Browsing Mathematics Theses and Dissertations by Title

Jasper, John, 1981 (University of Oregon, June , 2011)[more][less]Jasper, John, 1981 20110927T22:05:12Z 20110927T22:05:12Z 201106 http://hdl.handle.net/1794/11575 ix, 99 p. We characterize the diagonals of four classes of selfadjoint operators on infinite dimensional Hilbert spaces. These results are motivated by the classical SchurHorn theorem, which characterizes the diagonals of selfadjoint matrices on finite dimensional Hilbert spaces. In Chapters II and III we present some known results. First, we generalize the SchurHorn theorem to finite rank operators. Next, we state Kadison's theorem, which gives a simple necessary and sufficient condition for a sequence to be the diagonal of a projection. We present a new constructive proof of the sufficiency direction of Kadison's theorem, which is referred to as the Carpenter's Theorem. Our first original SchurHorn type theorem is presented in Chapter IV. We look at operators with three points in the spectrum and obtain a characterization of the diagonals analogous to Kadison's result. In the final two chapters we investigate a SchurHorn type problem motivated by a problem in frame theory. In Chapter V we look at the connection between frames and diagonals of locally invertible operators. Finally, in Chapter VI we give a characterization of the diagonals of locally invertible operators, which in turn gives a characterization of the sequences which arise as the norms of frames with specified frame bounds. This dissertation includes previously published coauthored material. Committee in charge: Marcin Bownik, Chair; N. Christopher Phillips, Member; Yuan Xu, Member; David Levin, Member; Dietrich Belitz, Outside Member en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2011; Mathematics Pure sciences SchurHorn theorem Diagonals Frames Selfadjoint operators Infinite dimensional versions of the SchurHorn theorem Thesis

Phan, Christopher Lee, 1980 (University of Oregon, June , 2009)[more][less]Phan, Christopher Lee, 1980 20100515T00:13:21Z 20100515T00:13:21Z 200906 http://hdl.handle.net/1794/10367 xi, 95 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. We investigate some homological properties of graded algebras. If A is an R algebra, then E (A) := Ext A ( R, R ) is an Ralgebra under the cup product and is called the Yoneda algebra. (In most cases, we assume R is a field.) A wellknown and widelystudied condition on E(A) is the Koszul property. We study a class of deformations of Koszul algebras that arises from the study of equivariant cohomology and algebraic groups and show that under certain circumstances these deformations are PoincaréBirkhoffWitt deformations. Some of our results involve the [Special characters omitted] property, recently introduced by Cassidy and Shelton, which is a generalization of the Koszul property. While a Koszul algebra must be quadratic, a [Special characters omitted] algebra may have its ideal of relations generated in different degrees. We study the structure of the Yoneda algebra corresponding to a monomial [Special characters omitted.] algebra and provide an example of a monomial [Special characters omitted] algebra whose Yoneda algebra is not also [Special characters omitted]. This example illustrates the difficulty of finding a [Special characters omitted] analogue of the classical theory of Koszul duality. It is wellknown that PoincaréBirkhoffWitt algebras are Koszul. We find a [Special characters omitted] analogue of this theory. If V is a finitedimensional vector space with an ordered basis, and A := [Special characters omitted] (V)/I is a connectedgraded algebra, we can place a filtration F on A as well as E (A). We show there is a bigraded algebra embedding Λ: gr F E (A) [Special characters omitted] E (gr F A ). If I has a Gröbner basis meeting certain conditions and gr F A is [Special characters omitted], then Λ can be used to show that A is also [Special characters omitted]. This dissertation contains both previously published and coauthored materials. Committee in charge: Brad Shelton, Chairperson, Mathematics; Victor Ostrik, Member, Mathematics; Christopher Phillips, Member, Mathematics; Sergey Yuzvinsky, Member, Mathematics; Van Kolpin, Outside Member, Economics en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2009; Koszul properties Noncommutative graded algebras Yoneda algebra Grobner bases Homological algebra Mathematics Algebra, Homological Algebra, Yoneda Koszul algebras Koszul and generalized Koszul properties for noncommutative graded algebras Thesis

