This Thesis is concerned with the study of the geometry and structure of the space of Kahler metrics representing a fixed cohomology class on a compact Kahler manifold. The first part of the Thesis is concerned with a problem of geometric quantization: Can the geometry of the infinite-dimensional space of Kahler metrics be approximated in terms of the geometry of the finite-dimensional spaces of Fubini-Study Bergman metrics sitting inside it? We restrict to toric varieties and prove the following result: Given a compact Riemannian manifold with boundary and a smooth map from its boundary into the space of toric Kahler metrics there exists a harmonic map from the manifold with these boundary values and, up to the first two derivatives, it is the limit of harmonic maps from the Riemannian manifold into the spaces of Bergman metrics. This generalizes previous work of Song-Zelditch on geodesics in the space of toric Kahler metrics. In the second part of the Thesis we propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamical systems on certain infinite-dimensional spaces. We illustrate the fruitfulness of this approach in the context of the Ricci flow as well as another flow on the space of Kahler metrics. We introduce and study dynamical systems related to the Ricci operator on the space of Kahler metrics that arise as discretizations of these flows. As an application, we address several questions in Kahler geometry related to canonical metrics, energy functionals, the Moser-Trudinger-Onofri inequality, Nadel-type multiplier ideal sheaves, and the structure of the space of Kahler metrics.

We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation σ = σ₁σ₂...σ_n defined as the set of indices i such that either i is odd and σ_i > σ_i+1, or i is even and σ_i < σ_i+1. We show that this statistic is equidistributed with the 3-descent set statistic on permutations ⌡ = σ₁σ₂...σ_n+1 with σ₁ = 1, defined to be the set of indices i such that the triple σ_iσ_i+1σ_i+2 forms an odd permutation of 3. We then introduce Mahonian inversion statistics corresponding to the two new variations of descents and show that the joint distributions of the resulting descent-inversion pairs are the same. We examine the generating functions involving alternating Eulerian polynomials, defined by analogy with the classical Eulerian polynomials using alternating descents. By looking at the number of alternating inversions in alternating (down-up) permutations, we obtain a new q- analog of the Euler number E_n and show how it emerges in a q-analog of an identity expressing E_n as a weighted sum of Dyck paths.

We also discuss polytopes relevant to the descent statistic. One such polytope is a "signed" version of the Pitman-Stanley parking function polytope, which can be viewed as a generalization of the chain polytope of the zigzag poset. We also discuss the family of descent polytopes, also known as order polytopes of ribbon posets, giving ways to compute their ƒ-vectors and looking further into their combinatorial structure.

The asymptotic behavior for the spectral measure of Kähler manifold has been studied by many authors in the context of Kähler quantization. It is well known that the spectral measure has an asymptotic expansion, while the coefficients of this expansion are not known even for very simple examples. In this thesis we study the spectral properties of Kähler manifold assuming the existence of some symmetry, i.e., a Hamiltonian action.

The main tool we will use is a function which we call the stability function. Roughly speaking, it is the function which compares quantum states before reduction with quantum states after reduction. We will study this function in detail, compute the function for many classes of Kähler manifolds, and apply it to study various spectral problems on Kähler quotients.

As for the spectral measure, we will give an explicit way to compute the coefficients in the asymptotic expansion for toric varietites. It turns out that the upstairs spectral measure in this case is described by an interesting integral transform which we will call the twisted Mellin transfrom. We will study both analytic and combinatorial aspects of this transfrom in the beginning of this thesis.

Preprojective algebras II_Q of quivers Q were introduced by Gelfand and Ponomarev in 1979 in order to provide a model of quiver representations (in the special case of finite Dynkin quivers). They showed that the Dynkin case, the preprojective algebra decomposes as the direct sum of all indecomposable representations of the quiver with multiplicity 1. Since then, preprojective algebras have found many other important applications, see e.g. to Kleinian singularities. In this thesis, I computed the Hochschild homology/cohomology of II_Q over C for quivers of type ADET, together with the cup product, and more generally, the calculus structure. I also computed the calculus structure for the centrally extended preprojective algebra, introduced by P. Etingof and E. Rains.

In this thesis, we apply combinatorial means for proving and generalizing classical determinantal identities. In Chapter 1, we present some historical background and discuss the algebraic framework we employ throughout the thesis. In Chapter 2, we construct a fundamental bijection between certain monomials that proves crucial for most of the results that follow. Chapter 3 studies the first, and possibly the best-known, determinantal identity, the matrix inverse formula, both in the commutative case and in some non-commutative settings (Cartier-Foata variables, right-quantum variables, and their weighted generalizations). We give linear-algebraic and (new) bijective proofs; the latter also give an extension of the Jacobi ratio theorem. Chapter 4 is dedicated to the celebrated MacMahon master theorem. We present numerous generalizations and applications. In Chapter 5, we study another important result, Sylvester's determinantal identity. We not only generalize it to non-commutative cases, we also find a surprising extension that also generalizes the master theorem. Chapter 6 has a slightly different, representation theory flavor; it involves representations of the symmetric group, and also Hecke algebras and their characters. We extend a result on immanants due to Goulden and Jackson to a quantum setting, and reprove certain combinatorial interpretations of the characters of Hecke algebras due to Ram and Remmel.