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- All Subjects: Quantum Mechanics
- Creators: Bremner, Andrew
Description
In 1959, Iwasawa proved that the size of the $p$-part of the class groups of a $\mathbb{Z}_p$-extension grows as a power of $p$ with exponent ${\mu}p^m+{\lambda}\,m+\nu$ for $m$ sufficiently large. Broadly, I construct conditions to verify if a given $m$ is indeed sufficiently large. More precisely, let $CG_m^i$ (class group) be the $\epsilon_i$-eigenspace component of the $p$-Sylow subgroup of the class group of the field at the $m$-th level in a $\mathbb{Z}_p$-extension; and let $IACG^i_m$ (Iwasawa analytic class group) be ${\mathbb{Z}_p[[T]]/((1+T)^{p^m}-1,f(T,\omega^{1-i}))}$, where $f$ is the associated Iwasawa power series. It is expected that $CG_m^i$ and $IACG^i_m$ be isomorphic, providing us with a powerful connection between algebraic and analytic techniques; however, as of yet, this isomorphism is unestablished in general. I consider the existence and the properties of an exact sequence $$0\longrightarrow\ker{\longrightarrow}CG_m^i{\longrightarrow}IACG_m^i{\longrightarrow}\textrm{coker}\longrightarrow0.$$ In the case of a $\mathbb{Z}_p$-extension where the Main Conjecture is established, there exists a pseudo-isomorphism between the respective inverse limits of $CG_m^i$ and $IACG_m^i$. I consider conditions for when such a pseudo-isomorphism immediately gives the existence of the desired exact sequence, and I also consider work-around methods that preserve cardinality for otherwise. However, I primarily focus on constructing conditions to verify if a given $m$ is sufficiently large that the kernel and cokernel of the above exact sequence have become well-behaved, providing similarity of growth both in the size and in the structure of $CG_m^i$ and $IACG_m^i$; as well as conditions to determine if any such $m$ exists. The primary motivating idea is that if $IACG_m^i$ is relatively easy to work with, and if the relationship between $CG_m^i$ and $IACG_m^i$ is understood; then $CG_m^i$ becomes easier to work with. Moreover, while the motivating framework is stated concretely in terms of the cyclotomic $\mathbb{Z}_p$-extension of $p$-power roots of unity, all results are generally applicable to arbitrary $\mathbb{Z}_p$-extensions as they are developed in terms of Iwasawa-Theory-inspired, yet abstracted, algebraic results on maps between inverse limits.
ContributorsElledge, Shawn Michael (Author) / Childress, Nancy (Thesis advisor) / Bremner, Andrew (Committee member) / Fishel, Susanna (Committee member) / Jones, John (Committee member) / Paupert, Julien (Committee member) / Arizona State University (Publisher)
Created2013
Description
In 1984, Sinnott used $p$-adic measures on $\mathbb{Z}_p$ to give a new proof of the Ferrero-Washington Theorem for abelian number fields by realizing $p$-adic $L$-functions as (essentially) the $Gamma$-transform of certain $p$-adic rational function measures. Shortly afterward, Gillard and Schneps independently adapted Sinnott's techniques to the case of $p$-adic $L$-functions associated to elliptic curves with complex multiplication (CM) by realizing these $p$-adic $L$-functions as $Gamma$-transforms of certain $p$-adic rational function measures. The results in the CM case give the vanishing of the Iwasawa $mu$-invariant for certain $mathbb{Z}_p$-extensions of imaginary quadratic fields constructed from torsion points of CM elliptic curves.
In this thesis, I develop the theory of $p$-adic measures on $mathbb{Z}_p^d$, with particular interest given to the case of $d>1$. Although I introduce these measures within the context of $p$-adic integration, this study includes a strong emphasis on the interpretation of $p$-adic measures as $p$-adic power series. With this dual perspective, I describe $p$-adic analytic operations as maps on power series; the most important of these operations is the multivariate $Gamma$-transform on $p$-adic measures.
