TBA

A moment-angle complex is a space determined by a finite simplicial complex and is allowed a coordinatewise torus action. This space and its quotient provide connections among combinatorial, algebra and topology. In this talk, we look at the topological side. That is, we will determine the homotopy type of the quotient space given by the moment-angle complex corresponding to the k-skeleton of a simplex under the diagonal action.

We construct a hill such that any two points are connected by a path of zero slope (except at the end points). Such a hill cannot be given by a continuously differentiable function but can be given by a once differentiable function.

Originated from the theory of connections on vector bundles over the spin manifolds, inner fluctuations of Dirac operator of spectral triples provide a mechanism through which the gauge fields can be implemented in the noncommutative geometry. In this talk, we shall go over the motivations and constructions that show why inner fluctuations can be interpreted as the gauge fields in the theory of spectral triples and how they generalize the gauge theory for the noncommutative geometries.