We construct a strong filtered graph $\Gamma_s(\Lambda)$ dependent on the dominant weight $\Lambda$, and a weak filtered graph $\Gamma_w(\Kcen)$ dependent on the canonical central element $\Kcen$ for an arbitrary Kac-Moody algebra $g$. In our construction, both graphs $(\Gamma_s(\Lambda), \Gamma_w(\Kcen))$ have the vertex set as the Weyl group of $g$, with the grading given by the length function. The edges of the graph $\Gamma_s(\La)$ are labeled versions of the $\lambda$-chain model of K-Chevalley rules for Kac-Moody flag manifolds as developed by Lenart and Shimozono, originally defined by Lenart and Postnikov. Meanwhile, the labels on $\Gamma_w(\Kcen)$ come from the dual multiplication map of K-cohomology of affine Grassmannian $Gr_G$. We conjecture that the strong filtered graph and weak filtered graph are dual, which means we get an identity when we apply the up and down operators on the vertices. We proved this identity except one case that where we call the chain is $j$-present. Our identity is similar to the Möbius construction of the dual filtered graph, as previously studied by Patrias and Pylyavskyy, and in fact, in the limit $n\rightarrow \infty$ of the $A^{(1)}_{n-1}$, our construction recovers their identity. We also expect to recover their combinatorics of Möbius deformation of the shifted Young's lattice in type $C^{(1)}_n$ as $n$ approaches infinity. / Doctor of Philosophy / In this thesis, we introduce a pair of graphs $(\Gamma_s(\La),\Gamma_w(\Kcen))$ motivated by the study of affine Schubert calculus. Affine Schubert calculus emerges as an extension and generalization of classical Schubert calculus, which involves questions such as determining the number of lines intersecting four lines in three-dimensional space. This type of questions can often be translated into computations aimed at finding the structure constants for the Schubert basis in the K-(co)homology of the flag varieties such as affine Grassmannian. These structure constants represent the coefficients of the Schubert basis in the product of the other two Schubert bases, all indexed by the Weyl group of the affine Lie algebra $g$. We define up and down operators on the vertices of graphs $(\Gamma_s(\Lambda), \Gamma_w(\Kcen))$, which are elements in Weyl group of $g$, utilizing the structure constants as essential components. We conjecture that, in general, and prove in certain cases, this approach yields new identities for these operators, leading us to define this pair of graphs as a dual filtered graph.

In this thesis the close relationship between the topological $K$-homology group of the spacetime manifold $X$ of string theory and D-branes in string theory is examined. An element of the $K$-homology group is given by an equivalence class of $K$-cycles $[M,E,\phi]$, where $M$ is a closed spin$^c$ manifold, $E$ is a complex vector bundle over $M$, and $\phi: M\rightarrow X$ is a continuous map. It is proposed that a $K$-cycle $[M,E,\phi]$ represents a D-brane configuration wrapping the subspace $\phi(M)$. As a consequence, the $K$-homology element defined by $[M,E,\phi]$ represents a class of D-brane configurations that have the same physical charge. Furthermore, the $K$-cycle representation of D-branes resembles the modern way of characterizing fundamental strings, in which the strings are represented as two-dimensional surfaces with maps into the spacetime manifold. This classification of D-branes also suggests the possibility of physically interpreting D-branes wrapping singular subspaces of spacetime, enlarging the known types of singularities that string theory can cope with. / Master of Science

