Mathematical Sciences: The K-Theory and Isomorphism Invariants of Toeplitz Algebras

群作用于空间的情况，如实数群随时间演化作用于物理系统的状态空间的情况，在数学及其应用中经常遇到。在物理学的例子中，系统的每一种可能的形态都位于一条从遥远的过去流向遥远的未来的线上。人们感兴趣的是总体上发生的事情，从长远来看，这可能很难仅仅从短时间间隔内发生的事情来推断。为了分析的目的，可用的信息可以用几种不同的方式进行编码：通过微分方程、向量场或叶形，或者像夏教授的研究项目那样，通过希尔伯特空间上的算子代数。夏教授将研究一种相当普遍的Toeplitz代数构造，这是最近在（非交换）动力系统中发展起来的；这与我们更熟悉的交叉积的概念有关。他的工作的具体目标是计算Toeplitz代数的k群及其相关的对易子理想，并看看Toeplitz代数是否是潜在动力系统的完全同构不变量。

The situation of a group acting on a space, for instance the group of real numbers acting by time evolution on the state space of a physical system, is encountered quite often in mathematics and its applications. In the physics example, every possible configuration of the system lies on a line flowing from remote past to distant future. One is interested in what happens overall, in the long run, which may be quite difficult to infer solely from what happens over short intervals of time. For purposes of analysis, the available information can be encoded in several different ways: by differential equations or vector fields or foliations or, as is the case in Professor Xia's research project, by an algebra of operators on Hilbert space. Professor Xia will study a rather general Toeplitz algebra construction that has recently been developed for (non- commutative) dynamical systems; this is related to the more familiar notion of a crossed product. The specific objectives of his work are to calculate the K-groups of Toeplitz algebras and their associated commutator ideals, and to see whether the Toeplitz algebra is a complete isomorphism invariant of the underlying dynamical system.