This thesis gives new significance to product measures, and in particular to the use of product measures to construct measures on $mathbb{Z}_p^2$ from measures on $mathbb{Z}_p$. I introduce a subring of pseudo-polynomial measures on $mathbb{Z}_p^2$ which is closed under the standard operations on measures, including the $Gamma$-transform. I obtain results on the Iwasawa-invariants of such pseudo-polynomial measures, and use these results to deduce certain continuity results for the $Gamma$-transform. As an application, I establish the vanishing of the Iwasawa $mu$-invariant of Yager's two-variable $p$-adic $L$-function from measure theoretic considerations.
In this thesis, I develop the theory of $p$-adic measures on $mathbb{Z}_p^d$, with particular interest given to the case of $d>1$. Although I introduce these measures within the context of $p$-adic integration, this study includes a strong emphasis on the interpretation of $p$-adic measures as $p$-adic power series. With this dual perspective, I describe $p$-adic analytic operations as maps on power series; the most important of these operations is the multivariate $Gamma$-transform on $p$-adic measures.
This thesis gives new significance to product measures, and in particular to the use of product measures to construct measures on $mathbb{Z}_p^2$ from measures on $mathbb{Z}_p$. I introduce a subring of pseudo-polynomial measures on $mathbb{Z}_p^2$ which is closed under the standard operations on measures, including the $Gamma$-transform. I obtain results on the Iwasawa-invariants of such pseudo-polynomial measures, and use these results to deduce certain continuity results for the $Gamma$-transform. As an application, I establish the vanishing of the Iwasawa $mu$-invariant of Yager's two-variable $p$-adic $L$-function from measure theoretic considerations.
ContributorsZinzer, Scott Michael (Author) / Childress, Nancy (Thesis advisor) / Bremner, Andrew (Committee member) / Fishel, Susanna (Committee member) / Jones, John (Committee member) / Spielberg, John (Committee member) / Arizona State University (Publisher)
Created2015
Description
In this thesis, I investigate the C*-algebras and related constructions that arise from combinatorial structures such as directed graphs and their generalizations. I give a complete characterization of the C*-correspondences associated to directed graphs as well as results about obstructions to a similar characterization of these objects for generalizations of directed graphs. Viewing the higher-dimensional analogues of directed graphs through the lens of product systems, I give a rigorous proof that topological k-graphs are essentially product systems over N^k of topological graphs. I introduce a "compactly aligned" condition for such product systems of graphs and show that this coincides with the similarly-named conditions for topological k-graphs and for the associated product systems over N^k of C*-correspondences. Finally I consider the constructions arising from topological dynamical systems consisting of a locally compact Hausdorff space and k commuting local homeomorphisms. I show that in this case, the associated topological k-graph correspondence is isomorphic to the product system over N^k of C*-correspondences arising from a related Exel-Larsen system. Moreover, I show that the topological k-graph C*-algebra has a crossed product structure in the sense of Larsen.
