The subject of fractal geometry has exploded over the past 40 years with the availability of computer generated images. It was seen early on that there are many interesting questions at the intersection of probability and fractal geometry. In this dissertation we will introduce two random models for constructing fractals and prove various facts about them.

The invariant theory of finite reflection groups has rich connections to geometry, topology, representation theory, and combinatorics. We consider finite reflection groups acting on vector spaces over fields of arbitrary characteristic, where many arguments of classical invariant theory break down. When the characteristic of the underlying field is positive, reflections may be nondiagonalizable. A group containing these so-called transvections has order which is divisible by the characteristic of the underlying field, so is in the modular setting. In this thesis, we examine the action on differential derivations, which include products of differential forms and derivations, and identify the structure of the set of invariants under the action of groups fixing a single hyperplane, groups with maximal transvection root spaces acting on vector spaces over prime fields, as well as special linear groups and general linear groups over finite fields.

The superline has one even and one odd coordinate. We consider the Lie superalgebra of contact vector fields on the superline. Its tensor density modules are a one-parameter family of deformations of the natural action on the ring of polynomials on the superline. They are parameterized by a complex number, and they are irreducible when this parameter is not zero. In this dissertation, we describe the annihilating ideals of these representations in the universal enveloping algebra of this Lie superalgebra by providing their generators. We also describe the intersection of all such ideals: the annihilator of the direct sum of the tensor density modules. The annihilating ideal of an irreducible non-zero left module is called a primitive ideal, and the space of all such ideals in the universal enveloping algebra is its primitive spectrum. The primitive spectrum is endowed with the Jacobson topology, which induces a topology on the annihilators of the tensor density modules. We conclude our discussion with a description of the annihilators as a topological space.

The Hochschild cohomology of an associative algebra records information about the deformations of that algebra, and hence the first step toward understanding its deformations is an examination of the Hochschild cohomology. In this dissertation, we use techniques from homological algebra, invariant theory, and combinatorics to analyze the Hochschild cohomology of skew group algebras arising from finite cyclic groups acting on polynomial rings over fields of arbitrary characteristic. These algebras are the natural semidirect product of the group ring with the polynomial ring. Many families of algebras arise as deformations of skew group algebras, such as symplectic reflection algebras and rational Cherednik algebras. We give an explicit description of the Hochschild cohomology governing graded deformations of skew group algebras for cyclic groups acting on polynomial rings. For skew group algebras, a description of the Hochschild cohomology is known in the nonmodular setting (i.e., when the characteristic of the field and the order of the group are coprime). However, in the modular setting (i.e., when the characteristic of the field divides the order of the group), much less is known, as techniques commonly used in the nonmodular setting are not available.

We investigate non-geometric sharp permutation groups of type {0,k} whose point stabilizers are certain Frobenius groups. We show that if a point stabilizer has a cyclic Frobenius kernel whose order is a power of a prime and Frobenius complement cyclic of prime order, then the point stabilizer is isomorphic to the symmetric group on 3 letters, and there is up to permutation isomorphism, one such permutation group. Further, we determine a significant structural description of non-geometric sharp permutation groups of type {0,k} whose point stabilizers are Frobenius groups with elementary abelian Frobenius kernel K and Frobenius complement L with |L| = |K|-1. As a result of this structural description, it is shown that the smallest non-solvable Frobenius group cannot be a point stabilizer in a non-geometric sharp permutation group of type {0,k}.

We give some background on the free part of the action of tuples of integers into 2. We will construct specific structures on this space, and then show that certain other structures cannot exist.

The lattice point problem in dynamical systems investigates the distribution of certain objects with some length property in the space that the dynamics is defined. This problem in different contexts can be interpreted differently. In the context of symbolic dynamical systems, we are trying to investigate the growth of N(T), the number of finite words subject to a specific ergodic length T, as T tends to infinity. This problem has been investigated by Pollicott and Urbański to a great extent. We try to investigate it further, by relaxing a condition in the context of deterministic dynamical systems. Moreover, we investigate this problem in the context of random dynamical systems. The method for us is considering the Fourier-Stieltjes transform of N(T) and expressing it via a Poincaré series for which the spectral gap property of the transfer operator, enables us to apply some appropriate Tauberian theorems to understand asymptotic growth of N(T). For counting in the random dynamics, we use some results from probability theory.

