Doctor of Philosophy / Department of Mathematics / John S. Maginnis / In this work, we find the module structure of the cohomology of the group of four by four upper triangular matrices (with ones on the diagonal) with entries from the field on three elements modulo its center. Some of the relations amongst the generators for the cohomology ring are also given. This cohomology is found by considering a certain split extension. We show that the associated Lyndon-Hochschild-Serre spectral sequence collapses at the second page by illustrating a set of generators for the cohomology ring from generating elements of the second page. We also consider two other extensions using more traditional techniques. In the first we introduce some new results giving degree four and five differentials in spectral sequences associated to extensions of a general class of groups and apply these to both the extensions.

Davis and Januszkiewicz introduced in 1981 a family of compact real manifolds, the Quasi-Toric Manifolds, with a group action by a torus, a direct product of circle (T) groups. Their manifolds have an orbit space which is a simple polytope with a distinct isotropy subgroup associated to each face of the polytope, subject to some consistency conditions. They defined a characteristic function which captured the properties of the isotropy subgroups, and showed that their manifolds can be classified by the polytope and characteristic function. They further showed that the cohomology ring of the manifold can be written down directly from properties derived from the polytope and the characteristic function. This work considers the question of how far the circle group T can be replaced by the group of unit quaternions Q in the construction and description of quasi-toric manifolds. Unlike T, the group Q is not commutative, so the actions of Q n on the product H n of the set of quaternions using quaternionic multiplication are studied in detail. Then, in direct analogy to the quasi-toric manifolds, a family of compact real manifolds, the Quoric Manifolds, is introduced which have an action by Q n, and whose orbit space is a polytope. A characteristic functor is defined on the faces of the polytope which captures the properties of the isotropy classes of the orbits of the action. It is shown that quoric manifolds can be classified in a manner similar to the quasi-toric manifolds, by the polytope and characteristic functor. A restricted family, the global quoric manifolds, which satisfy an additional condition are defined. It is shown that an infinite number of polytopes exist in any dimension over which a global quoric manifold can be defined. It is shown that any global quoric manifold can be described as a quotient space of a moment angle complex over the polytope, and that its integral cohomology ring can be calculated, taking a form analagous to that in the quasi-toric case.

Nilpotent Hessenberg varieties are a family of subvarieties of the flag variety, which include the Springer varieties, the Peterson variety, and the whole flag variety. In this thesis I give a geometric proof that the cohomology of the flag variety surjects onto the cohomology of the Peterson variety; I provide a combinatorial criterion for determing the singular loci of a large family of regular nilpotent Hessenberg varieties; and I describe the equivariant cohomology of any regular nilpotent Hessenberg variety whose cohomology is generated by its degree two classes.

Dans la première partie de cette thèse, nous établissons la Conjectured'amplitude de Debarre : Le fibré cotangent T_X* d'une intersection X =H_1 cap ... cap H_c de c >= N/2 hypersurfaces génériques H_i dansP^N de degrés élevés d_1, ..., d_c >> 1 est ample.Tout d'abord, nous élaborons une interprétation géométrique desdifférentielles symétriques sur les espaces projectifs. De cettemanière, nous reconstruisons les différentielles symétriques deBrotbek sur X, lorsque les équations définissantes des hypersurfacesH_1, ..., H_c sont de type Fermat généralisé. De plus, nous dévoilonsdes familles nouvelles de différentielles symétriques de degréinférieur sur toutes les intersections possibles de X avec deshyperplans de coordonnées.Ensuite, nous introduisons ce que nous appelons la Méthode desCoefficients Mobiles ainsi que le Coup du Produit afin d'accomplir unedémonstration de la conjecture d'amplitude de Debarre. De plus, nousobtenons une borne effective inférieure sur les degrés : d_1,...,d_c >=N^N^2. Enfin, grace à des résultats connus au sujet de la conjecturede Fujita, nous établissons que Sym^k T_X* est très ample pour tout k>= 64 (d_1 + ... + d_c)^2.Dans la seconde partie de cette thèse, nous étudions la Conjectured'amplitude généralisée de Debarre stipulant que sur un corpsalgébriquement clos K de caractéristique quelconque, sur une variétéK-projective lisse P de dimension N munie de c >= N/2 fibrés endroites très amples L_1, ..., L_c, pour tous degrés élevés d_1,...,d_c >= d_* >> 1, pour c hypersurfaces génériques H_i dans lessystèmes linéaires L_i^d_i, l'intersection complète X := H_1 cap ... capH_c possède un fibré cotangent T_X* qui est ample.Sur de telles intersections X, nous construisons ce que nous appelonsdes `formes différentielles symétriques de Brotbek généralisées', etnous établissons que si L_1, ..., L_c sont presque proportionnelsmutuellement, alors la conjecture d'amplitude généralisée de Debarreest valide. Notre méthode est effective, et dans le cas où L_1 = ... =L_c, nous obtenons la meme borne inférieure d_* = N^N^2 que dans lapremière partie.Ces deux travaux sont parus sur arxiv.org. / In the first part of this thesis, we establish the Debarre AmplenessConjecture: The cotangent bundle T_X^* of the intersection X = H_1cap ... cap H_c of c >= N/2 generic hypersurfaces H_i in P^N of highdegrees d_1, ..., d_c >> 1 is ample.First of all, we provide a geometric interpretation of symmetricdifferential forms in projective spaces. Thereby, we reconstructBrotbek's symmetric differential forms on X, where the defininghypersurfaces H_1, ..., H_c are generalized Fermat-type. Moreover, weexhibit unveiled families of lower degree symmetric differential formson all possible intersections of X with coordinate hyperplanes.Thereafter, we introduce what we call the `moving coefficients method'and the `product coup' to settle the Debarre Ampleness Conjecture. Inaddition, we obtain an effective lower degree bound: d_1, ...,d_c >=N^{N^2}. Lastly, thanks to known results about the Fujita Conjecture,we establish the very-ampleness of Sym^k T_X^* for all k >= 64 (d_1 +... + d_c)^2.In the second part, we study the General Debarre Ampleness Conjecture,which stipulates that, over an algebraically closed field K with anycharacteristic, on an N-dimensional smooth projective K-variety Pequipped with c >= N/2 very ample line bundles L_1, ..., L_c, for alllarge degrees d_1, ..., d_c >= d_* >> 1, for generic c hypersurfacesH_i in the complete linear system L_i^d_i, the complete intersection X:= H_1 cap ... cap H_c has ample cotangent bundle T_X^*.On such an intersection variety X, we construct what we call`generalized Brotbek's symmetric differential forms', and we establishthat, if L_1,...,L_c are almost proportional mutually, then theGeneral Debarre Ampleness Conjecture holds true. Our method iseffective, and in the case where L_1 = ... = L_c, we obtain the samelower degree bound d_* = N^{N^2} as in the first part.These two works have been posted on arxiv.org.