ContributorsPatani, Nura (Author) / Kaliszewski, Steven (Thesis advisor) / Quigg, John (Thesis advisor) / Bremner, Andrew (Committee member) / Kawski, Matthias (Committee member) / Spielberg, John (Committee member) / Arizona State University (Publisher)
Created2011
Description
The Quantum Harmonic Oscillator is one of the most important models in Quantum Mechanics. Analogous to the classical mass vibrating back and forth on a spring, the quantum oscillator system has attracted substantial attention over the years because of its importance in many advanced and difficult quantum problems. This dissertation deals with solving generalized models of the time-dependent Schrodinger equation which are called generalized quantum harmonic oscillators, and these are characterized by an arbitrary quadratic Hamiltonian of linear momentum and position operators. The primary challenge in this work is that most quantum models with timedependence are not solvable explicitly, yet this challenge became the driving motivation for this work. In this dissertation, the methods used to solve the time-dependent Schrodinger equation are the fundamental singularity (or Green's function) and the Fourier (eigenfunction expansion) methods. Certain Riccati- and Ermakov-type systems arise, and these systems are highlighted and investigated. The overall aims of this dissertation are to show that quadratic Hamiltonian systems are completely integrable systems, and to provide explicit approaches to solving the time-dependent Schr¨odinger equation governed by an arbitrary quadratic Hamiltonian operator. The methods and results established in the dissertation are not yet well recognized in the literature, yet hold for high promise for further future research. Finally, the most recent results in the dissertation correspond to the harmonic oscillator group and its symmetries. A simple derivation of the maximum kinematical invariance groups of the free particle and quantum harmonic oscillator is constructed from the view point of the Riccati- and Ermakov-type systems, which shows an alternative to the traditional Lie Algebra approach. To conclude, a missing class of solutions of the time-dependent Schr¨odinger equation for the simple harmonic oscillator in one dimension is constructed. Probability distributions of the particle linear position and momentum, are emphasized with Mathematica animations. The eigenfunctions qualitatively differ from the traditional standing waves of the one-dimensional Schrodinger equation. The physical relevance of these dynamic states is still questionable, and in order to investigate their physical meaning, animations could also be created for the squeezed coherent states. This will be addressed in future work.
ContributorsLopez, Raquel (Author) / Suslov, Sergei K (Thesis advisor) / Radunskaya, Ami (Committee member) / Castillo-Chavez, Carlos (Committee member) / Platte, Rodrigo (Committee member) / Arizona State University (Publisher)
Created2012
Description
The uncrossing partially ordered set $P_n$ is defined on the set of matchings on $2n$ points on a circle represented with wires. The order relation is $\tau'\leq \tau$ in $P_n$ if and only if $\tau'$ is obtained by resolving a crossing of $\tau$. %This partial order has been studied by Alman-Lian-Tran, Huang-Wen-Xie, Kenyon, and Lam. %The posets $P_n$ emerged from studies of circular planar electrical networks. Circular planar electrical networks are finite weighted undirected graphs embedded into a disk, with boundary vertices and interior vertices. By Curtis-Ingerman-Morrow and de Verdi\`ere-Gitler-Vertigan, the electrical networks can be encoded with response matrices. By Lam the space of response matrices for electrical networks has a cell structure, and this cell structure can be described by the uncrossing partial orders. %Lam proves that the posets can be identified with dual Bruhat order on affine permutations of type $(n,2n)$. Using this identification, Lam proves the poset $\hat{P}_n$, the uncrossing poset $P_n$ with a unique minimum element $\hat{0}$ adjoined, is Eulerian. This thesis consists of two sets of results: (1) flag enumeration in intervals in the uncrossing poset $P_n$ and (2) cyclic sieving phenomenon on the set $P_n$.
I identify elements in $P_n$ with affine permutations of type $(0,2n)$. %This identification enables us to explicitly describe the elements in $P_n$ with the elements in $\mathcal{MP}_n$.
Using this identification, I adapt a technique in Reading for finding recursions for the cd-indices of intervals in Bruhat order of Coxeter groups to the uncrossing poset $P_n$. As a result, I produce recursions for the cd-indices of intervals in the uncrossing poset $P_n$. I also obtain a recursion for the ab-indices of intervals in the poset $\hat{P}_n$, the poset $P_n$ with a unique minimum $\hat0$ adjoined. %We define an induced subposet $\mathcal{MP}_n$ of the affine permutations under Bruhat order.
Reiner-Stanton-White defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action on a finite set and a polynomial. Sagan observed the CSP on the set of non-crossing matchings with the $q$-Catalan polynomial. Bowling-Liang presented similar results on the set of $k$-crossing matchings for $1\leq k \leq 3$. In this dissertation, I focus on the set of all matchings on $[2n]:=\{1,2,\dots,2n\}$. I find the number of matchings fixed by $\frac{2\pi}{d}$ rotations for $d|2n$. I then find the polynomial $X_n(q)$ such that the set of matchings together with $X_n(q)$ and the cyclic group of order $2n$ exhibits the CSP.
