We develop and study the notion of a spherical supervariety, which is a generalization of the classical notion of a spherical variety in algebraic geometry. Spherical supervarieties are supervarieties admitting an action of a quasi-reductive group with an open orbit of a hyperborel subgroup. Three characterizations of spherical supervarieties are given: one which generalizes the Vinberg-Kimelfeld characterization of affine spherical varieties, another that extends the ideas of the affine case to the quasi-projective case, and finally one in terms of invariant rational functions which applies to any supervariety. Our characterization of affine spherical supervarieties leads to (non-constructive) existence theorems for finite-dimensional highest weight representations admitting certain coinvariants under spherical quasireductive subgroups.

Several interesting examples of spherical supervarieties are given. We present a classification of indecomposable spherical representations (for certain supergroups) and for each the description of its algebra of functions. Adjoint orbits of odd self-commuting elements are shown to be spherical in many cases, in particular for basic simple Lie superalgebras. We study group-graded supergroups and their spherical homogeneous supervarieties, showing in particular that the algebra of functions on an affine homogeneous supervariety is almost never completely reducible for such supergroups.

Finally we study the case of symmetric supervarieties and show that, despite their not always being spherical (in contrast to the classical case), we may under some circumstances guarantee the existence of an Iwasawa decomposition, which implies sphericity. The fixed points of automorphisms of generalized root systems coming from supersymmetric pairs are studied along the way. We use the Iwasawa decomposition to gain partial understanding of the structure of the space of functions as a representation. Finally, the case of a supergroup as a symmetric supervariety is discussed in detail, culminating in a description of the socle filtration and the Loewy layers of its space of functions.

The thesis comprises three largely independent projects undertaken during my stay at UC Berkeley, all revolving around the same mathematical objects: Cosemisimple Hopf algebras, regarded here as function algebras on linearly reductive quantum groups. We often specialize further to Hopf *-algebras coacting universally on finite-dimensional Hilbert spaces perhaps endowed with additional structure. Such a Hopf algebra is to be thought of as the algebra of representative functions on the compact quantum automorphism group of the respective structure.

Chapter 2 is based on [23]. The question of whether or not a Hopf algebra H is faithfully flat over a Hopf subalgebra A has received positive answers in several particular cases: when H is commutative, or cocommutative, or pointed, or when A contains the coradical of H. We prove the result for cosemisimple H, adding this latter class of Hopf algebras to those known to be faithfully flat over all Hopf subalgebras. We also show that the third term of the resulting ``exact sequence'' A -> H -> C is always a cosemisimple coalgebra, and that the expectation H -> A is positive when H is a CQG algebra.

Chapter 3 consists of material from [22], with earlier related results appearing in [24]. We define the notion of a (linearly reductive) center for a linearly reductive quantum group, and show that the quotient of a such a quantum group by its center is simple whenever its fusion semiring is free in the sense of Banica and Vergnioux. We also prove that the same is true of free products of quantum groups under very mild non-degeneracy conditions. Several natural families of compact quantum groups, some with non-commutative fusion semirings and hence very ``far from classical'', are thus seen to be simple. Examples include quotients of free unitary groups by their centers as well as quotients of quantum reflection groups by their centers.

In Chapter 4 we show that provided n is different from 3, the involutive Hopf *-algebra Au(n) coacting universally on an n-dimensional Hilbert space has enough finite-dimensional representations, in the sense that every non-zero element acts non-trivially in some finite-dimensional *-representation. This implies that the discrete quantum group with group algebra Au(n) is maximal almost periodic, i.e. it embeds in its quantum Bohr compactification; this answers a question posed by P. Soltan. We also prove analogous results for the involutive Hopf *-algebra Bu(n) coacting universally on an n-dimensional Hilbert space equipped with a non-degenerate bilinear form.

This thesis is a resolution of three related problems proposed by Yu. I. Manin and V. Kac for the so-called exceptional Lie superalgebras F(4) and G(3). The first problem posed by Kac (1978) is the problem of finding character and superdimension formulae for the simple modules. The second problem posed by Kac (1978) is the problem of classifying all indecomposable representations. The third problem posed by Manin (1981) is the problem of constructing the superanalogue of Borel-Weil-Bott theorem.

The first purpose of this dissertation is to introduce and develop a theory of toric stacks which encompasses and extends the notions of toric stacks defined by Lafforgue, Borisov-Chen-Smith, Fantechi-Mann-Nironi, Iwanari, and Satriano, as well as classical toric varieties. In addition to introducing a broader class of smooth toric stacks, the definition we introduce allows singularities.

The second purpose is to characterize toric stacks in a "bottom up" fashion, similar to the treatment of smooth toric Deligne-Mumford stacks by Fantechi-Mann-Nironi and the characterization of toric varieties as "abstract toric schemes" which are reduced, separated, and normal.