Harmonic Analysis and Non-commutative Geometry

“非对易几何与调和分析”项目正在研究利用非对易几何和算子K-理论技术的几个问题。基于半单李群的几何量子化思想，我们正在开发等变K-理论中的元素。这些结构统一并扩展了福克斯、哈斯克尔、朱格和卡斯帕罗夫发展的以前的结构。我们正在使用非对易几何来开发神经网络模型，我们希望它将比经典模型有几个优点。模型应该提供更高的计算效率，框架应该允许计算模型的大尺度结构的不变量。a.康纳斯和G.卡斯帕罗夫发展了新的数学领域，为研究本质上复杂的系统提供了工具。当将这些思想应用于几何和拓扑学中的数学问题时，已经取得了许多壮观的成功。这些技术正开始用于最初被认为超出理论范围的问题，这些新工具有望在广泛的应用中被证明是有用的。这个项目由两个使用非对易几何的程序组成。第一个项目是研究如何将对称性融入到这个新的几何学中。我们有新的技术，我们认为这些技术是自然的，将揭示几何对象的结构。第二个项目涉及使用非对易几何来开发大脑和大脑功能的模型，通常被称为神经网络。大脑固有的复杂结构一直是数学建模的主要绊脚石，我们的目标之一是应用一些非对易几何技术来获得神经功能的新模型。一个意想不到但令人高兴的发展是，数学系和我们校园里的神经科学界之间进行了认真的交流。在本科生、研究生和研究水平上的真正合作正在发展中，并有望丰富数学和生物。数学神经科学具有巨大的应用潜力(经典神经网络的应用已经证明了这一点)，这种潜力将随着计算能力的增加和我们对神经计算的理解的扩展而增加。

AbstractFoxThe project "Non-Commutative Geometry and Harmonic Analysis" is investigating several problems that use the techniques of non-commutative geometry and operator K-theory. We are developing elements in equivariant K-theory based on ideas from geometric quantization for semi-simple Lie groups. These constructions unify and extend previous constructions developed by Fox, Haskel, Julg, and Kasparov. We are using non-commutative geometry to develop models of neural networks, which we hope will have several advantages over the classical models. The models should provide for greater computational efficiency and the framework should allow for the computation of invariants of the large-scale structure of the model.A. Connes and G. Kasparov have developed new areas of mathematics that provide tools for studying systems that are inherently complex. There have been a number of spectacular successes of these ideas when applied to mathematical problems in geometry and topology. The techniques are starting to be used on problems that were originally considered outside the scope of the theory and there is promise that these new tools will prove useful in a huge range of applications. This project consists of two programs using non-commutative geometry. The first program is studying ways that symmetry can incorporated into this new geometry. We have new techniques that we believe are natural and will shed light on the structure of geometric objects. The second program involves the use of non-commutative geometry to develop models of the brain and brain functions, commonly known as neural networks. The inherently complex structure of the brain has been a major stumbling block for mathematical modeling and one of our goals is to apply some of the techniques of non-commutative geometry to obtain new models of neural functioning. An unexpected, but delightful, development has been serious communication between the mathematics department and the neuroscience community on our campus. A true collaboration at the undergraduate, graduate and research level is developing and promises to enrich both mathematics and biology. Mathematical neuroscience has a huge potential for applications (as already demonstrated by the applications of classical neural networks) which will increase as computational power increases and our understanding neural computation expands.