I identify elements in $P_n$ with affine permutations of type $(0,2n)$. %This identification enables us to explicitly describe the elements in $P_n$ with the elements in $\mathcal{MP}_n$.
Using this identification, I adapt a technique in Reading for finding recursions for the cd-indices of intervals in Bruhat order of Coxeter groups to the uncrossing poset $P_n$. As a result, I produce recursions for the cd-indices of intervals in the uncrossing poset $P_n$. I also obtain a recursion for the ab-indices of intervals in the poset $\hat{P}_n$, the poset $P_n$ with a unique minimum $\hat0$ adjoined. %We define an induced subposet $\mathcal{MP}_n$ of the affine permutations under Bruhat order.
Reiner-Stanton-White defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action on a finite set and a polynomial. Sagan observed the CSP on the set of non-crossing matchings with the $q$-Catalan polynomial. Bowling-Liang presented similar results on the set of $k$-crossing matchings for $1\leq k \leq 3$. In this dissertation, I focus on the set of all matchings on $[2n]:=\{1,2,\dots,2n\}$. I find the number of matchings fixed by $\frac{2\pi}{d}$ rotations for $d|2n$. I then find the polynomial $X_n(q)$ such that the set of matchings together with $X_n(q)$ and the cyclic group of order $2n$ exhibits the CSP.
ContributorsKim, Younghwan (Author) / Fishel, Susanna (Thesis advisor) / Bremner, Andrew (Committee member) / Czygrinow, Andrzej (Committee member) / Kierstead, Henry (Committee member) / Paupert, Julien (Committee member) / Arizona State University (Publisher)
Created2018
Description
Earth-system models describe the interacting components of the climate system and
technological systems that affect society, such as communication infrastructures. Data
assimilation addresses the challenge of state specification by incorporating system
observations into the model estimates. In this research, a particular data
assimilation technique called the Local Ensemble Transform Kalman Filter (LETKF) is
applied to the ionosphere, which is a domain of practical interest due to its effects
on infrastructures that depend on satellite communication and remote sensing. This
dissertation consists of three main studies that propose strategies to improve space-
weather specification during ionospheric extreme events, but are generally applicable
to Earth-system models:
Topic I applies the LETKF to estimate ion density with an idealized model of
the ionosphere, given noisy synthetic observations of varying sparsity. Results show
that the LETKF yields accurate estimates of the ion density field and unobserved
components of neutral winds even when the observation density is spatially sparse
(2% of grid points) and there is large levels (40%) of Gaussian observation noise.
Topic II proposes a targeted observing strategy for data assimilation, which uses
the influence matrix diagnostic to target errors in chosen state variables. This
strategy is applied in observing system experiments, in which synthetic electron density
observations are assimilated with the LETKF into the Thermosphere-Ionosphere-
Electrodynamics Global Circulation Model (TIEGCM) during a geomagnetic storm.
Results show that assimilating targeted electron density observations yields on
average about 60%–80% reduction in electron density error within a 600 km radius of
the observed location, compared to 15% reduction obtained with randomly placed
vertical profiles.
Topic III proposes a methodology to account for systematic model bias arising
ifrom errors in parametrized solar and magnetospheric inputs. This strategy is ap-
plied with the TIEGCM during a geomagnetic storm, and is used to estimate the
spatiotemporal variations of bias in electron density predictions during the
transitionary phases of the geomagnetic storm. Results show that this strategy reduces
error in 1-hour predictions of electron density by about 35% and 30% in polar regions
during the main and relaxation phases of the geomagnetic storm, respectively.
technological systems that affect society, such as communication infrastructures. Data
assimilation addresses the challenge of state specification by incorporating system
observations into the model estimates. In this research, a particular data
assimilation technique called the Local Ensemble Transform Kalman Filter (LETKF) is
applied to the ionosphere, which is a domain of practical interest due to its effects
on infrastructures that depend on satellite communication and remote sensing. This
dissertation consists of three main studies that propose strategies to improve space-
weather specification during ionospheric extreme events, but are generally applicable
to Earth-system models:
Topic I applies the LETKF to estimate ion density with an idealized model of
the ionosphere, given noisy synthetic observations of varying sparsity. Results show
that the LETKF yields accurate estimates of the ion density field and unobserved
components of neutral winds even when the observation density is spatially sparse
(2% of grid points) and there is large levels (40%) of Gaussian observation noise.
