Colloquium

We explore three directions extending the traditional theory of complex quadratic iterations to: (1) complex quadratic networks; (2) template iterations and (3) mutated systems. In all three cases, we define Julia and Mandelbrot sets, and describe how their properties are different than those of the traditional sets for single map iterations. In the case of networks, the dynamics of each node are driven by the same complex quadratic map f(z) = z2 +c, with coupling specified by the adjacency matrix Aij and weights gij , so that the system takes the form of an iteration in Cn. We discuss how the topology of the network Mandelbrot changes under perturbation of the network architecture, and how this can be used to create and analyze a fractal profile for a person’s brain network. In the case of templates, we study non-autonomous iterations of a pair of complex quadratic functions, fc0 and fc1 , applied according to a general binary symbolic sequence s. Here, the Mandelbrot set is a subset of the product space C2 ×{0, 1}∞. We discuss its slices in either subspace, present a Hausdorff convergence theorem and illustrate how the framework can be used in genetics, to understand the how cells differentiate and specialize. In the case of mutations, we insert an “erroneous” function fc0 to replace the intact replication function fc1 within a disk around a mutation epicenter. We analyze the perturbations to the Julia set induced by mutations with different parameter c0, different radius and different location. We discuss how this can be used to understand the role of mutations in cancer modeling research.

Zoom:  https://arizona.zoom.us/j/83738249833  Password:  applied (all lower case)