In this lecture we introduce a new transfinite dimension function for metric spaces which utilizes Henderson's topological D-dimension and ascribes to any metric space either an ordinal number or the symbol Ω. The construction of our function is motivated by that of Urbański's transfinite Hausdorff dimension, tHD. Henderson's dimension is a topological invariant, however, like Hausdorff dimension and tHD the function presented will be invariant under bi-Lipschitz continuous maps and generally not under homeomorphisms. We present some original results on D-dimension and build the general theory for the D-variant of transfinite Hausdorff dimension, \mathrm{t}_D\mathrm{HD}. In particular, we will show for any ordinal number α, existence of a metrizable space which has \mathrm{t}_D\mathrm{HD} greater than or equal to α and less than or equal to \omega_\tau, where τ is the least ordinal which satisfies α < \omega_\tau.

We prove the existence and nonexistence of radial solutions of singular semilinear equations Δu + k(x)f(u)=0 with boundary condition on the exterior of the ball with radius R>0 in ℝ^N such that lim r →∞ u(r)=0, where f: ℝ \ {0} →ℝ is an odd and locally Lipschitz continuous nonlinear function such that there exists a β >0 with f <0 on (0, β), f >0 on (β, ∞), and K(r) ~ r^-α for some α >0.

Rational Catalan combinatorics connects various Catalan numbers to the representation theory of rational Cherednik algebras for Coxeter and complex reflection groups. Lewis, Reiner, and Stanton seek a theory of rational Catalan combinatorics for the general linear group over a finite field. The finite general linear group is a modular reflection group that behaves like a finite Coxeter group. They conjecture a Hilbert series for a space of invariants under the action of this group using (q,t)-binomial coefficients. They consider the finite general linear group acting on the quotient of a polynomial ring by iterated powers of the irrelevant ideal under the Frobenius map. Often conjectures about reflection groups are solved by considering the local case of a group fixing one hyperplane and then extending via the theory of hyperplane arrangements to the full group. The Lewis, Reiner and Stanton conjecture had not previously been formulated for groups fixing a hyperplane. We formulate and prove their conjecture in this local case.

For decades, mathematical games have been used to explore various properties of particular sets. The Banach-Mazur game is the prototypical intersection game and its modifications by e.g., W. Schmidt and C. McMullen are used in number theory and many other areas of mathematics. We give a brief survey of a few of these modifications and their properties followed by our own modification. One of our main results is proving that this modification is equivalent to an important set theoretic game, called the perfect set game, developed by M. Davis.

Schmidt's game, and other similar intersection games have played an important role in recent years in applications to number theory, dynamics, and Diophantine approximation theory. These games are real games, that is, games in which the players make moves from a complete separable metric space. The determinacy of these games trivially follows from the axiom of determinacy for real games,ADR, which is a much stronger axiom than that asserting all integer games are determined, AD. One of our main results is a general theorem which under the hypothesis AD implies the determinacy of intersection games which have a property allowing strategies to be simplified. In particular, we show that Schmidt's (α,β,ρ) game on R is determined from AD alone, but on Rn for n≥3 we show that AD does not imply the determinacy of this game. We then give an application of simple strategies and prove that the winning player in Schmidt's (α,β,ρ) game on R has a winning positional strategy, without appealing to the axiom of choice. We also prove several other results specifically related to the determinacy of Schmidt's game. These results highlight the obstacles in obtaining the determinacy of Schmidt's game from AD

We show that any Borel action on a standard Borel space of a group which is topologically isomorphic to the sum of a countable abelian group with a countable sum of lines and circles induces an orbit equivalence relation which is hypersmooth. We also show that any Borel action of a second countable locally compact abelian group on a standard Borel space induces an orbit equivalence relation which is essentially hyperfinite, generalizing a result of Gao and Jackson for the countable abelian groups.

