This thesis examines several aspects of reduced decompositions in finite Coxeter groups. Effort is primarily concentrated on the symmetric group, although some discussions are subsequently expanded to finite Coxeter groups of types B and D.

In the symmetric group, the combined frameworks of permutation patterns and reduced decompositions are used to prove a new characterization of vexillary permutations. This characterization and the methods used yield a variety of new results about the structure of several objects relating to a permutation. These include its commutation classes, the corresponding graph of the classes, the zonotopal tilings of a particular polygon, and a poset defined in terms of these tilings. The class of freely braided permutations behaves particularly well, and its graphs and posets are explicitly determined.

The Bruhat order for the symmetric group is examined, and the permutations with boolean principal order ideals are completely characterized. These form an order ideal which is a simplicial poset, and its rank generating function is computed. Moreover, it is determined when the set of permutations avoiding a particular set of patterns is an order ideal, and the rank generating functions of these ideals are computed. The structure of the intervals and order ideals in this poset is elucidated via patterns, including progress towards understanding the relationship between pattern containment and subintervals in principal order ideals.

The final discussions of the thesis are on reduced decompositions in the finite Coxeter groups of types B and D. Reduced decompositions of the longest element in the hyperoctahedral group are studied, and expected values are calculated, expanding on previous work for the symmetric group. These expected values give a quantitative interpretation of the effects of the Coxeter relations on reduced decompositions of the longest element in this group. Finally, the Bruhat order in types B and D is studied, and the elements in these groups with boolean principal order ideals are characterized and enumerated by length.

Let G be a group and A be a ring. There is a stable equivalence of orthogonal spectra

between the topological Hochschild homology of the group algebra A[G] and the smash product of the topological Hochschild homology of A and the cyclic bar construction of G. This thesis generalizes this result to a twisted group algebra A^T[G]. As an A-module, A^T[G] = A[G], but the multiplication is given by ag ⋅ a′ g ′ = ag(a′) ⋅ gg′, where G acts on A from the left through ring automorphisms. The main result is given in terms of a variant THH^g(A) of the topological Hochschild spectrum that is equipped with a twisted cyclic structure inherited from the cyclic structure of the cyclic pointed space THH(A)[—]. We first define a parametrized orthogonal spectrum E(A,G) over the cyclic bar construction N^cy(G). We prove there is a stable equivalence of spectra between the associated Thom spectrum of E(A,G) and THH(A^T[G]). We then prove there is a stable equivalence of orthogonal spectra

where the wedge-sum on the left hand side ranges over the conjugacy classes of elements of G and the equivalence depends on a choice of representative
g ∈ ⟨ g ⟩ of every conjugacy class of elements in G.

In this thesis, we study the Ehrhart polynomials of different polytopes. In the 1960's Eugene Ehrhart discovered that for any rational d-polytope P; the number of lattice points, i(P,m), in the mth dilated polytope mP is always a quasi-polynomial of degree d in m; whose period divides the least common multiple of the denominators of the coordinates of the vertices of P. In particular, if P is an integral polytope, i(P,m) is a polynomial. Thus, we call i(P,m) the Ehrhart (quasi-)polynomial of P.

In the first part of my thesis, motivated by a conjecture given by De Loera, which gives a simple formula of the Ehrhart polynomial of an integral cyclic polytope, we define a more general family of polytopes, lattice-face polytopes, and show that these polytopes have the same simple form of Ehrhart polynomials. we also give a conjecture which connects my theorem to a well-known fact that the constant term of the Ehrhart polynomial of an integral polytope is 1. In the second part (joint work with Brian Osserman), we use Mochizuki's work in algebraic geometry to obtain identities for the number of lattice points in different polytopes. We also prove that Mochizuki's objects are counted by polynomials in the characteristic of the base field.

A crystal with a small miscut from a plane of symmetry results in a surface covered by steps of atomic height. In the absence of material deposition, crystal surfaces relax to become flat via the motion of steps, and can develop macroscopically flat regions called facets. For axisymmetric surface profiles, the steps are concentric circles with radii that satisfy a system of ODEs coupled because of step interactions. These "step-flow" ODEs have peculiar properties associated with the rapid motion of extremal steps and the formation of facets.

First, I will focus on predictions generated by the step-flow ODEs. These results concern properties of step bunching and - since crystals with an infinite number of steps have been studied frequently in the past - the effects of finite crystal height. In particular, I show that under certain conditions, step bunches can always form by choosing a suitable initial step configuration. The effect of finite crystal height is described quantitatively by tracking the facet expansion.

