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This thesis introduces the p-adic metric on the rational numbers. We then present the basic properties of this metric. Using this metric, we explore conic sections, viewed as equidistant sets. Lastly, we move on the sequences and series, and from there, we define p-adic expansions and the analytic completion of Q with respect to the p-adic metric, which leads to exploring some arithmetic properties of Qp.

<p>See abstract. </p>

In the 1970s and 1980s, Jean-Pierre Serre formulated a conjecture connecting two-dimensional Galois representations and modular forms. The conjecture came to be known as Serre's modularity conjecture. It was recently proved by Khare and Wintenberger in 2008. Serre's conjecture has various important consequences in number theory. Most notably, it played a key role in the proof of Fermat's last theorem. A natural question is, what is the analogue of Serre's conjecture for higher dimensional Galois representations? In 2002, Ash, Doud and Pollack formulated a precise statement for a higher dimensional analogue of Serre's conjecture. They also provided numerous computational examples as evidence for this generalized conjecture. We consider the three-dimensional version of the Ash-Doud-Pollack conjecture. We find specific examples of three-dimensional Galois representations and computationally verify the generalized conjecture in all these examples.

<p>In this thesis, for symplectic and split odd special orthogonal groups over a p-adic field, we provide an extensive account of the intersection of local Arthur packets. More specifically, following Atobe's reformulation of Moeglin's construction of local Arthur packets, we give precise algorithms to determine whether a given representation lies in any local Arthur packet. Furthermore, if the representation does lie in a local Arthur packet, we give a systematic approach to determine all the other local Arthur packets which also contain the representation. These algorithms are based on certain operators which act on certain construction data coming from Atobe's parametrization of the local Arthur packets.</p>

<p>This thesis is focused on determinantal rings in 2 different contexts. In Chapter 3 the homological properties of powers of determinantal ideals are studied. In particular the focus is on local cohomology of determinantal thickenings and we explicitly describe the $R$-module structure of some of these local cohomology modules. In Chapter 4 we introduce \textit{generalized diagonal} matrices, a class of sparse matrices which contain diagonal and upper triangular matrices. We study the ideals of minors of such matrices and describe their properties such as height, multiplicity, and Cohen-Macaulayness. </p>

We study the size of sets containing sums of two integer cubes such that their representation is unique and also fit between two consecutive integer cubes. We will try to write algorithms that efficiently calculate the size of these sets and also implement these algorithms in PythonTM. Although we will fail to find a non-iterative algorithm, we will find different ways of approximating the size of these sets. We will also find that techniques used in our failed algorithms can be used to calculate the number of integer lattice points inside a circle.

<p>See abstract. </p>

No description available.

<p>This thesis has three major topics. The first is on generalized multiplicities. The second is on height bounds for ideals of minors of matrices with a given rank. The last topic is on the ideal of minors of generic generalized diagonal matrices.</p> <p>In the first part of this thesis, we discuss various generalizations of Hilbert-Samuel multiplicity. These include the Buchsbaum-Rim multiplicity, mixed multiplicities, $j$-multiplicity, and $\varepsilon$-multiplicity. For $(R,m)$ a Noetherian local ring of dimension $d$ and $I$ a $m$-primary ideal in $R$, Lech showed the following bound for the Hilbert-Samuel multiplicity of $I$, $e(I) \leq d!\lambda(R/I)e(m)$. Huneke, Smirnov, and Validashti improved the bound to $e(mI) \leq d!\lambda(R/I)e(m)$. We generalize the improved bound to the Buchsbaum-Rim multiplicity and to mixed multiplicities. </p> <p>For the second part of the thesis we discuss bounds on heights of ideals of minors of matrices. A classical bound for these heights was shown by Eagon and Northcott. Bruns' bound is an improvement on the Eagon-Northcott bound taking into consideration the rank of the matrix. We prove an analogous bound to Bruns' bound for alternating matrices. We then discuss an open problem by Eisenbud, Huneke, and Ulrich that asks for height bounds for symmetric matrices given their rank. We show a few reduction steps and prove some small cases of this problem. </p> <p>Finally, for the last topic we explore properties of the ideal of minors of generic generalized diagonal matrices. Generalized diagonal matrices are matrices with two ladders of zeros in the bottom left and top right corners. We compute their initial ideals and give a description of the facets of their Stanley-Reisner complex. Using this description, we characterize when these ideals are Cohen-Macaulay. In the special case where the ladders of zeros are triangles, we compute the height and multiplicity</p>