Topic II proposes a targeted observing strategy for data assimilation, which uses
the influence matrix diagnostic to target errors in chosen state variables. This
strategy is applied in observing system experiments, in which synthetic electron density
observations are assimilated with the LETKF into the Thermosphere-Ionosphere-
Electrodynamics Global Circulation Model (TIEGCM) during a geomagnetic storm.
Results show that assimilating targeted electron density observations yields on
average about 60%–80% reduction in electron density error within a 600 km radius of
the observed location, compared to 15% reduction obtained with randomly placed
vertical profiles.
Topic III proposes a methodology to account for systematic model bias arising
ifrom errors in parametrized solar and magnetospheric inputs. This strategy is ap-
plied with the TIEGCM during a geomagnetic storm, and is used to estimate the
spatiotemporal variations of bias in electron density predictions during the
transitionary phases of the geomagnetic storm. Results show that this strategy reduces
error in 1-hour predictions of electron density by about 35% and 30% in polar regions
during the main and relaxation phases of the geomagnetic storm, respectively.
ContributorsDurazo, Juan, Ph.D (Author) / Kostelich, Eric J. (Thesis advisor) / Mahalov, Alex (Thesis advisor) / Tang, Wenbo (Committee member) / Moustaoui, Mohamed (Committee member) / Platte, Rodrigo (Committee member) / Arizona State University (Publisher)
Created2018
Description
Diophantine arithmetic is one of the oldest branches of mathematics, the search
for integer or rational solutions of algebraic equations. Pythagorean triangles are
an early instance. Diophantus of Alexandria wrote the first related treatise in the
fourth century; it was an area extensively studied by the great mathematicians of the seventeenth century, including Euler and Fermat.
The modern approach is to treat the equations as defining geometric objects, curves, surfaces, etc. The theory of elliptic curves (or curves of genus 1, which are much used in modern cryptography) was developed extensively in the twentieth century, and has had great application to Diophantine equations. This theory is used in application to the problems studied in this thesis. This thesis studies some curves of high genus, and possible solutions in both rationals and in algebraic number fields, generalizes some old results and gives answers to some open problems in the literature. The methods involve known techniques together with some ingenious tricks. For example, the equations $y^2=x^6+k$, $k=-39,\,-47$, the two previously unsolved cases for $|k|<50$, are solved using algebraic number theory and the ‘elliptic Chabauty’ method. The thesis also studies the genus three quartic curves $F(x^2,y^2,z^2)=0$ where F is a homogeneous quadratic form, and extend old results of Cassels, and Bremner. It is a very delicate matter to find such curves that have no rational points, yet which do have points in odd-degree extension fields of the rationals.
The principal results of the thesis are related to surfaces where the theory is much less well known. In particular, the thesis studies some specific families of surfaces, and give a negative answer to a question in the literature regarding representation of integers n in the form $n=(x+y+z+w)(1/x+1/y+1/z+1/w).$ Further, an example, the first such known, of a quartic surface $x^4+7y^4=14z^4+18w^4$ is given with remarkable properties: it is everywhere locally solvable, yet has no non-zero rational point, despite having a point in (non-trivial) odd-degree extension fields of the rationals. The ideas here involve manipulation of the Hilbert symbol, together with the theory of elliptic curves.
for integer or rational solutions of algebraic equations. Pythagorean triangles are
an early instance. Diophantus of Alexandria wrote the first related treatise in the
fourth century; it was an area extensively studied by the great mathematicians of the seventeenth century, including Euler and Fermat.
