As a systems scientist, biology looks all full of mysteries that are not understandable. A cell, the basic unit of life, consists of numerous molecules that highly interact with each other. Such interaction between molecules often results in paradoxical observations in many biological experiments. I was intrigued whether there exists any evolutionary design principle behind the puzzling dynamics of living systems. To unravel such a hidden design principle underlying complex phenomena, we need a systems biological approach by combining mathematical simulation and biochemical experimentation. In this talk, I will present the state space analysis of a molecular interaction network that is critical for cell fate determination and further discuss how to control such a network to change the cell fate as we want. The proposed state space analysis demonstrates that implementation of an attractor landscape to analyze a biological network is useful for gaining a better understanding of the complex network dynamics and the resulting cell fate determination.

Amenability is one of those properties of group that has many different characterizations. I will discuss what it means in terms of invariant means, random walks and C* algebras. If time permits, I will also describe some related notions such as property rapid decay in the C* algebra setting.

A symmetric matrix with complex entries may be diagonalized, so the corresponding quadratic form may be written as a sum of squares. There is a large variety of distinct sum of squares decompositions of the quadratic form. I shall present a compactification of this variety, and discuss and present old and new results on powersum decompositions for forms of higher degree.

We will discuss connections between three notions in 3-dimensional topology that are, roughly speaking, algebraic, topological, and analytic. These are: the left-orderability of the fundamental group of a 3-manifold M, the existence of certain codimension 1 foliations on M, and the Heegaard Floer homology of M.