In November 2014 Alexander Grothendieck passed away at the age of 86. There is no doubt that he was one of the greatest and most innovative mathematicians of the 20th century. After a bitter childhood, his meteoric ascent started in the Cartan Seminar in Paris, it led to a breakthrough while he worked in São Paulo, and to the Fields Medal. He introduced numerous new concepts and techniques, whi...

In this article, we introduced and explore the properties of two sets of functions: weighted pseudo periodic functions of class r, and weighted Stepanov-like pseudo periodic functions of class r. We show the existence and uniqueness of weighted pseudo periodic solution of class r that are solutions to neutral functional differential equations. Other applications to partial differential equation...

H. Esat Kondakci,1,* Lane Martin,1 Robert Keil,2 Armando Perez-Leija,3 Alexander Szameit,4 Ayman F. Abouraddy,1 Demetrios N. Christodoulides,1 and Bahaa E. A. Saleh1 1CREOL, The College of Optics & Photonics, University of Central Florida, Orlando, Florida 32816, USA 2Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, Innsbruck 6020, Austria 3Max-Born-Institut, Max-Born...

The derived group of a permutation representation, introduced by R.H. Crowell, unites many notions of knot theory. We survey Crowell’s construction, and offer new applications. The twisted Alexander group of a knot is defined. Using it, we obtain twisted Alexander modules and polynomials. Also, we extend a well-known theorem of Neuwirth and Stallings giving necessary and sufficient conditions f...

For a 3-manifold M , McMullen derived from the Alexander polynomial of M a norm on H(M,R) called the Alexander norm. He showed that the Thurston norm is an upper bound for the Alexander norm. He asked if these two norms were the same when M fibers over the circle. Here, I give examples that show this is not the case. This question relates to the faithfulness of the Gassner representations of th...