We extend the definition of conical representations for Riemannian symmetric space to a certain class of infinite-dimensional Riemannian symmetric spaces. Using an infinite-dimensional version of Weyl's Unitary Trick, there is a correspondence between smooth representations of infinite-dimensional noncompact-type Riemannian symmetric spaces and smooth representations of infinite-dimensional compact-type symmetric spaces. We classify all smooth conical representations which are unitary on the compact-type side. Finally, a new class of non-smooth unitary conical representations appears on the compact-type side which has no analogue in the finite-dimensional case. We classify these representations and show how to decompose them into direct integrals of irreducible conical representations.

This thesis investigates a construction in contact topology of Legendrian submanifolds called the Legendrian product. We investigate and compute invariants for these Legendrian submanifolds, including the Thurston-Bennequin invariant and Maslov class; Legendrian contact homology for the product of two Legendrian knots; and generating family homology.

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The Bellman method is a tool that can be used to solve a wide variety of problems from harmonic analysis. However, the problem of finding the Bellman function for a specific problem was originally only possible with a guess-and-check method, although that problem has since been mitigated. This paper applies the method pioneered by Vasily Vasyunin of finding the Bellman function directly to a problem presented in JanineWittwer’s survey article [5], and also explores how Bellman functions can be used to prove H˝older’s inequality.

Additive enumeration problems, such as counting the number of integer partitions, lie at the intersection of various branches of mathematics including combinatorics, number theory, and analysis. Extending partitions to integer unimodal sequences has also yielded interesting combinatorial results and asymptotic formulae, which form the subject of this thesis. Much like the important work of Hardy and Ramanujan proving the asymptotic formula for the partition function, Auluck and Wright gave similar formulas for unimodal sequences. Following the circle method of Wright, we provide the asymptotic expansion for unimodal sequences with odd parts. This is then generalized to a two-parameter family of mixed congruence relations, with parts on one side with parts on one side up to the peak satisfying r (mod m) and parts on the other side -r (mod m), and an asymptotic formula is provided. Techniques used in the proofs include Wright's circle method, modular transformations, and bounding of complex integrals.

The main problem in sampling theory is to reconstruct a function from its values (samples) on some discrete subset <font face=symbol>W</font> of its domain. However, taking samples on an appropriate sampling set <font face=symbol>W</font> is not always practical or even possible - for example, associated measuring devices may be too expensive or scarce. In the dynamical sampling problem, it is assumed that the sparseness of the sampling locations can be compensated by involving dynamics. For example, when f is the initial state of a physical process (say, change of temperature or air pollution), we can sample its values at the same sampling locations as time progresses, and try to recover f from the combination of these spatio-temporal samples. <br /> In our recent work, we have taken a more operator-theoretic approach to the dynamical sampling problem. We assume that the unknown function f is a vector in some Hilbert space H and A is a bounded linear operator on H. The samples are given in the form (Aⁿf,g) for 0 <font face=symbol>£</font> n<L(g) and g <font face=symbol>Ã</font> G, where G is a countable (finite or infinite) set of vectors in H, and the function L:G<font face=symbol>®</font> {1,2,...,<font face=symbol>Â¥</font>} represents the "sampling level." Then the main problem becomes to recover the unknown vector f <font face=symbol>Ã</font> H from the above measurements. <br /> The dynamical sampling problem has potential applications in plenacoustic sampling, on-chip sensing, data center temperature sensing, neuron-imaging, and satellite remote sensing, and more generally to Wireless Sensor Networks (WSN). It also has connections and applications to other areas of mathematics including C^*-algebras, spectral theory of normal operators, and frame theory. </body> </html>

The use and detection of symmetry is ubiquitous throughout modern mathematics. In the realm of low-dimensional topology, symmetry plays an increasingly significant role due to the fact that many of the modern invariants being developed are computationally expensive to calculate. If information is known about the symmetries of a link, this can be incorporated to greatly reduce the computation time. This manuscript will consider graphical techniques that are amenable to such methods. First, we discuss an obstruction to links being periodic, developed jointly with Dr. Khaled Qazaqzeh at Kuwait University, using a model developed by Caprau and Tipton. We will discuss useful corollaries of this new method that arise when applying the criterion to multi-component links, and give a survey of its effectiveness when applied to low-crossing links. The second part will investigate a structure that arose in the model of Caprau and Tipton, namely singular links. We first define an invariant of singular links. We then develop a method based on the work of Turaev and Ohtsuki that allows for the creation of operator invariants from R-matrices. Finally, we show that the invariant defined previously is the natural extension of the Kauffman Bracket, when viewed through this framework. In the final section we investigate torus links, and relate the values of the Tail of the Colored Jones Polynomials of links within this family. This chapter involves well-known q-series, first studied by Ramanujan, and an unexpected combinatorial series related to planar integer partitions. This work was inspired by two seemingly unrelated questions of Robert Osburn and Oliver Dasbach. The first asked how the tails of two different links might be related, once one recognized that their Tait graphs have some shared structure. The latter asked if there might be a deletion/contraction type formulation for the Tail of the Colored Jones Polynomial, as it relates to the Tait graph of a link. This work is being done jointly with Mustafa Hajij at the University of South Florida.

<p> Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation <i>H&tilde;</i>&mu;*(x; q,t) = <i>H&tilde;</i>_&mu;(<i>x; t,q</i>). We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (<i>q</i> = 0) when mu is a partition with at most three rows, and for the coefficients of the square-free monomials in X={x_1,x_2,...} for all shapes mu. We also provide a proof for the full relation in the case when mu is a hook shape, and for all shapes at the specialization <i>t</i> = 1. Our work in the Hall-Littlewood case reveals a new recursive structure for the cocharge statistic on words.</p>

<p> We study a viscoelastic model of the solid-solid phase change of olivine to its denser $\beta$-spinel state at high pressures and temperatures reachable in laboratory experiments matching conditions typical of Earth's mantle. Using a previously unknown technique, the equations are transformed to the problem of finding two complex analytic functions in the sample satisfying certain conditions on the outer boundary. The Sherman-Lauricella boundary integral equation is used in a numerical algorithm that eliminates the bottleneck of having to solve a large matrix equation at every timestep. The method is implemented and used to compute the solution of a number of non-axisymmetric test problems, some static and some dynamic in time. Next we develop an alternative formulation in which the Lam\'e equations of linearized elasticity are used to model the deformation of the two phases, and we allow for compressibility. The formulation is novel in that separate reference configurations are maintained for the core and shell regions of the sample that grow or shrink in time by accretion or removal at the boundary, one at the expense of the other. We then compare the behavior of the evolution of this system to the incompressible viscoelastic case and to an alternative elastic model. Finally, we study the stability of circular interfaces with axisymmetric initial data under the evolution equations. For various parameter values of the circular interface evolution, we find families of small perturbations of the circular interface and radial interface velocity jump that either grow or decay exponentially in time. In unstable cases, the growth rate increases without bound as the wave number of the perturbation increases. In stable cases, the evolution equations are well-posed until the interface leaves the stability regime, at which point the numerical solutions blow up in an oscillatory manner. Examples of stable and unstable behavior are presented.</p>