This document contains a proof that Bieberbach groups with finite abelianization are not connective (an E-theoretic property) and then uses this result to provide a characterization of connectivity in the case of Bieberbach groups.

Renormalization procedures for families of dynamical systems have been used to prove many interesting results. Examples of results include that the bifurcation rate for the attractors of an analytic one-parameter family of quadratic-like maps is universal for all such families, unique ergodicity for almost every interval exchange mapping, a unique ergodicity criterion for the vertical translation flow of a flat surface in terms of its ``renormalization dynamics", known as Masur's criterion, and the classification of circle diffeomorphisms up to $C^{\infty}$ conjugation. We introduce renormalization procedures for $C^{*}$-algebras and étale groupoids using the concepts of $C_{0}(X)$-algebras and Morita equivalence for the former, and groupoid bundles and groupoid equivalence, in the sense of Muhly, Renault and Williams, for the latter. We focus on proving analogs to Masur's criterion in both cases using $C^{*}$-algebraic methods. Applying our criterion to our examples of renormalization procedures provides a unique trace criterion for unital AF algebras extending the one provided by Treviño in the setting of flat surfaces and the one provided by Veech in the setting of interval exchange mappings. Also, we recover the old fact that rotation of the circle by an irrational angle is uniquely ergodic, and the new fact that interesting groupoids associated to certain iterated function systems, recently introduced by Korfanty, have unique invariant probability measures whenever they are minimal. Lastly, we show how an étale groupoid renormalization procedure arises from an étale groupoid which factors down onto a groupoid associated to its renormalization dynamics, whenever it is a local homeomorphism. / Graduate

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I define and analyze the Hensel-Steinitz algebra ����(��), a crossed product C∗-algebra associated with multiplication maps on continuous functions on the ring of ��-adic integers. In ����(��), I define an ideal and identify it with a known algebra. From this, I construct a short exact sequence conveying the structure of the algebras. I further identify smooth subalgebras within both ����(��) and its ideal, classify derivations on those algebras, and compare the classification with derivations on other smooth algebras. I also analyze the algebras associated with multiplication maps based on the multiplier being a root of unity, not a root of unity, or not invertible in the ��-adic integers. In the case of the multiplier being a root of unity and the quotient group therefore being finite, unexpected additional structure is found.

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This thesis has two distinguish parts. The first part concerns the study of n-ary algebras. A n-ary algebra is a vector space with a multiplication on n arguments. Classically the multiplications are binary, but the use of ternary multiplication in theoretical physic like for Nambu brackets led mathematicians to investigate these type of algebras. Two classes of n-ary algebras are fundamental: the associative n-ary algebras and the Lie n-ary algebras. We are interested by both classes. Concerning the associative n-ary algebras we are mostly interested in 3-ary partially associative 3-ary algebras, that is, algebras whose multiplication satisfies ((xyz)tu)+(x(yzt)u)+(xy(ztu))=0. This type is interesting because the previous woks on this subject was not distinguish the even and odd cases. We show in this thesis that the case n=3 can not be treated as the even cases. We investigate in detail the free partially associative 3-ary algebra on k generators. This algebra is graded and we compute the dimensions of the 7 first components. In the general case, we give a spanning set such as the sub family of non zero vector is a basis. The main consequences are the free partially associative 3-ary algebra is solvable. In the free commutative partially associative 3-ary algebra any product on 9 elements is trivial. The operad for partially associative 3-ary algebra do not satisfy the Koszul property. Then we study n-ary products on the tensors. The simplest example is given by a internal product of non square matrices. We can define a 3-ary product by taking A . ^tB . C. We show that we have to generalize a bit the definition of partial associativity for n-ary algebras. We then introduce the products -partially associative where  is a permutation of the symmetric group of degree n. Concerning the n-ary algebras, two classes have been defined: Filipov algebras (also called recently Lie-Nambu algebras) and some more general class, the n-Lie algebras. Filipov algebras are very important in the study of the mechanic of Nambu-Poisson, and is a particular case of the other. So to define an approach of Maurer-Cartan type, that is, define a scalar cohomology, we consider in this work Fillipov as n-Lie algebras and develop such a calculus in the n-Lie algebras frame work. We also give some classifications of n-ary nilpotent algebras. The last chapter of this part concerns my work in Master on the Poisson algebras on polynomials. We present link with the Lie algebras is clear. Thus we extend our study to Poisson algebras which associated Lie algebra is rigid and we apply these results to the enveloping algebras of rigid Lie algebras. The second part concerns intervals arithmetic. The interval arithmetic is used in a lot of problems concerning robotic, localization of parameters, and sensibility of inputs. The classical operations of intervals are based of the rule : the result of an operation of interval is the minimal interval containing all the result of this operation on the real elements of the concerned intervals. But these operations imply many problems because the product is not distributive with respect the addition. In particular it is very difficult to translate in the set of intervals an algebraic functions of a real variable. We propose here an original model based on an embedding of the set of intervals on an associative algebra. Working in this algebra, it is easy to see that the problem of non distributivity disappears, and the problem of transferring real function in the set of intervals becomes natural. As application, we study matrices of intervals and we solve the problem of reduction of intervals matrices (diagonalization, eigenvalues, and eigenvectors).

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