这种情况下，我希望证明不可能构建这样的对偶。在这个项目的过程中，我还期望研究与量子力学密切相关的数学对象（即算子代数及其推广）。这一计划的任何进展都将不仅有助于对拓扑群和代数的理解，而且将极大地刺激这些领域的研究，并将引起明显的“人才流失”。从对数学界的影响来看，对偶理论经常充当两个或多个数学领域之间的通道。因此，二元性具有在遥远地区的研究人员之间创造协同作用的潜力。

The concept of duality dates back to the time of the French mathematician Evariste Galois (1811-1832), who formulated the criterion for solubility of polynomial equations in a single variable by radicals (n-th roots). At the heart of Galois' solution lies a correspondence between the algebraic structures containing some of the solutions of a given polynomial (called "intermediate field extensions") and substructures of an algebraic structure invented by Galois that is typical to the polynomial (the "Galois group" of the polynomial). An important property of this correspondence is that it reverses the inclusion relation: The larger the intermediate field extension is, the smaller the corresponding substructure is, and vice versa. Although dualities frequently occur in mathematics, they are less often recognized as such. For instance, the starting point of affine algebraic geometry is a duality between sets zero-sets of polynomials in several variables ("algebraic sets") and sets of polynomials with special properties ("radical ideals"). The main feature common to all meaningful dualities is the collection of pairs of dual properties, that is, pairs (P,Q) such that an object satisfies P if and only if its dual possesses Q. The long-term goal of the proposed research is discovering a duality for topological groups with sufficiently many interesting pairs of properties (P,Q) of the above-mentioned type. I do not rule out the possibility that such duality does not exist, in which case, I hope to prove that it is impossible to construct such a duality. In the course of this project, I anticipate to also investigate mathematical objects that are closely related to quantum-mechanics (namely, operator algebras and their generalizations). Any progress in this program outlined will not only contribute to the understanding of topological groups and algebras, but will also considerably stimulate research in these areas, and will cause noticeable "brain drain." From the point of view of the effect on the mathematical community, duality theories often serve as tunnels between two or more otherwise remote areas of mathematics. Thus, dualities have the potential of creating synergy between researchers of distant areas.