Existence Of Complex Structures Research Articles

Existence of complex structures on decomposable Lie algebras

We provide the classification of the six-dimensional decomposable Lie algebras, with the dimension of the biggest indecomposable summand less than five, admitting complex structures.

Almost complex structures on spheres

In this paper we review the well-known fact that the only spheres admitting an almost complex structure are S2 and S6. The proof described here uses characteristic classes and the Bott periodicity theorem in topological K-theory. This paper originates from the talk “Almost Complex Structures on Spheres” given by the second author at the MAM1 workshop “(Non)-existence of complex structures on S6”, held in Marburg from March 27th to March 30th, 2017. It is a review paper, and as such no result is intended to be original. We tried to produce a clear, motivated and as much as possible self-contained exposition.

On the history of the Hopf problem

This short note serves as a historical introduction to the Hopf problem: This unsolved mathematical question was the subject of the Conference “MAM 1 — (Non-)Existence of Complex Structures on S6”, which took place at Philipps-Universität Marburg, Germany, between March 27th and March 30th, 2017. We also include a list of the participants to the conference and a group picture.

Notes on G2: The Lie algebra and the Lie group

These notes have been prepared for the Workshop on “(Non)-existence of complex structures on S6”, celebrated in Marburg in March, 2017. The material is not intended to be original. It contains a survey about the smallest of the exceptional Lie groups: G2, its definition and different characterizations as well as its relationship to the spheres S6 and S7. With the exception of the summary of the Killing–Cartan classification, this survey is self-contained, and all the proofs are provided. Although these proofs are well-known, they are scattered, some of them are difficult to find, and others require stronger background, while we will usually stick to linear algebra arguments. The approach is algebraical, working at the Lie algebra level most often. We analyze the complex Lie algebra (and group) of type G2 as well as the two real Lie algebras of type G2, the split and the compact one. The octonion algebra will play its role, but it is not the starting point. Also, both the 3-forms approach and the spinorial approach are viewed and connected. Special emphasis is put on relating all the viewpoints by providing precise models.

Hodge numbers of a hypothetical complex structure on S6

These are the notes for the talk “Hodge numbers of a hypothetical complex structure on S6” given by the author at the MAM1 “(Non)-existence of complex structures on S6” held in Marburg in March 2017. They are based on [12] and [27], where Hodge numbers and the dimensions of the successive pages of the Frölicher spectral sequence for S6 endowed with a hypothetical complex structure are investigated. We also add results from [20], where the Bott–Chern cohomology of hypothetical complex structures on S6 is studied. The material is not intended to be original.

S6 and the geometry of nearly Kähler 6-manifolds

We review results on and around the almost complex structure on S6, both from a classical and a modern point of view. These notes have been prepared for the Workshop “(Non)-existence of complex structures on S6” (Erste Marburger Arbeitsgemeinschaft Mathematik – MAM-1), held in Marburg in March 2017.

An efficient 4 way coupling CFD–DEM model for dense gas–solid particulate flows simulations

The goal of this paper is to illustrate how some recent numerical developments enable the simulation of dense two-phase flows. A coupled Computational Fluid Dynamic and Discrete Element Method (CFD–DEM) approach has been designed for the simulation of 3D fluidized beds. DEM is employed to model the granular particle phase while classical CFD is used to simulate the fluid flow by solving the Volume Averaged Navier-Stokes (VANS) equations. A special emphasize is put on the implementation of the velocity–pressure algorithm in the context of two-phase flows. By taking into consideration all particle–particle, fluid–particle and particle–wall interactions, a complete 4 way coupling model has been derived. Three dimensional computations of a single bubble formation as well as of the bubbly regime have been conducted. The simulations point out the existence of complex structures, such as the worm-like shape one, similar to those that have already been described in the literature.

ORTHOGONAL ALMOST COMPLEX STRUCTURES ON THE RIEMANNIAN PRODUCTS OF EVEN-DIMENSIONAL ROUND SPHERES

We discuss the integrability of orthogonal almost complex structures on Riemannian products of even-dimensional round spheres and give a partial answer to the question raised by E. Calabi concerning the existence of complex structures on a product manifold of a round 2-sphere and of a round 4-sphere.

The crystal structures of manganese and the actinide elements

The crystal structures of manganese and the actinide elements are discussed from the point of view of a recent theory of the electronic structure of body-centred cubic metals. It is shown that the existence of complex structures may be related to the possibility of Jahn-Teller distortions occurring in the body-centred cubic lattice.