The Lie algebra Vec(ℝ) of polynomial vector fields on the line acts naturally on ℂ[]. This action has a one-parameter family of deformations called the tensor density modules F_λ. The bounded indecomposable modules of Vec(ℝ) of length 2 composed of tensor density modules have been classified by Feigin and Fuchs. We present progress towards describing the annihilators of the unique indecomposable extension of F_λ by F_(λ+2) in the non-resonant case λ ≠ -½. We give the intersection of the annihilator and the subalgebra of lowest weight vectors of the universal enveloping algebra (Vec(ℝ)) of Vec(ℝ). This result is found by applying structural descriptions of the lowest weight vectors of (Vec(ℝ)).

First, we show that the Rothberger and 2-Rothberger games are equivalent. Then we adjust the former proof and introduce another game, the restricted Menger game, in order to obtain a broader result. This provides an answer in the context of Hausdorff spaces for an open question posed by Aurichi, Bella, and Dias.

Spreading models have become fundamental to the study of asymptotic geometry in Banach spaces. The existence of spreading models in every Banach space, and the so-called good sequences which generate them, was one of the first applications of Ramsey theory in Banach space theory. We use Ramsey theory and other techniques from infinitary combinatorics to examine some old and new questions concerning spreading models and good sequences. First, we consider the lp spreading model problem which asks whether a Banach space contains lp provided that every spreading model of a normalized block basic sequence of the basis is isometrically equivalent to lp. Next, using the Hindman-Milliken-Taylor theorem, we prove a new stabilization theorem for spreading models which produces a basic sequence all of whose normalized constant coefficient block basic sequences are good. When the resulting basic sequence is semi-normalized, all the spreading models generated by the above good sequences must be uniformly equivalent to lp or c0. Finally, we investigate the assumption that every normalized block tree on a Banach space has a good branch. This turns out to be a very strong assumption and is equivalent to the space being 1-asymptotic lp. We also show that the stronger assumption …

The non-resonant bounded uniserial representations of Vec(R) form a certain class of extensions composed of tensor density modules, all of whose subquotients are indecomposable. The problem of classifying the extensions with a given composition series is reduced via cohomological methods to computing the solution of a certain system of polynomial equations in several variables derived from the cup equations for the extension. Using this method, we classify all non-resonant bounded uniserial extensions of Vec(R) up to length 6. Beyond this length, all such extensions appear to arise as subquotients of extensions of arbitrary length, many of which are explained by the psuedodifferential operator modules. Others are explained by a wedge construction and by the pseudodifferential operator cocycle discovered by Khesin and Kravchenko.

In 1980 Feigin and Fuchs classified the length 2 bounded representations of Vec(R), the Lie algebra of polynomial vector fields on the line, as a result of their work on the cohomology of Vec(R). This dissertation is concerned mainly with the uniserial (completely indecomposable) representations of Vec(R) with a single Casimir eigenvalue and weights bounded below. Such representations are composed of irreducible representations with semisimple Euler operator action, bounded weight space dimensions, and weights bounded below. These are known to be the tensor density modules with lowest weight λ, for any non-zero complex number λ, and the trivial module C, with Vec(R) actions π_λ and π_C, respectively. Our proofs are cohomology arguments involving the first cohomology groups of Vec(R) with values in the space of homomorphisms between two irreducible representations. These results classify the finite length uniserial extensions, with a single Casimir eigenvalue, of admissible irreducible Vec(R) representations with weights bounded below. In almost every case there is at most one uniserial representation with a given composition series. However, in the case of an odd length extension with composition series {π_1,π_C,π_1,…,π_C,π_1}, there is a one-parameter family of extensions. We also give preliminary results on uniserial representations of the Virasoro Lie …

Rank one subshifts are dynamical systems generated by a regular combinatorial process based on sequences of positive integers called the cut and spacer parameters. Despite the simple process that generates them, rank one subshifts comprise a generic set and are the source of many counterexamples. As a result, measure theoretic rank one subshifts, called rank one transformations, have been extensively studied and investigations into rank one subshifts been the basis of much recent work. We will answer several open problems about rank one subshifts. We completely classify the maximal equicontinuous factor for rank one subshifts, so that this factor can be computed from the parameters. We use these methods to classify when large classes of rank one subshifts have mixing properties. Also, we completely classify the situation when a rank one subshift can be a factor of another rank one subshift.

