The talk to be presented is about the domain of mathematical computation which extends beyond modern calculus and classical analysis when numbers are not restricted to belong to a commutative field. It describes the dynamics of complexification, resulting in an endless remorphing of the computational landscape. Nonlinear computation weaves a colourful tapestry always in a state of becoming. In ...

We produce a one-parameter family of hyperplane arrangements that are counterexamples to the conjecture of Saito that the complexified complement of a free arrangement is K(n, 1 ). These arrangements are the restriction of a one-parameter family of arrangements that arose in the study of tilings of certain centrally symmetric octagons. This other family is discussed as well. I. Definitions and ...

Let M = G/K be a homogeneous Riemannian manifold with dimCGC = dimRG, where GC denotes the universal complexification of G. Under certain extensibility assumptions on the geodesic flow of M , we give a characterization of the maximal domain of definition in TM for the adapted complex structure and show that it is unique. For instance, this can be done for generalized Heisenberg groups and natur...

Neretin and Segal independently defined a semigroup of annuli with boundary parametrizations, which is viewed as a complexification of the group of diffeomorphisms of the circle. By extending the parametrizations to quasisymmetries, we show that this semigroup is a quotient of the Teichmüller space of doubly-connected Riemann surfaces by a Z action. Furthermore, the semigroup can be given a com...

Dirichlet–Neumann operators arise in many applications in the sciences, and this has inspired a number of studies on their analytical properties. In this paper we further investigate the analyticity properties of Dirichlet–Neumann operators as functions of the boundary shape. In particular, we study the size of the disc of convergence of their Taylor-series representation. For this we use a com...

The orthogonal and unitary calculi give a method to study functors from the category of real or complex inner product spaces based topological spaces. We construct between complexification-realification adjunction These allow for movement versions calculi, comparisons Taylor towers produced by both calculi. show that when inputted functor is weakly polynomial, tower restricted through realifica...

Abstract We study a system of equations on compact complex manifold, that couples the scalar curvature Kähler metric with spectral function first-order deformation structure. The comes from an infinite-dimensional reduction, which is hyperkähler reduction for particular choice function. main tool studying flat connection space deformations structure, allows to obtain formal complexification mom...

This paper investigates the role of ecological interactions in the evolution of complexity through the results of simulations. Webworld is a robust model of evolution in food webs. It is extended for variability of organism complexity under evolution. Statistical and network analysis indicates a clear tendency for complexification within the model, led by adaptations that initially disconnect s...

We consider the Segal–Bargmann transform for a noncompact symmetric space of the complex type. We establish isometry and surjectivity theorems for the transform, in a form as parallel as possible to the results in the compact case. The isometry theorem involves integration over a tube of radius R in the complexification, followed by analytic continuation with respect to R. A cancellation of sin...