The modern approach is to treat the equations as defining geometric objects, curves, surfaces, etc. The theory of elliptic curves (or curves of genus 1, which are much used in modern cryptography) was developed extensively in the twentieth century, and has had great application to Diophantine equations. This theory is used in application to the problems studied in this thesis. This thesis studies some curves of high genus, and possible solutions in both rationals and in algebraic number fields, generalizes some old results and gives answers to some open problems in the literature. The methods involve known techniques together with some ingenious tricks. For example, the equations $y^2=x^6+k$, $k=-39,\,-47$, the two previously unsolved cases for $|k|<50$, are solved using algebraic number theory and the ‘elliptic Chabauty’ method. The thesis also studies the genus three quartic curves $F(x^2,y^2,z^2)=0$ where F is a homogeneous quadratic form, and extend old results of Cassels, and Bremner. It is a very delicate matter to find such curves that have no rational points, yet which do have points in odd-degree extension fields of the rationals.
The principal results of the thesis are related to surfaces where the theory is much less well known. In particular, the thesis studies some specific families of surfaces, and give a negative answer to a question in the literature regarding representation of integers n in the form $n=(x+y+z+w)(1/x+1/y+1/z+1/w).$ Further, an example, the first such known, of a quartic surface $x^4+7y^4=14z^4+18w^4$ is given with remarkable properties: it is everywhere locally solvable, yet has no non-zero rational point, despite having a point in (non-trivial) odd-degree extension fields of the rationals. The ideas here involve manipulation of the Hilbert symbol, together with the theory of elliptic curves.
ContributorsNguyen, Xuan Tho (Author) / Bremner, Andrew (Thesis advisor) / Childress, Nancy (Committee member) / Jones, John (Committee member) / Quigg, John (Committee member) / Fishel, Susanna (Committee member) / Arizona State University (Publisher)
Created2019
Description
The dynamics of a fluid flow inside 2D square and 3D cubic cavities
under various configurations were simulated and analyzed using a
spectral code I developed.
This code was validated against known studies in the 3D lid-driven
cavity. It was then used to explore the various dynamical behaviors
close to the onset of instability of the steady-state flow, and explain
in the process the mechanism underlying an intermittent bursting
previously observed. A fairly complete bifurcation picture emerged,
using a combination of computational tools such as selective
frequency damping, edge-state tracking and subspace restriction.
The code was then used to investigate the flow in a 2D square cavity
under stable temperature stratification, an idealized version of a lake
with warmer water at the surface compared to the bottom. The governing
equations are the Navier-Stokes equations under the Boussinesq approximation.
Simulations were done over a wide range of parameters of the problem quantifying
the driving velocity at the top (e.g. wind) and the strength of the stratification.
Particular attention was paid to the mechanisms associated with the onset of
instability of the base steady state, and the complex nontrivial dynamics
occurring beyond onset, where the presence of multiple states leads to a
rich spectrum of states, including homoclinic and heteroclinic chaos.
A third configuration investigates the flow dynamics of a fluid in a rapidly
rotating cube subjected to small amplitude modulations. The responses were
quantified by the global helicity and energy measures, and various peak
responses associated to resonances with intrinsic eigenmodes of the cavity
and/or internal retracing beams were clearly identified for the first time.
A novel approach to compute the eigenmodes is also described, making accessible
a whole catalog of these with various properties and dynamics. When the small
amplitude modulation does not align with the rotation axis (precession) we show
that a new set of eigenmodes are primarily excited as the angular velocity
increases, while triadic resonances may occur once the nonlinear regime kicks in.
under various configurations were simulated and analyzed using a
spectral code I developed.
