This thesis is devoted to application of topological ideas to Galois theory. In the fi rst part we obtain a characterization of branching data that guarantee that a regular mapping from a Riemann surface to the Riemann sphere having this branching data is invertible in radicals. The mappings having such branching data are then studied with emphasis on those exceptional properties of these mappings that single them out among all mappings from a Riemann surface to the Riemann sphere. These results provide a framework for understanding an earlier work of Ritt on rational functions invertible in radicals. In the second part of the thesis we apply topological methods to prove lower bounds in Klein's resolvent problem, i.e. the problem of determining whether a given algebraic function of n variables is a branch of a composition of rational functions and an algebraic function of k variables. The main topological result here is that the smallest dimension of the base-space of a covering from which a given covering over a torus can be induced is equal to the minimal number of generators of the monodromy group of the covering over the torus. This result is then applied for instance to prove the bounds k is at least n/2 in Klein's resolvent problem for the universal algebraic function of degree n and the answer k = n for generic algebraic function of n variables of degree at least 2n.

Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / In the endeavor to study noncommutative algebraic geometry, Alex Rosenberg defined in [13] the spectrum of an Abelian category. This spectrum generalizes the prime spectrum of a commutative ring in the sense that the spectrum of the Abelian category R − mod is homeomorphic to the prime spectrum of R. This spectrum can be seen as the beginning of “categorical geometry”, and was used in [15] to study noncommutative algebriac geometry. In this thesis, we are concerned with geometries extending beyond traditional algebraic geometry coming from the algebraic structure of rings. We consider monoids in a monoidal category as the appropriate generalization of rings–rings being monoids in the monoidal category of Abelian groups. Drawing inspiration from the definition of the spectrum of an Abelian category in [13], and the exploration of it in [15], we define the spectrum of a monoidal category, which we will call the monoidal spectrum. We prove a descent condition which is the mathematical formalization of the statment “R − mod is the category of quasi-coherent sheaves on the monoidal spectrum of R − mod”. In addition, we prove a functoriality condidition for the spectrum, and show that for a commutative Noetherian ring, the monoidal spectrum of R − mod is homeomorphic to the prime spectrum of the ring R. In [1], Paul Balmer defined the prime tensor ideal spectrum of a tensor triangulated cat- gory; this can be thought of as the beginning of “tensor triangulated categorical geometry”. This definition is very transparent and digestible, and is the inspiration for the definition in this thesis of the prime tensor ideal spectrum of an monoidal Abelian category. It it shown that for a polynomial identity ring R such that the catgory R − mod is monoidal Abelian, the prime tensor ideal spectrum is homeomorphic to the prime ideal spectrum.

Doctor of Philosophy / Mathematics / Ilia Zharkov / A smooth affine hypersurface of complex dimension n is homotopy equivalent to a real n-dimensional cell complex. We describe a recipe of constructing such cell complex for the hypersurfaces of dimension 1 and 2, i.e. for curves and surfaces. We call such cell complex a skeleton of the hypersurface. In tropical geometry, to each hypersurface, there is an associated hypersurface, called tropical hypersurface given by degenerating a family of complex amoebas. The tropical hypersurface has a structure of a polyhedral complex and it is a base of a torus fibration of the hypersurface constructed by Mikhalkin. We introduce on the edges of a tropical hypersurface an orientation given by the gradient flow of some piece-wise linear function. With the help of this orientation, we choose some sections and fibers of the fibration.These sections and fibers constitute a cell complex and we prove that this complex is the skeleton by using decomposition of the coemoeba of a classical pair-of-pants. We state and prove our main results for the case of curves and surfaces in Chapters 4 and 5.

Master of Science / Department of Mathematics / Andrew G. Bennett / An online mathematics placement exam was administered to new freshmen enrolled at Kansas State University for the Fall of 2009. The purpose of this exam is to help determine which students are prepared for a college Calculus I or Calculus II course. Problems on the exam were analyzed and grouped together using different techniques including expert analysis and item response theory to determine which problems were similar or even relevant to placement. Student scores on the exam were compared to their performance on the final exam at the end of the course as well as ACT data. This showed how well the placement exam indicated which students were prepared. A model was created using ACT information and the new information from the placement exam that improved prediction of success in a college calculus course. The new model offers a significant improvement upon what the ACT data provides to advisers. Suggestions for improvements to the test and methodology are made based upon the analysis.

