In our thesis we study the algebras of differential operators in algebraic and geometric terms. We consider two problems in which the algebras of differential operators naturally arise. The first one deals with the algebraic structure of differential and pseudodifferential operators. We define the Krichever-Novikov type Lie algebras of differential operators and pseudodifferential symbols on Riemann surfaces, along with their outer derivations and central extensions. We show that the corresponding algebras of meromorphic differential operators and pseudodifferential symbols have many invariant traces and central extensions, given by the logarithms of meromorphic vector fields. We describe which of these extensions survive after passing to the algebras of operators and symbols holomorphic away from several fixed points. We also describe the associated Manin triples, emphasizing the similarities and differences with the case of smooth symbols on the circle. The second problem is related to the geometry of differential operators and its connection with representations of semi-simple Lie algebras. We show that the semiregular module, naturally associated with a graded semi-simple complex Lie algebra, can be realized in geometric terms, using the Brion's construction of degeneration of the diagonal in the square of the flag variety. Namely, we consider the Beilinson-Bernstein localization of the semiregular module and show that it is isomorphic to the D-module obtained by applying the Emerton-Nadler-Vilonen geometric Jacquet functor to the D-module of distributions on the square of the flag variety with support on the diagonal.

This thesis studies orbit equivalence in the almost continuous setting. Recently A. del Junco and A. Sahin obtained an almost continuous version of Dye’s theorem. They proved that any two ergodic measure-preserving homeomorphisms of Polish spaces are almost continuously orbit equivalent. One purpose of this thesis is to extend their result to all free actions of countable amenable groups. We also show that the cocycles associated with the constructed orbit equivalence are almost continuous. In the second part of the thesis we obtain an analogue of Dye’s reconstruction theorem for etale equivalence relations in the almost continuous setting. We introduce topological full groups of etale equivalence relations and show that if the topological full groups are isomorphic, then the equivalence relations are almost continuously orbit equivalent.

The first part of this thesis investigates the Gromov width of maximal dimensional coadjoint orbits of compact simple Lie groups. An upper bound for the Gromov width is provided for all compact simple Lie groups but only for those coadjoint orbits that satisfy a certain technical assumption, whereas the lower bound is proved only for groups of type A, but without the technical restriction. The two bounds use very different techniques: the proof of the upper bound uses more analytical tools, while the proof of the lower bound is more geometric. The second part of the thesis is a short report on a joint project with my supervisor, which was concerned with the relationship between two different definitions of orbifolds: one using Lie groupoids and the other involving diffeologies. The results are summarized in Chapter 5 of this text.

Any genus $g$ surface, $\Sigma_{g,n},$ with $n$ boundary components may be given a trinion decomposition: a realization of the surface as a union of $2g-2+n$ trinions glued together along $3g-3+n$ of their boundary circles. Together with the flows of Goldman, Jeffrey and Weitsman use the trinion boundary circles in a decomposition of $\Sigma_{g,n}$ to obtain a Hamiltonian action of a compact torus $(S^1)^{3g-3+n'} $ on an open dense subset of the moduli space of certain gauge equivalence classes of flat $SU(2)-$connections on $\Sigma_{g,n}.$ Jeffrey and Weitsman also provide a complete description of the moment polytopes for these torus actions, and we make use of this description to study the cohomology of associated toric varieties. While we are able to make use of the work of Danilov to obtain the integral (rational) cohomology ring in the smooth (orbifold) case, we show that the aforementioned toric varieties almost always possess singularities worse than those of an orbifold. In these cases we use an algorithm of Bressler and Lunts to recover the intersection cohomology Betti numbers using the combinatorial information provided by the corresponding moment polytopes. The main contribution of this thesis is a computation of the intersection cohomology Betti numbers for the toric varieties associated to trinion decomposed surfaces $\Sigma_{2,0},\Sigma_{2,1},\Sigma_{3,0}, \Sigma_{3,1}, \Sigma_{4,0},$ and $\Sigma_{4,1}.$