This code was validated against known studies in the 3D lid-driven
cavity. It was then used to explore the various dynamical behaviors
close to the onset of instability of the steady-state flow, and explain
in the process the mechanism underlying an intermittent bursting
previously observed. A fairly complete bifurcation picture emerged,
using a combination of computational tools such as selective
frequency damping, edge-state tracking and subspace restriction.
The code was then used to investigate the flow in a 2D square cavity
under stable temperature stratification, an idealized version of a lake
with warmer water at the surface compared to the bottom. The governing
equations are the Navier-Stokes equations under the Boussinesq approximation.
Simulations were done over a wide range of parameters of the problem quantifying
the driving velocity at the top (e.g. wind) and the strength of the stratification.
Particular attention was paid to the mechanisms associated with the onset of
instability of the base steady state, and the complex nontrivial dynamics
occurring beyond onset, where the presence of multiple states leads to a
rich spectrum of states, including homoclinic and heteroclinic chaos.
A third configuration investigates the flow dynamics of a fluid in a rapidly
rotating cube subjected to small amplitude modulations. The responses were
quantified by the global helicity and energy measures, and various peak
responses associated to resonances with intrinsic eigenmodes of the cavity
and/or internal retracing beams were clearly identified for the first time.
A novel approach to compute the eigenmodes is also described, making accessible
a whole catalog of these with various properties and dynamics. When the small
amplitude modulation does not align with the rotation axis (precession) we show
that a new set of eigenmodes are primarily excited as the angular velocity
increases, while triadic resonances may occur once the nonlinear regime kicks in.
ContributorsWu, Ke (Author) / Lopez, Juan (Thesis advisor) / Welfert, Bruno (Thesis advisor) / Tang, Wenbo (Committee member) / Platte, Rodrigo (Committee member) / Herrmann, Marcus (Committee member) / Arizona State University (Publisher)
Created2019
Description
The longstanding issue of how much time it takes a particle to tunnel through quantum barriers is discussed; in particular, the phenomenon known as the Hartman effect is reviewed. A calculation of the dwell time for two successive rectangular barriers in the opaque limit is given and the result depends on the barrier widths and hence does not lead to superluminal tunneling or the Hartman effect.
ContributorsMcDonald, Scott (Author) / Davies, Paul (Thesis director) / Comfort, Joseph (Committee member) / McCartney, M. R. (Committee member) / Barrett, The Honors College (Contributor)
Created2009-05
Description
Pierre de Fermat, an amateur mathematician, set upon the mathematical world a challenge so difficult it took 357 years to prove. This challenge, known as Fermat's Last Theorem, has many different ways of being expressed, but it simply states that for $n > 2$, the equation $x^n + y^n = z^n$ has no nontrivial solution. The first set of attempts of proofs came from mathematicians using the essentially elementary tools provided by number theory: the notable mathematicians were Leonhard Euler, Sophie Germain and Ernst Kummer. Kummer was the final mathematician to try to use essentially elementary number theory as the basis for his proof and even exclaimed that Fermat's Last Theorem could not be solved using number theory alone; Kummer claimed that greater mathematics would have to be developed in order to prove this ever-growing mystery. The 20th century arrives and two Japanese mathematicians, Goro Shimura and Yutaka Taniyama, shock the world by claiming two highly distinct branches of mathematics, elliptic curves and modular forms, were in fact one and the same. Gerhard Frey then took this claim to the extreme by stating that this claim, the Taniyama-Shimura conjecture, was the necessary link to finally prove Fermat's Last Theorem was true. Frey's statement was then validated by Kenneth Ribet by proving that the Frey Curve could not indeed be a modular form. The final piece of the puzzle placed, the English mathematician Andrew Wiles embarked on a 7 year journey to prove Fermat's Last Theorem as now the the proof of the theorem rested in his area of expertise, that being elliptic curves. In 1994, Wiles published his complete proof of Fermat's Last Theorem, putting an end to one of mathematics' greatest mysteries.
ContributorsBoyadjian, Hoveeg Krikor (Author) / Bremner, Andrew (Thesis director) / Jones, John (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2016-12