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Articles 1 - 11 of 11
Full-Text Articles in Algebra
Langlands Reciprocity And The Splitting Behavior Of Primes In Number Fields, Skip E. Moses
Langlands Reciprocity And The Splitting Behavior Of Primes In Number Fields, Skip E. Moses
Master's Theses
This thesis explores the evolution of reciprocity laws in number theory in order to provide a conceptual bridge between the classical ideas of quadratic reciprocity and the modern framework of the Langlands program. We develop the necessary algebraic background to understand how the splitting behavior of primes in number fields reflects deep arithmetic structure in ℚ. Starting with quadratic fields and cyclotomic extensions, we motivate the development of the Kronecker–Weber theorem and the characterization of abelian extensions of ℚ. We then introduce Artin reciprocity and show how it generalizes quadratic reciprocity through the formalism of Frobenius elements and Artin L-functions. …
Banach Algebras And The Gelfand Theory Of Group Algebras On Locally Compact Abelian Groups, James Gabriel Bonvanie
Banach Algebras And The Gelfand Theory Of Group Algebras On Locally Compact Abelian Groups, James Gabriel Bonvanie
Master's Theses
A Banach algebra is a complex algebra that is simultaneously a Banach space in which the norm is submultiplicative. Notably, $L^1(\mathbb{R})$ with the convolutional product is an Abelian, non-unital Banach algebra that admits an approximate identity. We rectify $L^1(\mathbb{R})$ lacking a unit via the unitization $L^1(\mathbb{R})\times\mathbb{C}$ with identity $(0,1)$. Unitization opens the discussion to the spectrum $\sigma(x)$ of a Banach algebra element, in which the spectrum is a nonempty, compact subset of the complex plane. The spectrum of an Abelian Banach algebra is fully characterized with multiplicative linear functionals, and we prove that the Fourier transform is the unique multiplicative …
Representation Theory And Its Applications In Physics, Max Varverakis
Representation Theory And Its Applications In Physics, Max Varverakis
Master's Theses
Representation theory, which encodes the elements of a group as linear operators on a vector space, has far-reaching implications in physics. Fundamental results in quantum physics emerge directly from the representations describing physical symmetries. We first examine the connections between specific representations and the principles of quantum mechanics. Then, we shift our focus to the braid group, which describes the algebraic structure of braids. We apply representations of the braid group to physical systems in order to investigate quasiparticles known as anyons. Finally, we obtain governing equations of anyonic systems to highlight the differences between braiding statistics and conventional Bose-Einstein/Fermi-Dirac …
Hyperbolic Groups And The Word Problem, David Wu
Hyperbolic Groups And The Word Problem, David Wu
Master's Theses
Mikhail Gromov’s work on hyperbolic groups in the late 1980s contributed to the formation of geometric group theory as a distinct branch of mathematics. The creation of hyperbolic metric spaces showed it was possible to define a large class of hyperbolic groups entirely geometrically yet still be able to derive significant algebraic properties. The objectives of this thesis are to provide an introduction to geometric group theory through the lens of quasi-isometry and show how hyperbolic groups have solvable word problem. Also included is the Stability Theorem as an intermediary result for quasi-isometry invariance of hyperbolicity.
Representations From Group Actions On Words And Matrices, Joel T. Anderson
Representations From Group Actions On Words And Matrices, Joel T. Anderson
Master's Theses
We provide a combinatorial interpretation of the frequency of any irreducible representation of Sn in representations of Sn arising from group actions on words. Recognizing that representations arising from group actions naturally split across orbits yields combinatorial interpretations of the irreducible decompositions of representations from similar group actions. The generalization from group actions on words to group actions on matrices gives rise to representations that prove to be much less transparent. We share the progress made thus far on the open problem of determining the irreducible decomposition of certain representations of Sm × Sn arising from group actions on matrices.
Groups Of Non Positive Curvature And The Word Problem, Zoe Nepsa
Groups Of Non Positive Curvature And The Word Problem, Zoe Nepsa
Master's Theses
Given a group $\Gamma$ with presentation $\relgroup{\scr{\scr{A}}}{\scr{R}}$, a natural question, known as the word problem, is how does one decide whether or not two words in the free group, $F(\scr{\scr{A}})$, represent the same element in $\Gamma$. In this thesis, we study certain aspects of geometric group theory, especially ideas published by Gromov in the late 1980's. We show there exists a quasi-isometry between the group equipped with the word metric, and the space it acts on. Then, we develop the notion of a CAT(0) space and study groups which act properly and cocompactly by isometries on these spaces, such groups …
Unique Signed Minimal Wiring Diagrams And The Stanley-Reisner Correspondence, Vanessa Newsome-Slade
Unique Signed Minimal Wiring Diagrams And The Stanley-Reisner Correspondence, Vanessa Newsome-Slade
Master's Theses
Biological systems are commonly represented using networks consisting of interactions between various elements in the system. Reverse engineering, a method of mathematical modeling, is used to recover how the elements in the biological network are connected. These connections are encoded using wiring diagrams, which are directed graphs that describe how elements in a network affect one another. A signed wiring diagram provides additional information about the interactions between elements relating to activation and inhibition. Due to cost concerns, it is optimal to gain insight into biological networks with as few experiments and data as possible. Minimal wiring diagrams identify the …
Van Kampen Diagrams And Small Cancellation Theory, Kelsey N. Lowrey
Van Kampen Diagrams And Small Cancellation Theory, Kelsey N. Lowrey
Master's Theses
An Introduction To Fröberg's Conjecture, Caroline Semmens
An Introduction To Fröberg's Conjecture, Caroline Semmens
Master's Theses
The goal of this thesis is to make Fröberg's conjecture more accessible to the average math graduate student by building up the necessary background material to understand specific examples where Fröberg's conjecture is true.
Exploring Mathematical Strategies For Finding Hidden Features In Multi-Dimensional Big Datasets, Tri Duong, Fang Ren, Apurva Mehta
Exploring Mathematical Strategies For Finding Hidden Features In Multi-Dimensional Big Datasets, Tri Duong, Fang Ren, Apurva Mehta
STAR Program Research Presentations
With advances in technology in brighter sources and larger and faster detectors, the amount of data generated at national user facilities such as SLAC is increasing exponentially. Humans have a superb ability to recognize patterns in complex and noisy data and therefore, data is still curated and analyzed by humans. However, a human brain is unable to keep up with the accelerated pace of data generation, and as a consequence, the rate of new discoveries hasn't kept pace with the rate of data creation. Therefore, new procedures to quickly assess and analyze the data are needed. Machine learning approaches are …
Characterization Of The Drilling Via The Vibration Augmenter Of Rotary-Drills And Sound Signal Processing Of Impacted Pipe As A Potential Water Height Assessment Tool, Nicholas Morris
STAR Program Research Presentations
The focus of the internship has been on two topics: a) Characterize the drilling performance of a novel percussive augmenter – this drill was developed by the JPL’s Advanced Technologies Group and its performance was characterized; and b) Examine the feasibility of striking a pipe as a means of assessing the water height inside the pipe. The purpose of this investigation is to examine the possibility of using a simple method of applying impacts to a pipe wall and determining the water height from the sonic characteristic differences including damping, resonance frequencies, etc. Due to multiple variables that are relevant …