Buildings: An Exposition Detailing Construction and Theorems

In the mid to late twentieth century, Jacques Tits’ work in the area of Lie groups and Lie algebras caused him to develop the construct of a building. Since then the topic has expanded to be viewed from a range of different perspectives and has proven useful in a range of other fields of mathematical research. The topic of buildings brings together several areas of mathematics, including combinatorics, incidence geometry, and Coxeter groups, to name but a few. Buildings are used by many as a vehicle to understanding properties of some of the more complex and unworkable groups that one may wish to understand. This is done through having a building upon which a group can act. In this thesis we will see that the building itself can be considered the fundamental object, and motivate ideas that by taking buildings in different geometries we are able to find new examples of groups. There are a variety of ways from which one may approach the study of buildings, each with its own benefits and shortcomings. This thesis provides an introduction to the topic of buildings showing a geometric based construction with recurring examples

Measure transfer and SS-adic developments for subshifts

Based on previous work of the authors, to any $S$-adic development of a subshift $X$ a "directive sequence" of commutative diagrams is associated, which consists at every level $n \geq 0$ of the measure cone and the letter frequency cone of the level subshift $X_n$ associated canonically to the given $S$-adic development. The issuing rich picture enables one to deduce results about $X$ with unexpected directness. For instance, we exhibit a large class of minimal subshifts with entropy zero that all have infinitely many ergodic probability measures. As a side result we also exhibit, for any integer $d \geq 2$, an $S$-adic development of a minimal, aperiodic, uniquely ergodic subshift $X$, where all level alphabets ${\cal A}_n$ have cardinality $d\,$, while none of the $d-2$ bottom level morphisms is recognizable in its level subshift $X_n \subset {\cal A}_n^\mathbb Z$

SS-adic expansions related to continued fractions (Natural extension of arithmetic algorithms and S-adic system)

"Natural extension of arithmetic algorithms and S-adic system". July 20～24, 2015. edited by Shigeki Akiyama. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.We consider S-adic expansions associated with continued fraction algorithms, where an S-adic expansion corresponds to an infinite composition of substitutions. Recall that a substitution is a morphism of the free monoid. We focus in particular on the substitutions associated with regular continued fractions (Sturmian substitutions), and with Arnoux-Rauzy, Brun, and Jacobi{Perron (multidimensional) continued fraction algorithms. We also discuss the spectral properties of the associated symbolic dynamical systems under a Pisot type assumption

Analysis of hyperbolic metamaterials using the differential method

In this work hyperbolic metamaterials will be analyzed using the differential method. The first part is dedicated to the theoretical development of the differential method. This method is used to describe the electric and magnetic fields diffracted by a structure of any geometry or material. This particular work will be limited to structures which geometry is invariant along one spatial direction. Then, the development of a code in C++ language based on the differential method is explained. The code is used to describe the diffracted fields in terms of the scattering matrix of the structure. Finally, this code is utilized to analyzed hyperbolic metamaterials...En el presente trabajo se analizar´a metamateriales hiperb´olicos empleando el m´etodo diferencial. En primer lugar, se desarrolla la teor´ıa detr´as del m´etodo diferencial. Este m´etodo sirve para describir los campos el´ectricos y magn´eticos difractados por una estructura de cualquier geometr´ıa y material. Para este trabajo se consideran ´unicamente estructuras invariantes en una direcci´on espacial. Despu´es, se explica el desarrollo de un c´odigo en lenguaje C++ basado en el m´etodo diferencial para describir los campos difractados a trav´es de la matriz de scattering de la estructura considerada. Finalmente, se emplea el c´odigo para analizar metamateriales hiperb´olicos..

Combinatorics of Pisot Substitutions

Siirretty Doriast

Real Analysis, Harmonic Analysis, and Applications

The workshop focused on important developments within the last few years in real and harmonic analysis, including polynomial partitioning and decoupling as well as significant concurrent progress in the application of these for example to number theory and partial differential equations

Homotopic and Geometric Galois Theory (online meeting)

In his "Letter to Faltings'', Grothendieck lays the foundation of what will become part of his multi-faceted legacy to arithmetic geometry. This includes the following three branches discussed in the workshop: the arithmetic of Galois covers, the theory of motives and the theory of anabelian Galois representations. Their geometrical paradigms endow similar but complementary arithmetic insights for the study of the absolute Galois group $\mathrm{G}_{\mathbb{Q}}$ of the field of rational numbers that initially crystallized into a functorially group-theoretic unifying approach. Recent years have seen some new enrichments based on modern geometrical constructions - e.g. simplicial homotopy, Tannaka perversity, automorphic forms - that endow some higher considerations and outline new geometric principles. This workshop brought together an international panel of young and senior experts of arithmetic geometry who sketched the future desire paths of homotopic and geometric Galois theory