The PR-tree data structure is introduced to characterize the sets of path-tree models of path graphs. We further characterize the sets of directed path-tree models of directed path graphs with a slightly restricted form of the PR-tree called the Strong PR-tree. Additionally, via PR-trees and Strong PR-trees, we characterize path graphs and directed path graphs by their Split Decompositions. Two distinct approaches (Split Decomposition and Reduction) are presented to construct a PR-tree that captures the path-tree models of a given graph G = (V, E) with n = |V| and m = |E|. An implementation of the split decomposition approach is presented which runs in O(nm) time. Similarly, an implementation of the reduction approach is presented which runs in O(A(n + m)nm) time (where A(s) is the inverse of Ackermann’s function arising from Union-Find [40]). Also, from a PR-tree, an algorithm to construct a corresponding Strong PR-tree is given which runs in O(n + m) time. The sizes of the PR-trees and Strong PR-trees produced by these approaches are O(n + m) with respect to the given graph. Furthermore, we demonstrate that an implicit form of the PR-tree and Strong PR-tree can be represented in O(n) space.

In this thesis we develop the notion of LA-Courant algebroids, the infinitesimal analogue of multiplicative Courant algebroids. Specific applications include the integration of q- Poisson (d, g)-structures, and the reduction of Courant algebroids. We also introduce the notion of pseudo-Dirac structures, (possibly non-Lagrangian) subbundles W ⊆ E of a Courant algebroid such that the Courant bracket endows W naturally with the structure of a Lie algebroid. Specific examples of pseudo-Dirac structures arise in the theory of q-Poisson (d, g)-structures.

Let G be a simple algebraic group. Labelled trivalent graphs called webs can be used to produce invariants in tensor products of minuscule representations. For each web, a configuration space of points in the affine Grassmannian is constructed. This configuration space gives a natural way of calculating the invariant vectors coming from webs. In the case of G = SL_3, non-elliptic webs yield a basis for the invariant spaces. The non-elliptic condition, which is equivalent to the condition that the dual diskoid of the web is CAT(0), is explained by the fact that affine buildings are CAT(0). In the case of G = SL_n, a sufficient condition for a set of webs to yield a basis is given. Using this condition and a generalization of a technique by Westbury, a basis is constructed for SL_n. Due to the geometric Satake correspondence there exists another natural basis of invariants, the Satake basis. This basis arises from the underlying geometry of the affine Grassmannian. There is an upper unitriangular change of basis from the basis constructed above to the Satake basis. An example is constructed showing that the Satake, web and dual canonical basis of the invariant space are all different. The natural action of rotation on tensor factors sends invariant space to invariant space. Since the rotation of web is still a web, the set of vectors coming from webs is fixed by this action. The Satake basis is also fixed, and an explicit geometric and combinatorial description of this action is developed.

Let $F$ be a non-Archimedean local field and $G$ be the general linear group $\mathrm{GL}_n$ over $F$. Bushnell and Henniart described the essentially tame local Langlands correspondence of $G(F)$ using rectifiers, which are certain characters defined on tamely ramified elliptic maximal tori of $G(F)$. They obtained such result by studying the automorphic induction character identity. We relate this formula to the spectral transfer character identity, based on the theory of twisted endoscopy of Kottwitz, Langlands, and Shelstad. In this article, we establish the following two main results. (i) To show that the automorphic induction character identity is equal to the spectral transfer character identity when both are normalized by the same Whittaker data. (ii) To express the essentially tame local Langlands correspondence using admissible embeddings constructed by Langlands-Shelstad $\chi$-data and to relate Bushnell-Henniart's rectifiers to certain transfer factors.

One of the most famous problem of finding optimal weight to maximize an agent's expected terminal utility in finance literature is Merton's optimal portfolio problem. Classic solution to this problem is given by stochastic Hamilton-Jacobi-Bellman Equation where we briefly review it in chapter 1. Similar idea has found many applications in other finance literatures and we will focus on its application to the high-frequency trading using limit orders in this thesis. In [1], major analysis using the constant volatility arithmetic Brownian motion stock price model with exponential utility function is described. We re-analyze the solution of HJB equation in this case using different asymptotic expansion. And then, we extend the model to the regime-switching volatility model to capture the status of market more accurately.

The bounded Fatou components for certain quadratic polynomials are attached to each other at the boundary and form chain-like structures called ``bubble rays". In the context of mating quadratic polynomials, these bubble rays can serve as a replacement for external rays. The main objective of this thesis is to apply this idea to the mating of starlike quadratics.