In this dissertation, we study morphisms of smooth complex projective varieties to the projective line with at most two singular fibres. We show that if such a morphism has at most one singular fibre, then the domain of the morphism is uniruled and the morphism admits algebraic sections. We reach the same conclusions, but with algebraic genus zero multisections instead of algebraic sections, if the morphism has at most two singular fibres and the first Chern class of the domain of the morphism is supported in a single fibre of the morphism. To achieve these result, we use action completed symplectic cohomology groups associated to compact subsets of convex symplectic domains. These groups are defined using Pardon's virtual fundamental chains package for Hamiltonian Floer cohomology. In the above setting, we show that the vanishing of these groups implies the existence of unirulings and (multi)sections.

Given a reduced cooperad C, we consider the 2-colored operad Cyl(C) which governs diagrams U: V -> W, where V, W are Cobar(C)-algebras, and U is an infinity-morphism. We then investigate the deformation complexes of Cyl(C) and Cobar(C). Our main result is that the restriction maps between between the deformation complexes Der'(Cyl(C)) and Der'(Cobar(C)) are homotopic quasi-isomorphisms of filtered Lie algebras. We show how this result may be applied to modifying diagrams of homotopy algebras by derived automorphism. We then recall that Tamarkin's construction gives us a map from the set of Drinfeld associators to the homotopy classes of Lie infinity quasi-isomorphisms for Hochschild cochains of a polynomial algebra. Due to results of V. Drinfeld and T. Willwacher, both the source and the target of this map are equipped with natural actions of the Grothendieck-Teichmueller group GRT. We use our earlier results to prove that this map from the set of Drinfeld associators to the set of homotopy classes of Lie infinity quasi-isomorphisms for Hochschild cochains is GRT-equivariant. / Mathematics

<p dir="ltr">We first introduce several families of monoidal categories with plethysm products as their monoidal products and use this to describe operadic structures as plethysm monoids. In order to link this approach with the classical theory, we give a generalization of the Baez-Dolan plus construction. We then show that an operadic structure can be defined as a plethysm monoid if its associated Feynman category is a plus construction of a unique factorization category.</p>

xi, 95 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We investigate some homological properties of graded algebras. If A is an R -algebra, then E (A) := Ext A ( R, R ) is an R-algebra under the cup product and is called the Yoneda algebra. (In most cases, we assume R is a field.) A well-known and widely-studied condition on E(A) is the Koszul property. We study a class of deformations of Koszul algebras that arises from the study of equivariant cohomology and algebraic groups and show that under certain circumstances these deformations are Poincaré-Birkhoff-Witt deformations. Some of our results involve the [Special characters omitted] property, recently introduced by Cassidy and Shelton, which is a generalization of the Koszul property. While a Koszul algebra must be quadratic, a [Special characters omitted] algebra may have its ideal of relations generated in different degrees. We study the structure of the Yoneda algebra corresponding to a monomial [Special characters omitted.] algebra and provide an example of a monomial [Special characters omitted] algebra whose Yoneda algebra is not also [Special characters omitted]. This example illustrates the difficulty of finding a [Special characters omitted] analogue of the classical theory of Koszul duality. It is well-known that Poincaré-Birkhoff-Witt algebras are Koszul. We find a [Special characters omitted] analogue of this theory. If V is a finite-dimensional vector space with an ordered basis, and A := [Special characters omitted] (V)/I is a connected-graded algebra, we can place a filtration F on A as well as E (A). We show there is a bigraded algebra embedding Λ: gr F E (A) [Special characters omitted] E (gr F A ). If I has a Gröbner basis meeting certain conditions and gr F A is [Special characters omitted], then Λ can be used to show that A is also [Special characters omitted]. This dissertation contains both previously published and co-authored materials. / Committee in charge: Brad Shelton, Chairperson, Mathematics; Victor Ostrik, Member, Mathematics; Christopher Phillips, Member, Mathematics; Sergey Yuzvinsky, Member, Mathematics; Van Kolpin, Outside Member, Economics

Ce mémoire a deux objectifs principaux. Premièrement de développer et interpréter les groupes de cohomologie de Hochschild de basse dimension et deuxièmement de borner la dimension cohomologique des k-algèbres par dessous; montrant que presque aucune k-algèbre commutative est quasi-libre. / The aim of this master’s thesis is two-fold. Firstly to develop and interpret the low dimensional Hochschild cohomology of a k-algebra and secondly to establish a lower bound for the Hochschild cohomological dimension of a k-algebra; showing that nearly no commutative k-algebra is quasi-free.