The second part of my talk concerns the numerical solution of these ODEs, as a means of studying the morphological relaxation of crystal surfaces. Two main issues arise from the step system. First, steps near the facet move much faster than other steps, meaning that using the same time resolution for all steps in the crystal is inefficient. Second, when steps cluster tightly together in bunches (step bunching), the ODEs become very stiff to integrate. I will present an integration algorithm which "detects" the formation of step bunches and rapidly moving steps, and tracks their positions without significantly compromising the efficiency of integration for the majority of steps.

Finally, I will talk about a PDE model of surface relaxation, focusing on the issue of boundary conditions. A boundary condition that incorporates the discreteness of steps at the facet is implemented and is shown to give good agreement with the step-flow data for a wide range of step-interaction parameter values.

In Part I we will study the quantifier rank spectrum of sentences of L_w1,w. We will show that there are scattered sentences with models of arbitrarily high but bounded quantifier rank. We will also consider the case of weakly and almost scattered sentences as well as suggest some conjectures.

In Part II we will look at a new method of induction in the case of sheaves. We will then use this method to generalize the classical proof of the Suslin-Kleene Separation Theorem to the context of sheaves on a partial Grothendieck topology.

Vaught's Conjecture questions the number of models of a complete, countable theory and is one of the questions that have shaped modern model theory. Since it was posed by Vaught in 1959, partial results have been achieved by analyzing the Scott Hierarchy.

In Morley's ground-breaking paper giving a positive result towards Vaught's Conjecture, he introduces the notion of scattered theories. Sacks uses a generalization of these theories, called weakly-scattered theories, to produce further results. He introduces a tree hierarchy, called the Raw Hierarchy, which is similar to the Scott Hierarchy and enumerates the models of a weakly scattered theory.

We effectively construct a predecessor function for the type definitions in the Raw Hierarchy for any weakly-scattered theory. Using this predecessor function, we improve a recent bounding result by Sacks for weakly-scattered theories.

The algebraic structure of the K-theory of a topological space is described by the more general notion of a lambda ring. We show how computations in a lambda ring are facilitated by the use of Adams operations, which are ring homomorphisms, and apply this principle to understand the algebraic structure.

In a torsion free ring the Adams operations completely determine the lambda ring. This principle can be used to determine the K-theory of an infinite loop space functorially in terms of the K-theory of the corresponding spectrum. In particular we obtain a description of the K-theory of the infinite loop space tmf in terms of Katz's ring of divided congruences of modular forms. At primes greater than 3 we can also relate this to a Hecke algebra.

In this thesis, we describe the asymptotic behavior of complete Ricci-flat Kahler metrics on open manifolds that can be compactified by adding a smooth, ample divisor. This result provides an answer to a question addressed by Tian and Yau in \cite{TY1}, therefore refining the main result in that paper.

In this thesis, I consider hyper-Kähler manifolds of complex dimension 4 which are fibrations. It is known that the fibers are abelian varieties and the base is P². We assume that the general fiber is isomorphic to a product of two elliptic curves. We are able to relate this class of hyper-Kähler fibrations to already known examples. We prove that such a hyper-Kähler manifold is deformation equivalent to a Hilbert scheme of two points on a K3 surface.

We study degenerated complex Monge-Ampere equations over Projective manifolds. The degeneration is mainly on the cohomology class which is K\ahler in classic cases. Our main results concern the case when this class is semi-ample and big with certain generalizations to more general cases.

Two kinds of arguments are used. One is maximum principle argument. The other one makes use of pluripotential theory. The first two parts of this thesis are devoted to separated discussions using them. At last, we combine the efforts to give further applications. Future problems are also mentioned in the last part.

We define a new family of Gromov-Witten type invariants based on intersection theory on the moduli space of pseudoholomorphic curves of arbitrary genus with boundary in a Lagrangian submanifold. We assume the Lagrangian submanifold arises as the fixed points of an anti-symplectic involution. These invariants recover Welschinger's invariants in genus 0. Furthermore, we calculate the invariant for the real quintic threefold in genus 0 and degree 1 to be 30.

We define and study the structure of SUSY Lie conformal and vertex algebras. This leads to effective rules for computations with superfields. Given a strongly conformal SUSY vertex algebra V and a supercurve X, we construct a vector bundle V ^r_X on X, the fiber of which, is isomorphic to V. Moreover, the state-field correspondence of V canonically gives rise to (local) sections of these vector bundles. We also define chiral algebras on any supercurve X, and show that the vector bundle V ^r_X, corresponding to a SUSY vertex algebra, carries the structure of a chiral algebra.