Master of Science / Department of Mathematics / Marianne Korten / Distribution theory is an important tool in studying partial differential equations. Distributions are linear functionals that act on a space of smooth test functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. There are different possible choices for the space of test functions, leading to different spaces of distributions. In this report, we take a look at some basic theory of distributions and their Fourier transforms. And we also solve some typical exercises at the end.

Master of Science / Department of Mathematics / Ivan Blank / Charles N. Moore / This paper will examine some mathematical properties and models of inhomogeneous materials. By deriving models for elastic energy and heat flow we are able to establish equations that arise in the study of divergence form uniformly elliptic partial differential equations. In the late 1950's DeGiorgi and Nash showed that weak solutions to our partial differential equation lie in the Holder class. After fixing the dimension of the space, the Holder exponent guaranteed by this work depends only on the ratio of the eigenvalues. In this paper we will look at a specific geometry and show that the Holder exponent of the actual solutions is bounded away from zero independent of the eigenvalues.

In this thesis we use a diffeo-geometric framework based on manifolds hat are locally modeled on ``convenient'' vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold M, we construct a weak symplectic structure on each leaf I_w of a foliation of the space of compact oriented isotropic submanifolds in M equipped with top degree forms of total measure 1. These forms are called weightings and such manifolds are said to be weighted. We show that this symplectic structure on the particular leaves consisting of weighted Lagrangians is equivalent to a heuristic weak symplectic structure of Weinstein. When the weightings are positive, these symplectic spaces are symplectomorphic to reductions of a weak symplectic structure of Donaldson on the space of embeddings of a fixed compact oriented manifold into M. When M is compact, by generalizing a moment map of Weinstein we construct a symplectomorphism of each leaf I_w consisting of positive weighted isotropics onto a coadjoint orbit of the group of Hamiltonian symplectomorphisms of M equipped with the Kirillov-Kostant-Souriau symplectic structure. After defining notions of Poisson algebras and Poisson manifolds, we prove that each space I_w can also be identified with a symplectic leaf of a Poisson structure. Finally, we discuss a kinematic description of spaces of weighted submanifolds.

This thesis studies the pre-quantization of quasi-Hamiltonian group actions from a cohomological viewpoint. The compatibility of pre-quantization with symplectic reduction and the fusion product are established, and are used to understand the necessary and sufficient conditions for the pre-quantization of M(G,S), the moduli space of at flat G-bundles over a closed surface S. For a simply connected, compact, simple Lie group G, M(G,S) is known to be pre-quantizable at integer levels. For non-simply connected G, however, integrality of the level is not sufficient for pre-quantization, and this thesis determines the obstruction, namely a certain 3-dimensional cohomology class, that places further restrictions on the underlying level. The levels that admit a pre-quantization of the moduli space are determined explicitly for all non-simply connected, compact, simple Lie groups G. Partial results are obtained for the case of a surface S with marked points. Also, it is shown that via the bijective correspondence between quasi-Hamiltonian group actions and Hamiltonian loop group actions, the corresponding notions of prequantization coincide.

Doctor of Philosophy / Department of Mathematics / David Yetter / In this paper we give a new generalization of the Khovanov homology. The construction begins with a Frobenius-algebra-like object in a category of graded vector-spaces with an anyonic braiding, with most of the relations weaken to hold only up to phase. The construction of Khovanov can be adapted to give a new link homology theory from such data. Both Khovanov's original theory and the odd Khovanov homology of Oszvath, Rassmusen and Szabo arise from special cases of the construction in which the braiding is a symmetry.

Master of Science / Department of Mathematics / Natasha Rozhkovskaya / “Incomplete open cubes” is one of the major projects of the artist Sol Lewitt. It consists of a collection of frame structures and a presentation of their diagrams. Each structure in the project is a cube with some edges removed so that the structure remains three-dimensional and connected Structures are considered to be identical if one can be transformed into another by a space rotation (but not reflection). The list of incomplete cubes consists of 122 structures. In this project, the concept of incomplete cubes was formulated in the language of graph theory. This allowed us to compare the problem posed by the artist with the similar questions of graph theory considered during the last decades. Classification of Incomplete cubes was then refined using the language of combinatorics. The list produced by the artist was then checked to be complete. And lastly, properties of Incomplete cubes in the list were studied.