Dans cette thèse, on donne une définition de la K-théorie topologique des espaces non-commutatifs de Kontsevich (c'est-à-dire des dg-catégories) définis sur les nombres complexes. L'introduction de ce nouvel invariant initie la recherche des invariants de nature topologique des espaces non-commutatifs, comme "simplifications" des invariants algébriques (K-théorie algébrique, homologie cyclique, périodique comme étudiés dans les travaux de Tsygan, Keller). La motivation principale vient de la théorie de Hodge non-commutative au sens de Katzarkov--Kontsevich--Pantev. En géométrie algébrique, la partie rationnelle de la structure de Hodge est donnée par la cohomologie de Betti rationnelle, qui est la cohomologie rationnelle de l'espace des points complexes du schéma. La recherche d'un espace associé à une dg-catégorie trouve une première réponse avec le champ (défini par Toën--Vaquié) classifiant les dg-modules parfaits sur cette dg-catégorie. La définition de la K-théorie topologique a pour ingrédient essentiel le foncteur de réalisation topologique des préfaisceaux en spectres sur le site des schémas de type fini sur les complexes. La partie connective de la K-théorie semi-topologique peut être définie comme la réalisation topologique du champ en monoïdes commutatifs des dg-modules parfaits. Cependant pour atteindre la K-théorie négative, on réalise le préfaisceau donné par la K-théorie algébrique non-connective. Un de nos résultats principaux énonce l'existence d'une équivalence naturelle entre ces deux définitions dans le cas connectif. On montre que la réalisation topologique du préfaisceau de K-théorie algébrique connective pour la dg-catégorie unité donne le spectre de K-théorie topologique usuel. Puis que c'est aussi vrai pour la K-théorie algébrique non-connective, en utilisant la propriété de restriction aux lisses de la réalisation topologique. En outre, cette propriété de restriction aux schémas lisses nécessite de montrer une généralisation de la descente propre cohomologique de Deligne, dans le cadre homotopique non-abélien.La K-théorie topologique est alors définie en localisant par rapport à l'élément de Bott. Cette définition repose donc sur des résultats non-triviaux. On montre alors que le caractère de Chern de la K-théorie algébrique vers l'homologie périodique se factorise par la K-théorie topologique, donnant un candidat naturel pour la partie rationnelle d'une structure de Hodge non-commutative sur l'homologie périodique, ceci étant énoncé sous la forme de la conjecture du réseau. Notre premier résultat de comparaison concerne le cas d'un schéma lisse de type fini sur les complexes -- la conjecture du réseau est alors vraie pour de tels schémas. On montre ensuite que cette conjecture est vraie dans le cas des algèbres associatives de dimension finie. / In this thesis, we give a definition of a topological K-theory of Kontsevich's non-commutative spaces (i.e. of dg-categories) defined over complex numbers. The introduction of this invariant initiates the quest for topological invariants of non-commutative spaces, which are considered as "simplifications" of algebraic ones like algebraic K-theory, cyclic homology, periodic homology as studied by Tsygan, Keller. The main motivation comes from non-commutative Hodge theory in the sense of Katzarkov--Kontsevich--Pantev. In algebraic geometry, the rational part of the Hodge structure is given by rational Betti cohomology, which is the rational cohomology of the underlying space of complex points. The existence of a space associated to a dg-category admits a first answer given by the stack (defined by Toën--Vaquié) classifying perfect dg-modules over this dg-category. The essential ingredient in the definition of the topological K-theory is the topological realization functor of spectral presheaves on the site of complex schemes of finite type. The connective part of the semi-topological K-theory can then be definied as the topological realization of the stack of perfect dg-modules over the space, together with its commutative monoid structure up to homotopy. But to deal with negative K-groups, we realize the presehaf given by non-connective algebraic K-theory. One of our main results relies the two previous definition in the connective case. We show that the topological realization of the presheaf of connective algebraic K-theory for the unit dg-category is equivalent to the usual topological K-theory spectrum. We show this is also true in the non-connective case, using a property of restriction to smooth schemes. This last property leads us to show a generalization of Deligne's proper cohomological descent to the homotopical non-abelian setting. This enables us to define topological K-theory by inverting the Bott element. We point out that the process of the definition involves non-trivial results. We then show that the Chern character from algebraic K-theory to periodic homology factorizes through topological K-theory, giving a natural candidate for the rational part of a non-commutative Hodge structure on the periodic homology of a smooth and proper dg-category. This last claim is written in the form of a conjecture : the lattice conjecture. Our first comparison result deals with the case of a smooth scheme of finite type over complex numbers -- we show the lattice conjecture holds for dg-categories of perfect complexes. We also show this conjecture is true in the case of finite dimensional associative algebras.

