Consider a semisimple linear algebraic group G over an arbitrary field F, and a projective homogeneous G-variety X. The geometry of such varieties has been a consistently active subject of research in algebraic geometry for decades, with significant contributions made by Grothendieck, Demazure, Tits, Panin, and Merkurjev, among others. An effective tool for the classification of these varieties is the notion of a cohomological (or alternatively, a motivic) invariant. Two such invariants are the set of Tits algebras of G defined by J. Tits, and the J-invariant of G defined by Petrov, Semenov, and Zainoulline. Quéguiner-Mathieu, Semenov and Zainoulline discovered a connection between these invariants, which they developed through use of the second Chern class map. The first goal of the present thesis is to extend this connection through the use of higher Chern class maps. Our main technical tool is the Steinberg basis, which provides explicit generators for the γ-filtration on the Grothendieck group K_0(X) in terms of characteristic classes of line bundles over X. As an application, we establish a connection between the J-invariant and the Tits algebras of a group G of inner type E6. The second goal of this thesis is to relate the indices of the Tits algebras of G to nontrivial torsion elements in the γ-filtration on K_0(X). While the Steinberg basis provides an explicit set of generators of the γ-filtration, the relations are not easily computed. A tool introduced by Zainoulline called the twisted γ-filtration acts as a surjective image of the γ-filtration, with explicit sets of both generators and relations. We use this tool to construct torsion elements in the degree 2 component of the γ-filtration for groups of inner type D2n. Such a group corresponds to an algebra A endowed with an orthogonal involution having trivial discriminant. In the trialitarian case (i.e. type D4), we construct a specific element in the γ-filtration which detects splitting of the associated Tits algebras. We then relate the non-triviality of this element to other properties of the trialitarian triple such as decomposability and hyperbolicity.

<p> In his monograph [Ne], Nekovar studies cohomological invariants of big Galois representations and looks at the variations of Selmer groups attached to intermediate number fields in a commutative p-adic Lie extension. In view of the formulation of the "main conjecture" for noncommutative extensions, it seems natural to extend the theory to a noncommutative p-adic Lie extension. This thesis will serve as a first step in an extension of this theory, namely, we will develop duality theorems over a noncommutative p-adic Lie extension which are extensions of Tate local duality, Poitou-Tate global duality and Grothendieck duality. </p> / Thesis / Doctor of Philosophy (PhD)

Nosso objetivo nesse trabalho é estudar o comportamento dos operadores de transferência em f0, 1gN2 , um associado ao deslocamento horizontal (se1 ) e outro associado ao deslocamento vertical (se2 ). Construímos uma equação cohomológica para fins de ampliar a gama de funções às quais os operadores de transferência se aplicam. Estudamos também o comportamento do operador de transferência obtido pela composição dos dois operadores citados e, em condições de comutatividade, encontramos um autovalor e uma autofunção associada, ambos estritamente positivos, e uma automedida para o operador dual, associada ao mesmo autovalor. Tal automedida é um estado de equilíbrio. Além disso, estudamos algumas propriedades ergódicas de transformações de blocos. / In this work we study the behavior of the transfer operators in f0, 1gN2 , one associated with horizontal shift (se1 ) and other associated with vertical shift (se2 ). We build a cohomological equation for the purpose of expanding the range of functions to which the transfer operators apply. We also study the behavior of the transfer operator obtained by the composition of the two cited operators and, in the conditions of commutativity, we find an eigenvalue and an associated eigenfunction, both strictly positive, and an eigen measuse for the dual operator, associated with the same eigenvalue. This eigen measure is an equilibrium state. Furthermore, we study some ergodic properties of block transformations.

Nosso objetivo nesse trabalho é estudar o comportamento dos operadores de transferência em f0, 1gN2 , um associado ao deslocamento horizontal (se1 ) e outro associado ao deslocamento vertical (se2 ). Construímos uma equação cohomológica para fins de ampliar a gama de funções às quais os operadores de transferência se aplicam. Estudamos também o comportamento do operador de transferência obtido pela composição dos dois operadores citados e, em condições de comutatividade, encontramos um autovalor e uma autofunção associada, ambos estritamente positivos, e uma automedida para o operador dual, associada ao mesmo autovalor. Tal automedida é um estado de equilíbrio. Além disso, estudamos algumas propriedades ergódicas de transformações de blocos. / In this work we study the behavior of the transfer operators in f0, 1gN2 , one associated with horizontal shift (se1 ) and other associated with vertical shift (se2 ). We build a cohomological equation for the purpose of expanding the range of functions to which the transfer operators apply. We also study the behavior of the transfer operator obtained by the composition of the two cited operators and, in the conditions of commutativity, we find an eigenvalue and an associated eigenfunction, both strictly positive, and an eigen measuse for the dual operator, associated with the same eigenvalue. This eigen measure is an equilibrium state. Furthermore, we study some ergodic properties of block transformations.

