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 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.

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.

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.

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.

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}.

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.

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 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 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.