Nodal domains, spectral minimal partitions, and their relation to Aharonov-Bohm operators

This survey is a short version of a chapter written by the first two authors in the book [A. Henrot, editor. Shape optimization and spectral theory. Berlin: De Gruyter, 2017] (where more details and references are given) but we have decided here to put more emphasis on the role of the Aharonov-Bohm operators which appear to be a useful tool coming from physics for understanding a problem motivated either by spectral geometry or dynamics of population. Similar questions appear also in Bose-Einstein theory. Finally some open problems which might be of interest are mentioned.Comment: arXiv admin note: substantial text overlap with arXiv:1506.0724

The Dynamics of Twisted Tent Maps

This paper is a study of the dynamics of a new family of maps from the complex plane to itself, which we call twisted tent maps. A twisted tent map is a complex generalization of a real tent map. The action of this map can be visualized as the complex scaling of the plane followed by folding the plane once. Most of the time, scaling by a complex number will "twist" the plane, hence the name. The "folding" both breaks analyticity (and even smoothness) and leads to interesting dynamics ranging from easily understood and highly geometric behavior to chaotic behavior and fractals.Comment: 87 pages. This is my Ph.D. thesis from IUPU