Dans cette thèse, on construit le morphisme terme constant pour les groupes de cohomologie l-adique à supports compacts des champs classifiants des G-Chtoucas. Ensuite on définit la partie cuspidale de ces groupes de cohomologie et on montre qu'elle est de dimension finie. De plus, on montre que la partie cuspidale coïncide avec la partie Hecke-finie au sens rationnel. / In this thesis, we construct the constant term morphism for the l-adic cohomology groups with compact supports of the classifying stacks of the G-Shtukas. Then we define the cuspidal part of these cohomology groups and we prove that it is of finite dimension. Moreover, we show that the cuspidal part coincides with the Hecke-finite part in the rational sense.

of the Doctoral Thesis Operadic Resolutions of Diagrams by Martin Doubek We study resolutions of the operad AC describing diagrams of a given shape C in the category of algebras of a given type A. We prove the conjecture by Markl on constructing the resolution out of resolutions of A and C, at least in a certain restricted setting. For associative algebras, we make explicit the cohomology theory for the diagrams and recover Gerstenhaber-Schack diagram cohomology. In general, we show that the operadic cohomology is Ext in the category of operadic modules. 1

In this paper, we examine the conjecture of Avner Ash, Darrin Doud, David Pollack, and Warren Sinnott relating Galois representations to the mod p cohomology of congruence subgroups of the general linear group of n dimensions over the integers. We present computational evidence for this conjecture (the ADPS Conjecture) for the case n = 3 by finding Galois representations which appear to correspond to cohomology eigenclasses predicted by the ADPS Conjecture for the prime p, level N, and quadratic nebentype. The examples include representations which appear to be attached to cohomology eigenclasses which arise from D8, S3, A5, and S5 extensions. Other examples include representations which are reducible as sums of characters, representations which are symmetric squares of two-dimensional representations, and representations which arise from modular forms, as predicted by Jean-Pierre Serre for n = 2.

No description available.

We want to take a look at the first cohomology group H^1(G, l^2(G)), in particular when G is locally-finite. First, though, we discuss some results about the space H^1(G, C G) for G locally-finite, as well as the space H^1(G, l^2(G)) when G is finitely generated. We show that, although in the case when G is finitely generated the embedding of C G into l^2(G) induces an embedding of the cohomology groups H^1(G, C G) into H^1(G, l^2(G)), when G is countably-infinite locally-finite, the induced homomorphism is not an embedding. However, even though the induced homomorphism is not an embedding, we still have that H^1(G, l^2(G)) neq 0 when G is countably-infinite locally-finite. Finally, we give some sufficient conditions for H^1(G,l^2(G)) to be zero or non-zero. / Master of Science

We give a description of the moment graph for Bott-Samelson varieties in arbitrary Lie type. We use this, along with curve neighborhoods and explicit moduli space computations, to compute a presentation for the small quantum cohomology ring of a particular Bott-Samelson variety in Type A. / Ph. D. / Since the early 1990’s, the study of quantum cohomology has been a fascinating, and fruitful field of research with connections to physics, representation theory, and combinatorics. The quantum cohomology of a space X encodes enumerative information about how many curves intersect certain subspaces of X; these counts are called Gromov-Witten invariants. For some spaces X, including the class of spaces we consider here, this count is only ”virtual” and negative Gromov-Witten invariants may arise. In this dissertation, we study the quantum cohomology of Bott-Samelson varieties. These spaces arise frequently in applications to representation theory and combinatorics, however their quantum cohomology was previously unexplored. The first of our three main theorems describes the moment graph for Bott-Samelson varieties. This is a description of what all the possible curves, stable under certain symmetries, exist in a Bott-Samelson variety. Our second main theorem is a technical result which enables us to compute some GromovWitten invariants directly. Finally, our third main theorem is a description of the quantum cohomology for a certain three-dimensional Bott-Samelson variety.