Nosso objetivo nesse trabalho é estudar o comportamento dos operadores de transferência em f0, 1gN2 , um associado ao deslocamento horizontal (se1 ) e outro associado ao deslocamento vertical (se2 ). Construímos uma equação cohomológica para fins de ampliar a gama de funções às quais os operadores de transferência se aplicam. Estudamos também o comportamento do operador de transferência obtido pela composição dos dois operadores citados e, em condições de comutatividade, encontramos um autovalor e uma autofunção associada, ambos estritamente positivos, e uma automedida para o operador dual, associada ao mesmo autovalor. Tal automedida é um estado de equilíbrio. Além disso, estudamos algumas propriedades ergódicas de transformações de blocos. / In this work we study the behavior of the transfer operators in f0, 1gN2 , one associated with horizontal shift (se1 ) and other associated with vertical shift (se2 ). We build a cohomological equation for the purpose of expanding the range of functions to which the transfer operators apply. We also study the behavior of the transfer operator obtained by the composition of the two cited operators and, in the conditions of commutativity, we find an eigenvalue and an associated eigenfunction, both strictly positive, and an eigen measuse for the dual operator, associated with the same eigenvalue. This eigen measure is an equilibrium state. Furthermore, we study some ergodic properties of block transformations.

Doctor of Philosophy / Department of Mathematics / Yan Soibelman / Given a quiver Q with/without potential, one can construct an algebra structure on the cohomology of the moduli stacks of representations of Q. The algebra is called Cohomological Hall algebra (COHA for short). One can also add a framed structure to quiver Q, and discuss the moduli space of the stable framed representations of Q. Through these geometric constructions, one can construct two representations of Cohomological Hall algebra of Q over the cohomology of moduli spaces of stable framed representations. One would get the double of the representations of Cohomological Hall algebras by putting these two representations together. This double construction implies that there are some relations between Cohomological Hall algebras and some other algebras. In this dissertation, we focus on the quiver without potential case. We first define Cohomological Hall algebras, and then the above construction is stated under some assumptions. We computed two examples in detail: A₁-quiver and Jordan quiver. It turns out that A₁-COHA and its double representations are related to the half infinite Clifford algebra, and Jordan-COHA and its double representations are related to the infinite Heisenberg algebra. Then by the fact that the underlying vector spaces of these two COHAs are isomorphic to each other, we get a COHA version of Boson-Fermion correspondence.

Doctor of Philosophy / Department of Mathematics / Yan S. Soibelman / The motivic Donaldson-Thomas theory of 2-dimensional Calabi-Yau categories can be induced from the theory of 3-dimensional Calabi-Yau categories via dimensional reduction. The cohomological Hall algebra is one approach to the motivic Donaldson-Thomas invariants. Given an arbitrary quiver one can construct a double quiver, which induces the preprojective algebra. This corresponds to a 2-dimensional Calabi-Yau category. One can further construct a triple quiver with potential, which gives rise to a 3-dimensional Calabi-Yau category. The critical cohomological Hall algebra (critical COHA for short) is defined for a quiver with potential. Via the dimensional reduction we obtain the cohomological Hall algebra (COHA for short) of the preprojective algebra. We prove that a subalgebra of this COHA consists of a semicanonical basis, thus is related to the generalized quantum groups. Another approach is motivic Hall algebra, from which an integration map to the quantum torus is constructed. Furthermore, a conjecture concerning some invariants of 2-dimensional Calabi-Yau categories is made. We investigate the correspondence between the A∞-equivalent classes of ind-constructible 2-dimensional Calabi-Yau categories with a collection of generators and a certain type of quivers. This implies that such an ind-constructible category can be canonically reconstructed from its full subcategory consisting of the collection of generators.