Long ago, Mahowald computed the ko homology of the classifying space of the group ∑ ₂. This computation has has a number of interesting and important ramifications, as it has been used in such varied contexts as demonstrating the survival of an infinite family of elements in the stable stem readily expressible in the Adams spectral sequence.

In this talk, we will generalize this result to all primes, indicating how the compute the eo _p—1 homology of Β∑ _p at the prime p. We will show that such groups are computable using an Adams spectral sequence which naturally generalizes that for ko homology, and we will prove that the Hopf algebras involved are of the form conjectured by the author and Andre Henriques.

This thesis studies surfaces in $\mathbb{R}^3$ which minimize area, subject to a fixed boundary and to a free boundary with length constraint. Based on physical experiments, I make two conjectures. First, I conjecture that minimizers supported on generic wires have finitely many surface components. I approach this conjecture by proving that surface components of near-wire minimizers are Lipschitz graphs in wire Frenet coordinates, and appear near maxima of wire curvature. Second, I conjecture and prove that surface components of near-wire minimizers are $C^1$ at corners where the thread touches the wire interior. Moreover, the limit of the surface normal field is the Frenet binormal of the wire at the corner point. This shows local wire geometry dominates global wire geometry in influencing the surface corner. Third, I show that these two conjectures are related: assuming additional regularity up to the corner, the finiteness conjecture follows.

Let A be an A ∞ ring spectrum. We give an explicit construction of topological Hochschild homology and cohomology of A using the Stasheff associahedra and another family of polyhedra called cyclohedra. Using this construction we can then study how THH(A) varies over the moduli space of A ∞ structures on A, a problem which seems largely intractible using strictly associative replacements of A. We study how topological Hochschild cohomology of any 2-periodic Morava K-theory varies over the moduli space of A∞ structures and show that in the generic case, when a certain matrix describing the multiplication is invertible, the result is the corresponding E-theory. If this matrix is not invertible, the result is some finite extension of Morava E-theory, and exactly which extension we get depends on the A ∞ structure.

To make sense of our constructions, we first set up a general framework for enriching a subcategory of the category of noncommutative sets over a category C using products of the objects of a non-∑ operad P in C. By viewing the simplicial category as a subcategory of the category of noncommutative sets in two different ways, we obtain two generalizations of simplicial objects. For the operad given by the Stasheff associahedra we obtain a model for the 2-sided bar construction in the first case and the cyclic bar and cobar construction in the second case. Using either the associahedra or the cyclohedra in place of the geometric simplices we can define the geometric realization of these objects.

Binary search trees (BSTs) are a class of simple data structures used to store and access keys from an ordered set. They have been around for about half a century. Despite their ubiquitous use in practical programs, surprisingly little is known about their optimal performance. No polynomial algorithm is known to compute the best BST for a given sequence of key accesses, and before our work, no sub O(log(n))-competitive online BST data structures were known to exist.

In this thesis, we describe tango trees, a novel BST data structure that is O(log log n)-competitive. We also describe a new geometric problem equivalent to computing optimal offline BSTs. This model lets us prove that a linear access algorithm described by Munro in 2000 is optimal within a constant factor. We then use the geometric model to describe a new class of lower bounds that includes both of the major earlier lower bounds for the performance of offline BSTs, and construct an optimal bound in this new class.

We report the results of a combined experimental and theoretical investigation of the dynamics of a journal bearing rolling down an incline under the influence of gravity; specifically, we examined a cylinder suspended in a viscous fluid housed within a cylindrical shell. Particular attention is given to elucidating the physical process that causes the bearing to roll down a ramp much slower than its solid counterpart. We performed a series of experiments in which the inner cylinder density and the fluid viscosity were varied. Three distinct types of behavior we observed: slight rocking without rolling, slow rolling, and accelerating without bound. A hydrodynamic model is developed and then simplified in the lubrication limit. The rocking and accelerating modes are readily observed in our model. In the rocking solutions, potential and kinetic energy are dissipated in the fluid as the inner cylinder approaches its equilibrium position. In the accelerating solutions, the whole system moves as a solid body so that no dissipation can occur and all potential energy is converted into kinetic energy. The model cannot predict the slow rolling motion because it does not include any restoring force on the inner cylinder that allows potential energy to be converted into heat without removing all of the kinetic energy by holding it a fixed distance away from the outer wall. To rationalize this slow rolling motion, we incorporate surface roughness and cavitation into the model. Both of these effects provide a restoring force on the inner cylinder that holds it away from the outer wall and so provides a bound on the heat dissipation. Finally, our analysis indicates that the forces generated by cavitation are not sufficiently large to create the observed motion, thereby implying that surface roughness is the more likely cause.