We study the K-theory and cohomology of spaces of aperiodic and repetitive tilings with finite local complexity. Given such a tiling, we build a spectral sequence converging to its K-theory and define a new cohomology (PV cohomology) that appears naturally in the second page of this spectral sequence. This spectral sequence can be seen as a generalization of the Leray-Serre spectral sequence and the PV cohomology generalizes the cohomology of the base space of a Serre fibration with local coefficients in the K-theory of its fiber. We prove that the PV cohomology of such a tiling is isomorphic to the Cech cohomology of its hull. We give examples of explicit calculations of PV cohomology for a class of 1-dimensional tilings (obtained by cut-and-projection of a 2-dimensional lattice). We also study the groupoid of the transversal of the hull of such tilings and show that they can be recovered: 1) from inverse limit of simpler groupoids (which are quotients of free categories generated by finite graphs), and 2) from an inverse semi group that arises from PV cohomology. The underslying Delone set of punctures of such tilings modelizes the atomics positions in an aperiodic solid at zero temperature. We also present a study of (classical and harmonic) vibrational waves of low energy on such solids (acoustic phonons). We establish that the energy functional (the "matrix of spring constants" which describes the vibrations of the atoms around their equilibrium positions) behaves like a Laplacian at low energy.

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Le propos de cette étude est de modéliser une méthodologie de “design augmenté” appliqué à la conception de produit innovant dans un environnement de PME. Cette approche inclut la Théorie C-K dans ce contexte d'innovation de rupture.En général, le processus de design industriel consiste à parcourir quatre pôles complémentaires :1/ la phase d'ego-design, où le designer conceptualise un besoin utilisateur,2/ la phase de techno-design, où le designer et l'ingénieur trouvent des solutions pour matérialiser ce concept,3/ la phase d'éco-design, où les acteurs sociaux concernés l'autorisent et4/ la phase d'ergo-design, où l'utilisateur final adopte le produit final.Une réflexion méthodologique conduit à la modélisation d'un raisonnement innovant de “design augmenté” (où les acteurs principaux sont remplacés par un cortège d'intervenants).Cette méthodologie a été expérimentée dans un contexte de PME. Grâce à l'utilisation du modèle, le management de projet de “design augmenté” fut couronné de succès. Cependant, d'autres validations, plus complexes encore, seront utiles pour sécuriser cette modélisation. L'exploitation de cette approche, par la variété de ses supports, doit guider le chef de projet innovant, en PME, dans le processus général d'innovation de rupture. / The purpose of this study is to model an ‘enhanced design' methodology applied to the conception of an innovative product in a SME environment. This approach includes C-K theory in a context of disruptive innovation.In general, the industrial design process consists of four major steps:1/ the ego-design phase, where the designer conceptualizes a user need,2/ the techno-design phase, where designer and engineer find solutions to materialize the concept,3/ the eco-design phase, where social actors involved authorize it and then4/ the ergo-design phase, where the user adopt the final product.A methodological reflection leads to the modeling of the innovative ‘enhanced design' reasoning (where major actors are replaced by a bunch of various stakeholders).The specific SME's case was successful. Using the model, the enhanced design project management was efficient. But some more complex application cases would help secure it. Using this approach, with appropriate information, should guide the SME design project manager in the general radical innovation process.

An n-parameter Fibonacci AF-algebra is determined by a constant incidence matrix K of a special form. The form of the matrix K is defined by a given n-parameter Fibonacci sequence. We compute the K-theory of certain Fibonacci AF-algebra, and relate their K-theory to the K-theory of an AF-algebra defined by incidence matrices that are the transpose of K.

Este trabalho busca explorar a definição dada por Connes em [Con01] do caráter de Chern para a geometria não-comutativa. Construímos os funtores K 0 e K 1 com os principais resultados para demonstrarmos a Sequência Exata de Seis Termos e a Sequência de Mayer-Vietoris. Calculamos os grupos de K-teoria de algumas álgebras de operadores pseudo-diferenciais clássicos de ordem zero. Posteriormente usamos as sequências exatas para calcular explicitamente o caráter de Chern-Connes nos C -sistemas dinâmicos. / This work intends to explore the definition given by Connes in [Con01] of the Chern charac- ter for noncommutative geometry. We construct the functors K 0 and K 1 with the main results to demonstrate the Exact Sequence of Six Terms and the Sequence of Mayer Vietoris. We compute the K-groups of some algebras of classical zero-order pseudo-differential operators. We then use the exact sequences to explicitly calculate the Chern-Connes Character of C -dynamic systems.

An elliptic theory is constructed for operators acting in subspaces defined via even pseudodifferential projections. Index formulas are obtained for operators on compact manifolds without boundary and for general boundary value problems. A connection with Gilkey's theory of η-invariants is established.

An elliptic theory is constructed for operators acting in subspaces defined via even pseudodifferential projections. Index formulas are obtained for operators on compact manifolds without boundary and for general boundary value problems. A connection with Gilkey's theory of η-invariants is established.