Dans cette thèse, on étudie deux thèmes, a priori différents mais qui rentrent dans le cadre des systèmes dynamiques : les échanges d’intervalles, la résolution d’équations cohomologiques et la description explicite des distributions invariantes par certains difféomorphismes d’un groupe de Lie compact.1 - On établit un critère d'existence de fonctions propres continues non constantes pour les échangesd'intervalles, c'est-à-dire de non mélange faible topologique. On construit pour tout entier m > 3des échanges de m intervalles de rang 2 uniquement ergodiques et non topologiquement faiblementmélangeants. Nous répondons aussi à une question de Ferenczi et Zamboni. On construit aussi pourtout entier pair m ≥ 4 des échanges de m intervalles possédant des valeurs propres irrationnelles et desvaleurs propres rationnelles (avec fonctions propres associées continues par morceaux) et qui sont soituniquement ergodiques, soit non minimaux.2 - On montre qu’un échange d’intervalles affine, dont les pentes sont des puissances d’un mêmeentier n, et dont les coupures et leurs images sont des rationnels , a une dynamique très simple : toutesses orbites sont propres et il possède une orbite périodique ou un cycle périodique.3 - On traite deux questions d’analyse sur un groupe de Lie connexe compact G. i) Soient a ∈ Get γ le difféomorphisme de G donné par γ(x) = ax (translation `a gauche par a). On donne lesconditions nécessaires et suffisantes pour que l’équation cohomologique f − f ◦ γ = g admette dessolutions dans l’espace de Fréchet C∞(G) des fonctions complexes C∞ sur G. ii) Lorsque G est le toreTn, on détermine explicitement les distributions sur Tn invariantes par un automorphisme affine γ i.e.γ(x) = Ax + a avec A ∈ GL(n, Z) et a ∈ Tn.4 - On donne des résultats obtenus dans 3) une application aux déformations infinitésimales d’unfeuilletage obtenu par suspension d’une translation d’un groupe de Lie compact. / In this thesis, we study two subjects, which are priori different but are within the scopeof dynamical systems: interval exchange, the resolution of cohomological equationsand the explicit description of invariant distributions by a diffeomorphism on a compactLie group.1. We prove a criterion for the existence of continuous non constant eigenfunc-tions for interval exchange transformations which are non topologically weakly mixing.We first construct, for any m > 3, uniquely ergodic interval exchange transforma-tions of Q-rank 2 with irrational eigenvalues associated to continuous eigenfunctionswhich are not topologically weakly mixing; this answers a question of Ferenczi andZamboni [5]. Moreover we construct, for any even integer m ≥ 4, interval exchangetransformations of Q-rank 2 with both irrational eigenvalues (associated to continuouseigenfunctions) and non trivial rational eigenvalues (associated to piecewise continu-ous eigenfunctions); these examples can be chosen to be either uniquely ergodic ornon minimal.2. We prove that an affine interval exchange, whose slopes are integer powers ofthe same integer n, and whose cuts and their images are rational, has a very simpledynamic: all its orbits are proper and it has a periodic orbit or a periodic cycle.3. A third section deals with two analytic questions on a connected compact Liegroup G. i) Let a ∈ G and denote by γ the diffeomorphism of G given by γ(x) = ax(left translation by a). We give necessary and sufficient conditions for the existenceof solutions of the cohomological equation f − f ◦ γ = g on the Fr´echet space C∞(G)of complex C∞ functions on G. ii) When G is the torus Tn, we compute explicitly thedistributions on Tn invariant by an affine automorphism γ, that is, γ(x) = Ax+a withA ∈ GL(n, Z) and a ∈ Tn.4. We apply the results of the preceding section to describe the infinitesimaldeformations of a foliation obtained by suspension of a translation associated to anelement on a compact Lie group.

Working over an arbitrary field F of characteristic not 2, we consider linear algebraic groups over F. We view these as functors, represented by finitely generated F-Hopf algebras, from the category of commutative, associative, F-algebras Alg_F, to the category of groups. Classical examples of these groups, such as the special linear group SL_n are split, however there are also linear algebraic groups arising from central simple F-algebras which are non-split. For example, associated to a non-split central simple F-algebra A of degree n is a non-split special linear group SL(A). It is well known that central simple algebras are twisted forms of matrix algebras. This means that over the separable closure of F, denoted F_sep, we have A⊗_F F_sep ∼= M_n(F_sep) and that there is a twisted Gal(F_sep/F)-action on M_n(F_sep) whose fixed points are A. We show that a similar method of twisted Galois descent can be used to obtain all non-split semisimple linear algebraic groups associated to central simple algebras as fixed points within their split counterparts. In particular, these techniques can be used to construct the spin and half-spin groups Spin(A, τ ) and HSpin(A, τ ) associated to a central simple F-algebra of degree 4n with orthogonal involution. Furthermore, we develop a theory of twisted Galois descent for Hopf algebras and show how the fixed points obtained this way are the representing Hopf algebras of our non-split groups. Returning to the view of group schemes as functors, we discuss how the group schemes we consider are sheaves on the étale site of Alg_F whose stalks are Chevalley groups over local, strictly Henselian F-algebras. This allows us to use the generators and relations presentation of Chevalley groups to explicitly describe group scheme morphisms. After showing how the Kronecker tensor product of matrices induces maps between simply connected groups, we give an explicit description of these maps in terms of Chevalley generators. This allows us to compute the kernel of these new maps composed with standard isogenies and thereby construct new tensor product maps between non-simply connected split groups. These new maps are Gal(F_sep/F)-morphisms and so we apply our techniques of twisted Galois descent to also obtain new tensor product morphisms between non-split groups schemes. Finally, we use one of our new split tensor product maps to compute the degree three cohomological invariants of HSpin_4n for all n.

Mirror symmetry is a phenomenon from physics that has inspired a lot of interesting mathematics. In the Landau-Ginzburg setting, we have two constructions, the A and B models, which are created based on a choice of an affine singularity with a group of symmetries. Both models are vector spaces equipped with multiplication and a pairing (making them Frobenius algebras), and they are also Frobenius manifolds. We give a result relating stabilization of singularities in classical singularity to its counterpart in the Landau-Ginzburg setting. The A model comes from so-called FJRW theory and can be de fined up to a full cohomological field theory. The structure of this model is determined by a generating function which requires the calculation of certain numbers, which we call correlators. In some cases the their values can be computed using known techniques. Often, there is no known method for finding their values. We give new computational methods for computing concave correlators, including a formula for concave genus-zero, four-point correlators and show how to extend these results to find other correlator values. In many cases these new methods give enough information to compute the A model structure up to the level of Frobenius manifold. We give the FJRW Frobenius manifold structure for various choices of singularities and groups.