Mathematical Sciences: K-Theory and Cyclic Cohomology Related to Operator Algebras

夏目教授的研究方向是非交换几何中的k理论和循环上同。他将研究几何学中产生的算子代数。他提出的问题的解决方案将阐明这两个领域的相互作用。他将考虑的一些具体问题是：(1)确定了闭Riemann曲面上基本群的扭曲群C*-代数的微分同态群的结构，(2)确定了Baum-Connes猜想对于几乎不完整的余维叶是否成立，(3)将解析k理论中的Godbillon-Vey映射的消失定理推广到更高余维的情况。（iv）建立单连通可解李群正则循环环的傅里叶反演公式。这个项目的一般领域是算子代数和几何。运算符可以被认为是无限的复数矩阵。特殊类型的运算符通常放在一个代数中，自然地称为运算符代数。在其他问题中，夏目教授将研究在分析封闭曲面时出现的算子代数。

Professor Natsume's project is in the area of K-theory and cyclic cohomology in noncommutative geometry. He will study operator algebras arising from geometry. Solutions to the problems he poses will illuminate the area of interaction of the two fields. Some of the specific problems he will consider are: (i) determine the structure of the group of diffeomorphisms of a twisted group C*-algebra of the fundamental group of a closed Riemann surface, (ii) determine whether the Baum-Connes conjecture is valid for codimension one foliations which are almost without holonomy, (iii) generalize the vanishing theorem of the Godbillon-Vey map in analytical K-theory to higher codimension cases, and (iv) establish a Fourier inversion formula for the canonical cyclic cocycle of a simply connected solvable Lie group. The general area of this project is operator algebras and geometry. Operators can be thought of as infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. Among other problems, Professor Natsume will study operator algebras which arise in the analysis of closed surfaces.