The first project deals with generalizing the definition of zeroth derived functors to work for any abelian category. The classical definitions of zeroth derived functors require existence of injectives or pro jectives. In this project, we give definitions of the zeroth derived functors that do not require the existence of injectives or projectives. The new definitions result in generalized definitions of projective and injective stabilization of functors. The category of coherent functors is shown to admit a zeroth right derived functor. An interesting result of this fact is a counterpart to the Yoneda lemma...
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*E*_{6}, *F*_{4}, *G*_{2}, and select cases of type *E*_{7} to confirm the shape of the expected resolutions as well as some geometric properties of the orbit closures.

In the first part of this thesis, using the parametrization of cluster variables by their g-vectors explicitly computed by S.-W. Yang and A. Zelevinsky, we extend the original construction of generalized associahedra by F. Chapoton, S. Fomin and A. Zelevinsky to any choice of acyclic initial cluster, and compare it to the one given by C. Hohlweg, C. Lange, and H. Thomas...
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We investigate the properties of coordinate rings of orbit closures for quivers of type $A_3$ by considering the desingularization given by Reineke. We construct explicit minimal free resolutions of the defining ideals of the orbit closures thus giving us a minimal set of generators for the defining ideal. The resolution allows us to read off some...
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In Part I, we study the simple Lie group of type $G_2$ from the cluster point of view. We prove a conjecture of Geiss, Leclerc and Schröer, relating the geometry of the partial flag varieties to cluster algebras in the case of $G_2$.

In Part II, we establish a concrete relationship between certain infinite dimensional quantum groups, namely the quantum loop algebra $U_{ \hbar}(L \mathfrak{g})$ and the Yangian $Y_{ \hbar}(\mathfrak{g})$, associated with a simple Lie algebra $\mathfrak{g}$....
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**In this dissertation we prove local existence and uniqueness of solutions of the focusing modified Korteweg - de Vries equation u_{t} + u2u_{x} + u_{xxx} = 0 in classes of unbounded functions that admit an asymptotic expansion at infinity in decreasing powers of x. We show that an asymptotic solution differs from a genuine solution by a smooth function that is of Schwartz class with respect to x and that solves a generalized version of the focusing mKdV equation. The latter equation is solved by discretization methods.**

In Part I we associate to each ideal triangulation of a bordered surface with marked points a quiver with potential, in such a way that whenever two ideal triangulations are related by a flip of an arc, the respective quivers with potentials are related by a mutation with respect to the flipped arc. We prove that if the surface has non-empty boundary, then the quivers with potentials...
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Based on this realization, we give combinatorial formulas for $F$-polynomials in cluster algebras of classical types in terms of the weighted paths in certain directed graphs. As a consequence we prove the positivity of $F$-polynomials...
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Both the comparability graph and the Hasse diagram for a polytope inherit nicely the rank function associated with a polytope. Using this rank function, we can study specific subgraphs of each graph, by restricting to vertices...
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