Harmonic Analysis and Non-commutative Geometry

非对易几何与调和分析研究课题是利用非对易几何与算子K理论的技巧来研究几个问题。我们正在开发的元素，在等变K理论的基础上的想法，从几何量子化的半单李群。这些构造统一并扩展了Fox、Haskel、Julg和Kasparov以前提出的构造。 我们正在使用非交换几何来开发神经网络模型，我们希望它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规模结构的不变量。Connes和G.卡斯帕罗夫开发了新的数学领域，为研究本质上复杂的系统提供了工具。当这些思想应用于几何和拓扑学的数学问题时，已经取得了一些惊人的成功。这些技术开始被用于最初被认为超出理论范围的问题，并且有希望证明这些新工具在广泛的应用中是有用的。这个项目包括两个程序使用非交换几何。第一个项目是研究如何将对称性融入新的几何学中。我们有新的技术，我们相信是自然的，将揭示几何物体的结构。第二个程序涉及使用非交换几何来开发大脑和大脑功能的模型，通常称为神经网络。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非交换几何的一些技术来获得神经功能的新模型。 一个意想不到但令人愉快的发展是我们校园数学系和神经科学界之间的认真沟通。在本科生，研究生和研究水平的真正合作正在发展，并承诺丰富数学和生物学。数学神经科学具有巨大的应用潜力（正如经典神经网络的应用所证明的那样），随着计算能力的增加和我们对神经计算的理解的扩展，这种潜力将会增加。

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.