Approaching some open problems in algebraic combinatorics and in C*-algebras theory using von Neumann algebras.

研究计划是在算子代数领域，这可以被视为C*-代数和冯诺依曼代数的研究。这两种代数都是在20年代末到30年代初引入的。 该建议的一部分是致力于调查的一些开放的问题，C*-代数的非同构到他们的对立代数。所提出的研究的基本哲学是，为了构造具有某些期望性质的C*-代数的示例，可以在von Neumann代数设置而不是C*-代数设置中工作。我打算研究的具体问题涉及在C*-代数理论中起重要作用的概念，即核性和纯无限C*-代数。有了这个建议，我打算提供一些见解埃利奥特的分类程序的C*-代数，通过生产的例子，C*-代数这是不知道的。此外，我计划研究一些与自由量子正交群和量子置换群有关的公开问题。自由量子群是一个新的研究领域，至今仍是一个相当神秘的对象。由于自由量子群及其冯诺依曼代数在数学物理中有重要的应用，本文的目的是更好地理解与自由量子群相关的冯诺依曼代数的性质和结构。最后，我计划用子因子理论证明代数组合学中的一个猜想。近年来，人们对拟对称函数越来越感兴趣。拟对称函数是多项式环上在对称群的某种作用下不变量的函数。这个作用诱导了Temperley-Lieb代数在多项式环上的忠实作用.该猜想证明了Temperley-Lieb代数的作用到Fuss-Catalan代数的忠实作用存在唯一的扩张。解决这一猜想将是起点的研究功能是不变的作用下的福斯-加泰罗尼亚代数。希望这项研究将作为有趣的和丰富的结果，研究拟对称函数已经。

The research proposal is in the area of Operator Algebras, which can be viewed as the study of C*-algebras and von Neumann algebras. Both kinds of algebras were introduced between the late '20s and the early '30s. The proposal is partly devoted to the investigation of some open questions on C*-algebras non-isomorphic to their opposite algebra. The underlying philosophy of the proposed research is that to construct an example of a C*-algebra with some desired property, one can work in a von Neumann algebra setting instead of a C*-algebra setting. The specific problems that I intend to investigate deal with concepts that play a major role in the theory of C*-algebras, namely nuclearity and purely infinite C*-algebras. With this proposal I intend to provide some insights into Elliott's classification program for C*-algebras, by producing examples of C*-algebras which were not known before. In addition, I plan to investigate some open problems related to free quantum orthogonal groups and quantum permutation groups. Free quantum group are a recent area of research, and are still quite mysterious objects. The objective of this work is to gain a better understanding of the properties and structure of the von Neumann algebras associated to free quantume groups, since free quantum groups and their von Neumann algebras have important applications to mathematical physics. Lastly, I plan to prove a conjecture in algebraic combinatorics using subfactor theory. There has been an increased interest in recent years in quasi-symmetric functions. Quasi-symmetric functions are functions which are invariants under a certain action of the symmetric group on the ring of polynomials. This action induces a faithful action of the Temperley- Lieb algebra on the ring of polynomials. The conjecture claims that there is a unique extensions of the action of the Temperley-Lieb algebra to a faithful action of the Fuss-Catalan algebra. Solving this conjecture will be the starting point of the study of functions which are invariant under the action of the Fuss-Catalan algebra. Hopefully this study will be as interesting and as rich in results as the study of quasi-symmetric functions has been.