The Cuntz Semigroup and the Classification of C*-algebras

该奖项提供支持，以支付美国参与者在2011年1月至7月期间举办的"关于C *-代数分类的Cuntz半群"项目期间访问巴塞罗那数学研究中心（CRM）的费用。绝大多数奖励资金将为年轻的美国数学家提供机会，包括研究生和博士后研究员，参加CRM计划的重要部分。特别是，该奖项将支持这些年轻的数学家参加题为"动力系统，C *-代数和分类"的高级课程，该课程将于2011年6月14日至23日举行。该奖项由国际科学与工程办公室共同资助。最近发现Cuntz半群--大致上是C *-代数上可数生成Banach模的同构类的半群--包含了大量的信息，从而进一步证实了通过K-理论不变量对C *-代数进行分类的方法。这个半群的显式计算导致了极小动力系统的C *-代数的分类定理。分类，C *-代数和动力学之间的联系已经进一步加强了证明康托集上顺从群作用的超有限性的进展。另一个新的前沿在描述集合论和算子代数的界面上打开了，其中各种等价关系（自然包括同构）的Borel复杂性允许人们量化C *-代数理论中各种分类问题的复杂性。因此，C~*-代数理论中的Cuntz半群和分类问题是泛函分析中一个活跃而迅速发展的分支，它将提供许多研究机会。该奖项提供了一个很好的机会，年轻的数学家在C *-代数领域的研究兴趣，接触到该领域的最新发展，与来自其他国家的研究人员进行互动，并可能启动未来的合作。除其他外，该奖项将有助于该领域未来的活力，并希望导致进一步的应用和与其他数学领域的联系。

This award provides support to defray expenses of US participants to visit the Centre de Recerca Matematica (CRM) in Barcelona during the program "The Cuntz Semigroup on the Classification of C*-algebras" which is being held in the period January through July 2011. The vast majority of the award funds will provide the opportunity for young US mathematicians, including graduate students and postdoctoral fellows, to attend significant portions of the CRM program. In particular, the award will support the attendance of these young mathematicians at an advanced course entitled "Dynamical Systems, C*-algebras and Classification," which will be held during June 14-23, 2011. The award is co-funded by the Office of International Science and Engineering.The classification of C*-algebras via K-theoretic invariants has recently been invigorated by the discovery that the Cuntz semigroup - roughly, a semigroup of isomorphism classes of countably generated Banach modules over a C*-algebra - contains an enormous amount of information. Explicit calculation of this semigroup has led to classification theorems for C*-algebras of minimal dynamical systems, among others. The connections between classification, C*-algebras, and dynamics have been further strengthened by progress on proving the hyperfiniteness of amenable group actions on the Cantor set. Another new front has opened at the interface of descriptive set theory and operator algebras, where the Borel complexity of various equivalence relations (including, naturally, isomorphism) allows one to quantify the complexity of various classification problems in C*-algebra theory. Thus, the Cuntz semigroup and classification problems in C*-algebra theory form an active and rapidly developing branch of functional analysis, one which will afford many research opportunities. The award provides a very good opportunity for young mathematicians with research interests in the area of C*-algebras to be exposed to the latest developments in the field, to interact with researchers from other countries, and to possibly initiate future collaborations. Among other things, this award will contribute to the future vitality of the field, and hopefully lead to further applications and connections with other areas of mathematics.