Perlmutter, Nathan (University of Oregon, August 18, 2015)[more][less]Botvinnik, Boris Perlmutter, Nathan 20150818T23:01:18Z 20150818T23:01:18Z 20150818 http://hdl.handle.net/1794/19241 Let n > 1. We prove a homological stability theorem for the diffeomorphism groups of (4n+1)dimensional manifolds, with respect to forming the connected sum with (2n1)connected, (4n+1)dimensional manifolds that are stably parallelizable. Our techniques involve the study of the action of the diffeomorphism group of a manifold M on the linking form associated to the homology groups of M. In order to study this action we construct a geometric model for the linking form using the intersections of embedded and immersed Z/kmanifolds. In addition to our main homological stability theorem, we prove several results regarding disjunction for embeddings and immersions of Z/kmanifolds that could be of independent interest. en_US University of Oregon All Rights Reserved. Algebraic Topology Diffeomorphism Groups Differential Topology Singularity Theory Surgery Theory Linking Forms, Singularities, and Homological Stability for Diffeomorphism Groups of Odd Dimensional Manifolds Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

Zhang, Tan, 1969 (University of Oregon, 2000)[more][less]Zhang, Tan, 1969 20080210T03:23:11Z 20080210T03:23:11Z 2000 0599845562 http://hdl.handle.net/1794/150 Adviser: Peter B. Gilkey. ix, 128 leaves A print copy of this title is available through the UO Libraries under the call number: MATH QA613 .Z43 2000 Relative to a nondegenerate metric of signature (p, q), an algebraic curvature tensor is said to be IP if the associated skewsymmetric curvature operator R(π) has constant eigenvalues and if the kernel of R(π) has constant dimension on the Grassmanian of nondegenerate oriented 2planes. A pseudoRiemannian manifold with a nondegenerate indefinite metric of signature (p, q) is said to be IP if the curvature tensor of the LeviCivita connection is IP at every point; the eigenvalues are permitted to vary with the point. In the Riemannian setting (p, q) = (0, m), the work of Gilkey, Leahy, and Sadofsky and the work of Ivanov and Petrova have classified the IP metrics and IP algebraic curvature tensors if the dimension is at least 4 and if the dimension is not 7. We use techniques from algebraic topology and from differential geometry to extend some of their results to the Lorentzian setting (p, q) = (1, m – 1) and to the setting of metrics of signature (p, q) = (2, m – 2). 5667358 bytes 1473 bytes 177540 bytes application/pdf text/plain text/plain en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2000 Manifolds (Mathematics) Metric spaces Curvature Operator algebras Eigenvalues Manifolds with indefinite metrics whose skewsymmetric curvature operator has constant eigenvalues Thesis

Walsh, Mark, 1976 (University of Oregon, June , 2009)[more][less]Walsh, Mark, 1976 20100312T01:15:37Z 20100312T01:15:37Z 200906 http://hdl.handle.net/1794/10265 x, 164 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. We study the topology of the space of metrics of positive scalar curvature on a compact manifold. The main tool we use for constructing such metrics is the surgery technique of Gromov and Lawson. We extend this technique to construct families of positive scalar curvature cobordisms and concordances which are parametrised by Morse functions and later, by generalised Morse functions. We then use these results to study concordances of positive scalar curvature metrics on simply connected manifolds of dimension at least five. In particular, we describe a subspace of the space of positive scalar curvature concordances, parametrised by generalised Morse functions. We call such concordances GromovLawson concordances. One of the main results is that positive scalar curvature metrics which are GromovLawson concordant are in fact isotopic. This work relies heavily on contemporary Riemannian geometry as well as on differential topology, in particular pseudoisotopy theory. We make substantial use of the work of Eliashberg and Mishachev on wrinkled maps and of results by Hatcher and Igusa on the space of generalised Morse functions. Committee in charge: Boris Botvinnik, Chairperson, Mathematics; James Isenberg, Member, Mathematics; Hal Sadofsky, Member, Mathematics; Christopher Phillips, Member, Mathematics; Michael Kellman, Outside Member, Chemistry en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2009; Scalar curvature Morse functions Concordances Wrinkled maps Mathematics Metrics of positive scalar curvature and generalised Morse functions Thesis

