We give a description of the moment graph for Bott-Samelson varieties in arbitrary Lie type. We use this, along with curve neighborhoods and explicit moduli space computations, to compute a presentation for the small quantum cohomology ring of a particular Bott-Samelson variety in Type A. / Ph. D. / Since the early 1990’s, the study of quantum cohomology has been a fascinating, and fruitful field of research with connections to physics, representation theory, and combinatorics. The quantum cohomology of a space X encodes enumerative information about how many curves intersect certain subspaces of X; these counts are called Gromov-Witten invariants. For some spaces X, including the class of spaces we consider here, this count is only ”virtual” and negative Gromov-Witten invariants may arise. In this dissertation, we study the quantum cohomology of Bott-Samelson varieties. These spaces arise frequently in applications to representation theory and combinatorics, however their quantum cohomology was previously unexplored. The first of our three main theorems describes the moment graph for Bott-Samelson varieties. This is a description of what all the possible curves, stable under certain symmetries, exist in a Bott-Samelson variety. Our second main theorem is a technical result which enables us to compute some GromovWitten invariants directly. Finally, our third main theorem is a description of the quantum cohomology for a certain three-dimensional Bott-Samelson variety.

Let X be the affine flag manifold of Lie type A_n-1⁽¹⁾ where n ≥ 3 and let W_aff be the associated affine Weyl group. The moment graph for X encodes the torus fixed points (which are elements of the affine Weyl group W_aff and the torus stable curves in X. Given a fixed point u ∈ W_aff and a degree d = (d₀, d₁, ..., d_n−1) ∈ ℤ_≥0ⁿ, the combinatorial curve neighborhood is the set of maximal elements in the moment graph of X which can be reached from u′ ≤ u by a chain of curves of total degree ≤ d. In this thesis we give combinatorial formulas and algorithms for calculating these elements. / Doctor of Philosophy / The study of curves on flag manifolds is motivated by questions in enumerative geometry and physics. To a space of curves and incidence conditions one can associate a combinatorial object called the ‘combinatorial curve neighborhood’. For a fixed degree d and a (Schubert) cycle, the curve neighborhood consists of the set of all elements in the Weyl group which can be reached from the given cycle by a path of fixed degree in the moment graph of the flag manifold, and are also Bruhat maximal with respect to this property. For finite dimensional flag manifolds, a description of the curve neighborhoods helped answer questions related to the quantum cohomology and quantum K theory rings, and ultimately about enumerative geometry of the flag manifolds. In this thesis we study the situation of the affine flag manifolds, which are infinite dimensional. We obtain explicit combinatorial formulas and recursions which characterize the curve neighborhoods for flag manifolds of affine Lie type A. Among the conclusions, we mention that, unlike in the finite dimensional case, the curve neighborhoods have more than one component, although all components have the same length. In general, calculations reflect a close connection between the curve neighborhoods and the Lie combinatorics of the affine root system, especially the imaginary roots.