In this paper we explore coloring theorems for the reals, its quotients, cardinals, and their combinations. This work is done under the scope of the axiom of determinacy. We also explore generalizations of Mycielski's theorem and show how these can be used to establish coloring theorems. To finish, we discuss the strange realm of long unions.

In this work, we concern ourselves with particular topics in Polish space theory. We first consider the space A(U) of complex-analytic functions on an open set U endowed with the usual topology of uniform convergence on compact subsets. With the operations of point-wise addition and point-wise multiplication, A(U) is a Polish ring. Inspired by L. Bers' algebraic characterization of the relation of conformality, we show that the topology on A(U) is the only Polish topology for which A(U) is a Polish ring for a large class of U. This class of U includes simply connected regions, simply connected regions excluding a relatively discrete set of points, and other domains of usual interest. One thing that we deduce from this is that, even though C has many different Polish field topologies, as long as it sits inside another Polish ring with enough complex-analytic functions, it must have its usual topology. In a different direction, we show that the bounded complex-analytic functions on the unit disk admits no Polish topology for which it is a Polish ring. We also study the Lie ring structure on A(U) which turns out to be a Polish Lie ring with the usual topology. In this case, …

We present a novel theorem of Borel Combinatorics that sheds light on the types of continuous functions that can be defined on the Cantor space. We specifically consider the part X=F(2ᴳ) from the Cantor space, where the group G is the additive group of integer pairs ℤ². That is, X is the set of aperiodic {0,1} labelings of the two-dimensional infinite lattice graph. We give X the Bernoulli shift action, and this action induces a graph on X in which each connected component is again a two-dimensional lattice graph. It is folklore that no continuous (indeed, Borel) function provides a two-coloring of the graph on X, despite the fact that any finite subgraph of X is bipartite. Our main result offers a much more complete analysis of continuous functions on this space. We construct a countable collection of finite graphs, each consisting of twelve "tiles", such that for any property P (such as "two-coloring") that is locally recognizable in the proper sense, a continuous function with property P exists on X if and only if a function with a corresponding property P' exists on one of the graphs in the collection. We present the theorem, and give several applications.

In this thesis, we discuss three main projects which are related to Polish groups and their actions on standard Borel spaces. In the first part, we show that the complexity of the classification problem of continua is Borel bireducible to a universal orbit equivalence relation induce by a Polish group on a standard Borel space. In the second part, we compare the relative complexity of various types of classification problems concerning subspaces of [0,1]^n for all natural number n. In the last chapter, we give a topological characterization theorem for the class of locally compact two-sided invariant non-Archimedean Polish groups. Using this theorem, we show the non-existence of a universal group and the existence of a surjectively universal group in the class.

This thesis is an exloration and exposition of a highly efficient shallow neural network algorithm called word2vec, which was developed by T. Mikolov et al. in order to create vector representations of a language vocabulary such that information about the meaning and usage of the vocabulary words is encoded in the vector space structure. Chapter 1 introduces natural language processing, vector representations of language vocabularies, and the word2vec algorithm. Chapter 2 reviews the basic mathematical theory of deterministic convex optimization. Chapter 3 provides background on some concepts from computer science that are used in the word2vec algorithm: Huffman trees, neural networks, and binary cross-entropy. Chapter 4 provides a detailed discussion of the word2vec algorithm itself and includes a discussion of continuous bag of words, skip-gram, hierarchical softmax, and negative sampling. Finally, Chapter 5 explores some applications of vector representations: word categorization, analogy completion, and language translation assistance.