Stewart, Allen (University of Oregon, September 29, 2014)[more][less]Vologodksy, Vadim Stewart, Allen 20140929T17:52:14Z 20140929T17:52:14Z 20140929 http://hdl.handle.net/1794/18418 We prove a formula expressing the motivic integral of a K3 surface over C((t)) with semistable reduction in terms of the associated limit mixed Hodge structure. Secondly, for every smooth variety over a complete discrete valuation field we define an analogue of the monodromy pairing, constructed by Grothendieck in the case of Abelian varieties, and prove that our monodromy pairing is a birational invariant of the variety. Finally, we propose a conjectural formula for the motivic integral of maximally degenerate K3 surfaces over an arbitrary complete discrete valuation field and prove this conjecture for Kummer K3 surfaces. This dissertation includes previously published coauthored material. en_US University of Oregon All Rights Reserved. Motivic Integral of K3 Surfaces over a NonArchimedean Field Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

Bartz, Jeremiah (University of Oregon, October 3, 2013)[more][less]Yuzvinsky, Sergey Bartz, Jeremiah 20131003T23:32:26Z 20131003T23:32:26Z 20131003 http://hdl.handle.net/1794/13252 In this dissertation, a method for producing multinets from a net in P^3 is presented. Multinets play an important role in the study of resonance varieties of the complement of a complex hyperplane arrangement and very few examples are known. Implementing this method, numerous new and interesting examples of multinets are identified. These examples provide additional evidence supporting the conjecture of Pereira and Yuzvinsky that all multinets are degenerations of nets. Also, a complete description is given of proper weak multinets, a generalization of multinets. en_US University of Oregon All Rights Reserved. hyperplane arrangements multiarrangements multinets nets pencil of plane curves resonance varieities Multinets in P^2 and P^3 Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

Wang, LiAn (University of Oregon, 2012)[more][less]Bownik, Marcin Wang, LiAn Wang, LiAn 20121026T04:04:21Z 20121026T04:04:21Z 2012 http://hdl.handle.net/1794/12429 We extend the theory of singular integral operators and multiplier theorems to the setting of anisotropic Hardy spaces. We first develop the theory of singular integral operators of convolution type in the anisotropic setting and provide a molecular decomposition on Hardy spaces that will help facilitate the study of these operators. We extend two multiplier theorems, the first by Taibleson and Weiss and the second by Baernstein and Sawyer, to the anisotropic setting. Lastly, we characterize the Fourier transforms of Hardy spaces and show that all multipliers are necessarily continuous. en_US University of Oregon All Rights Reserved. Fourier analysis Hardy spaces Harmonic analysis Multiplier Theorems on Anisotropic Hardy Spaces Electronic Thesis or Dissertation

Vanderpool, Ruth, 1980 (University of Oregon, June , 2009)[more][less]Vanderpool, Ruth, 1980 20100305T01:33:36Z 20100305T01:33:36Z 200906 http://hdl.handle.net/1794/10244 vii, 54 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. We investigate the existence of a stable homotopy category (SHC) associated to the category of p complete abelian groups [Special characters omitted]. First we examine [Special characters omitted] and prove [Special characters omitted] satisfies all but one of the axioms of an abelian category. The connections between an SHC and homology functors are then exploited to draw conclusions about possible SHC structures for [Special characters omitted]. In particular, let [Special characters omitted] denote the category whose objects are chain complexes of [Special characters omitted] and morphisms are chain homotopy classes of maps. We show that any homology functor from any subcategory of [Special characters omitted] containing the padic integers and satisfying the axioms of an SHC will not agree with standard homology on free, finitely generated (as modules over the p adic integers) chain complexes. Explicit examples of common functors are included to highlight troubles that arrise when working with [Special characters omitted]. We make some first attempts at classifying small objects in [Special characters omitted]. Committee in charge: Hal Sadofsky, Chairperson, Mathematics; Boris Botvinnik, Member, Mathematics; Daniel Dugger, Member, Mathematics; Sergey Yuzvinsky, Member, Mathematics; Elizabeth Reis, Outside Member, Womens and Gender Studies en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2009; Stable homotopy Pcomplete abelian groups Homology functor Abelian Mathematics Nonexistence of a stable homotopy category for pcomplete abelian groups Thesis