In this thesis we study the homological behaviour of configuration spaces as the number of objects in the configuration goes to infinity. For unordered configurations of distinct points (possibly equipped with some internal parameters) in a connected, open manifold it is a well-known result, going back to G. Segal and D. McDuff in the 1970s, that these spaces enjoy the property of homological stability. In Chapter 2 we prove that this property also holds for so-called oriented configuration spaces, in which the points of a configuration are equipped with an ordering up to even permutations. There are two important differences from the unordered setting: the rate (or slope) of stabilisation is strictly slower, and the stabilisation maps are not in general split-injective on homology. This can be seen by some explicit calculations of Guest-Kozlowski-Yamaguchi in the case of surfaces. In Chapter 3 we refine their calculations to show that, for an odd prime p, the difference between the mod-p homology of the oriented and the unordered configuration spaces on a surface is zero in a stable range whose slope converges to 1 as p goes to infinity. In Chapter 4 we prove that unordered configuration spaces satisfy homological stability with respect to finite-degree twisted coefficient systems, generalising the corresponding result of S. Betley for the symmetric groups. We deduce this from a general “twisted stability from untwisted stability” principle, which also applies to the configuration spaces studied in the next chapter. In Chapter 5 we study configuration spaces of submanifolds of a background manifold M. Roughly, these are spaces of pairwise unlinked, mutually isotopic copies of a fixed closed, connected manifold P in M. We prove that if the dimension of P is at most (dim(M)−3)/2 then these configuration spaces satisfy homological stability w.r.t. the number of copies of P in the configuration. If P is a sphere this upper bound on its dimension can be increased to dim(M)−3.

A group Γ is of type F_n for some n ≥ 1 if it has a classifying complex with finite n-skeleton. These properties generalise the classical notions of finite generation and finite presentability. We investigate the higher finiteness properties for fibre products of groups.

We introduce the definition of 1-cotilting object in a Grothendieck category and investigate its relation to the analogue of the standard definition of 1-cotilting module. The 1-cotilting quasi-coherent sheaves on a Noetherian scheme are stud- ied in particular: using the classification of hereditary torsion pairs in the category of quasi-coherent sheaves on a Noetherian scheme X, to each hereditary torsion- free class F that is generating we assign a 1-cotilting quasi-coherent sheaf whose 1-cotilting class is F. This provides a family of pairwise non-equivalent 1-cotilting quasi-coherent sheaves which are parametrized by specialization closed subsets of X avoiding the set of associated points of a chosen generator of the category of quasi-coherent sheaves. In many cases (e.g. for separated schemes), this set of avoided points can be chosen as the set of associated points of the scheme. 1

Consideremos A e B duas álgebras de Artin tais que é uma extensão cindida de A pelo ideal Q, onde é um ideal nilpotente de B. Estudamos algumas propriedades homológicas das categorias modA e modB, tais como dimensão projetiva e injetiva. A partir disso mostramos que se B pertence a uma das seguintes classes: hereditária, laura, fracamente shod, shod, quase inclinada, colada à esquerda, colada à direita ou disfarçada; então A pertence a mesma classe. Além disso, restringindo nosso estudo para álgebras de dimensão finita sobre um corpo algebricamente fechado, comparamos as respectivas aljavas ordinárias, bem como suas apresentações. Finalmente, após caracterizarmos o ideal Q, exibimos alguns exemplos de extensões no contexto de álgebras de caminhos com relações, que mostram que A pode ser de uma das classes citadas sem que B o seja / Let A and B be two Artin algebras such that B is a split-by-nilpotent extension of A by Q, were Q is a nilpotent ideal of B. We study some homological properties of the categories mod A and mod B such that the projetive and the injetive dimensions of their objects. Using this we show that if B belongs to one of this classes: hereditary, laura, weakly shod, shod, quasi-tilted, left glued, right glued or concealed; then A belongs to same class. Moreover restricting our study to finite dimensional algebras over algebraically closed fields, we compare the ordinary quivers and presentations of the corresponding algebras. Finally, after giving a characterization of ideal Q as above, we exhibit some exemples of split extensions in the context of path algebras bounded by relations, which shows that A can be one of the above cited algebras without B so

This thesis is on the representation theory of finite groups. Specifically, it is about finding connections between fusion and universal deformation rings. Two elements of a subgroup N of a finite group Γ are said to be fused if they are conjugate in Γ, but not in N. The study of fusion arises in trying to relate the local structure of Γ (for example, its subgroups and their embeddings) to the global structure of Γ (for example, its normal subgroups, quotient groups, conjugacy classes). Fusion is also important to understand the representation theory of Γ (for example, through the formula for the induction of a character from N to Γ). Universal deformation rings of irreducible mod p representations of Γcan be viewed as providing a universal generalization of the Brauer character theory of these mod p representations of Γ. It is the aim of this thesis to connect fusion to this universal generalization by considering the case when Γ is an extension of a finite group G of order prime to p by an elementary abelian p-group N of rank 2. We obtain a complete answer in the case when G is a dihedral group, and we also consider the case when G is abelian. On the way, we compute for many absolutely irreducible FpΓ-modules V, the cohomology groups H2(Γ,HomFp(V,V) for i = 1, 2, and also the universal deformation rings R(Γ,V).