The aim of this thesis is to give a self-contained introduction to the hyperbolic geometry and the theory of discrete subgroups of PSL(2,R), and to generalize the results on equivariant functions. We show that there is a deep relation between the geometry of the group and some analytic and algebraic properties of these functions. In addition, we provide some applications of equivariant functions consisting of new results as well as providing new and simple proofs to classical results on automorphic forms.

The aim of this thesis is to give a self-contained introduction to the hyperbolic geometry and the theory of discrete subgroups of PSL(2,R), and to generalize the results on equivariant functions. We show that there is a deep relation between the geometry of the group and some analytic and algebraic properties of these functions. In addition, we provide some applications of equivariant functions consisting of new results as well as providing new and simple proofs to classical results on automorphic forms.

Equivariant localisation is based on exploiting certain symmetries of some systems, generally represented by a non-free action of a Lie group on a manifold, to reduce the dimensionality of integral calculations that commonly appear in theoretical physics. In this work we present Cartan's model of equivariant cohomology in different scenarios, such as differential manifolds, symplectic manifolds or vector bundles and we reproduce the main corresponding localisation results.

The aim of this thesis is to give a self-contained introduction to the hyperbolic geometry and the theory of discrete subgroups of PSL(2,R), and to generalize the results on equivariant functions. We show that there is a deep relation between the geometry of the group and some analytic and algebraic properties of these functions. In addition, we provide some applications of equivariant functions consisting of new results as well as providing new and simple proofs to classical results on automorphic forms.

The aim of this thesis is to give a self-contained introduction to the hyperbolic geometry and the theory of discrete subgroups of PSL(2,R), and to generalize the results on equivariant functions. We show that there is a deep relation between the geometry of the group and some analytic and algebraic properties of these functions. In addition, we provide some applications of equivariant functions consisting of new results as well as providing new and simple proofs to classical results on automorphic forms.

In 2003, Johannes Kellendonk and Ian Putnam introduced pattern equivariant cohomology for tilings. In 2006, Betseygail Rand defined a type of pattern equivariant cohomology that incorporates rotational symmetry, using representation of the rotation group. In this doctoral thesis we study the relationship between these two types of pattern equivariant cohomology, showing exactly how to calculate one from the other in the case in which the rotation group is a finitely generated abelian group of free rank 1. We apply our result by calculating the cohomology of the pinwheel tiling.

If G is a nite group and if X is a G-space, then a Bredon RO(G)-graded equivariantcohomology theory is dened on X. Furthermore, if X is a G-manifold, thereexists a natural Čech hypercohomology theory on X. While Bredon RO(G)-gradedcohomology is important in the theoretical aspects, the Čech cohomology is indispensablewhen computing the cohomology groups. The purpose of this dissertation is toconstruct an isomorphism between these two types of cohomology theories so that theinterplay becomes deeper between the theory and concretely computing cohomologygroups of classical objects. Also, with the aid of Čech cohomology, we can naturallyextend the Bredon cohomology to the more generalized Deligne cohomology.In order to construct such isomorphism, on one hand, we give a new constructionof Bredon RO(G)-graded equivariant cohomology theory from the sheaf-theoreticviewpoint. On the other hand, with Illman's theorem of smooth G-triangulation ofa G-manifold, we extend the existence of good covers from the nonequivariant tothe equivariant case. It follows that, associated to an equivariant good cover of aG-manifold X, there is a bounded spectral sequence converging to Čech hypercohomologywhose E1 page is isomorphic to the E1 page of a Segal spectral sequence whichconverges to the Bredon RO(G)-graded equivariant cohomology. Furthermore, Thisisomorphism is compatible with the structure maps in the two spectral sequences. So there is an induced isomorphism between two limiting objects, which are exactly theČech hypercohomology and the Bredon RO(G)-graded equivariant cohomology.We also apply the above results to real varieties and obtain a quasi-isomorphismbetween two commonly used complexes of presheaves.

If G is a nite group and if X is a G-space, then a Bredon RO(G)-graded equivariantcohomology theory is dened on X. Furthermore, if X is a G-manifold, thereexists a natural Čech hypercohomology theory on X. While Bredon RO(G)-gradedcohomology is important in the theoretical aspects, the Čech cohomology is indispensablewhen computing the cohomology groups. The purpose of this dissertation is toconstruct an isomorphism between these two types of cohomology theories so that theinterplay becomes deeper between the theory and concretely computing cohomologygroups of classical objects. Also, with the aid of Čech cohomology, we can naturallyextend the Bredon cohomology to the more generalized Deligne cohomology.In order to construct such isomorphism, on one hand, we give a new constructionof Bredon RO(G)-graded equivariant cohomology theory from the sheaf-theoreticviewpoint. On the other hand, with Illman's theorem of smooth G-triangulation ofa G-manifold, we extend the existence of good covers from the nonequivariant tothe equivariant case. It follows that, associated to an equivariant good cover of aG-manifold X, there is a bounded spectral sequence converging to Čech hypercohomologywhose E1 page is isomorphic to the E1 page of a Segal spectral sequence whichconverges to the Bredon RO(G)-graded equivariant cohomology. Furthermore, Thisisomorphism is compatible with the structure maps in the two spectral sequences. So there is an induced isomorphism between two limiting objects, which are exactly theČech hypercohomology and the Bredon RO(G)-graded equivariant cohomology.We also apply the above results to real varieties and obtain a quasi-isomorphismbetween two commonly used complexes of presheaves.

We compute the RO(C2)-graded Bredon cohomology of certain families of real and complex C2-equivariant Grassmannians.

Our aim is to study the role of omnioriented quasitoric manifolds in equivariant complex bordism. These are a well-behaved class of even-dimensional smooth closed manifolds with the action of a half-dimensional compact torus and an equivariant stably complex structure. They are beneficial objects to work with as they can be described completely in terms of combinatorial data.We include an overview of equivariant complex bordism, highlighting the relationship between localisation and restriction to fixed point data. By keeping in mind the particularly interesting case when the group in question is the compact torus, we revisit work found in [BPR10], reinterpreting and expanding certain results relating to the universal toric genus.We then consider oriented torus graphs of stably complex torus manifolds and classify these using a boundary operator on exterior polynomials related to geometric equivariant complex bordism classes of the manifolds. We also extend the connected sum construction of quasitoric pairs which allows for a more general notion of the equivariant connected sum of omnioriented quasitoric manifolds.We then consider whether an equivariant version of Buchstaber and Ray’s result in [BR98] holds; that is, does there exist an omnioriented quasitoric manifold in every geometric equivariant complex bordism class in which they naturally exist? We conjecture that this is true showing that we have a combinatorial model for such objects and exhibiting low-dimensional examples.