Kloefkorn, Tyler (University of Oregon, September 29, 2014)[more][less]Shelton, Brad Kloefkorn, Tyler 20140929T17:46:46Z 20140929 http://hdl.handle.net/1794/18372 This dissertation studies new connections between combinatorial topology and homological algebra. To a finite ranked poset Γ we associate a finitedimensional quadratic graded algebra RΓ. Assuming Γ satisfies a combinatorial condition known as uniform, RΓ is related to a wellknown algebra, the splitting algebra AΓ. First introduced by Gelfand, Retakh, Serconek and Wilson, splitting algebras originated from the problem of factoring noncommuting polynomials. Given a finite ranked poset Γ, we ask a standard question in homological algebra: Is RΓ Koszul? The Koszulity of RΓ is related to a combinatorial topology property of Γ known as CohenMacaulay. One of the main theorems of this dissertation is: If Γ is a finite ranked cyclic poset, then Γ is CohenMacaulay if and only if Γ is uniform and RΓ is Koszul. We also define a new generalization of CohenMacaulay: weakly CohenMacaulay. The class of weakly CohenMacaulay finite ranked posets includes posets with disconnected open subintervals. We prove: if Γ is a finite ranked cyclic poset, then Γ is weakly CohenMacaulay if and only if RΓ is Koszul. Finally, we address the notion of numerical Koszulity. We show that there exist algebras RΓ that are numerically Koszul but not Koszul and give a general construction for such examples. This dissertation includes unpublished coauthored material. en_US University of Oregon All Rights Reserved. CohenMacaulay Koszul Splitting Algebras On Algebras Associated to Finite Ranked Posets and Combinatorial Topology: The Koszul, Numerically Koszul and CohenMacaulay Properties Electronic Thesis or Dissertation 20150329 Ph.D. doctoral Department of Mathematics University of Oregon

Giusti, Chad David, 1978 (University of Oregon, June , 2010)[more][less]Giusti, Chad David, 1978 20101203T22:07:48Z 20101203T22:07:48Z 201006 http://hdl.handle.net/1794/10869 viii, 57 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. We introduce a new finitecomplexity knot theory, the theory of plumbers' knots, as a model for classical knot theory. The spaces of plumbers' curves admit a combinatorial cell structure, which we exploit to algorithmically solve the classification problem for plumbers' knots of a fixed complexity. We describe cellular subdivision maps on the spaces of plumbers' curves which consistently make the spaces of plumbers' knots and their discriminants into directed systems. In this context, we revisit the construction of the Vassiliev spectral sequence. We construct homotopical resolutions of the discriminants of the spaces of plumbers knots and describe how their cell structures lift to these resolutions. Next, we introduce an inverse system of unstable Vassiliev spectral sequences whose limit includes, on its E ∞  page, the classical finitetype invariants. Finally, we extend the definition of the Vassiliev derivative to all singularity types of plumbers' curves and use it to construct canonical chain representatives of the resolution of the Alexander dual for any invariant of plumbers' knots. Committee in charge: Dev Sinha, Chairperson, Mathematics; Hal Sadofsky, Member, Mathematics; Arkady Berenstein, Member, Mathematics; Daniel Dugger, Member, Mathematics; Andrzej Proskurowski, Outside Member, Computer & Information Science en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2010; Plumbers' knots Vassiliev derivatives Finitecomplexity knots Spectral sequences Alexander dual Canonical chains Mathematics Theoretical mathematics Plumbers' knots and unstable Vassiliev theory Thesis

Brandl, MaryKatherine, 1963 (University of Oregon, 2001)[more][less]Brandl, MaryKatherine, 1963 20080210T04:19:31Z 20080210T04:19:31Z 2001 0493364234 http://hdl.handle.net/1794/147 Adviser: Brad Shelton. viii, 49 leaves A print copy of this title is available through the UO Libraries under the call number: MATH LIB. QA251.3 .B716 2001 We examine a family of twists of the complex polynomial ring on n generators by a nonsemisimple automorphism. In particular, we consider the case where the automorphism is represented by a single Jordan block. The multiplication in the twist determines a Poisson structure on affine nspace. We demonstrate that the primitive ideals in the twist are parameterized by the symplectic leaves associated to this Poisson structure. Moreover, the symplectic leaves are determined by the orbits of a regular unipotent subgroup of the complex general linear group. 1894196 bytes 1473 bytes 51748 bytes 53191 bytes application/pdf text/plain text/plain text/plain en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2001 Polynomial rings Poisson algebras Noncommutative rings Primitive and Poisson spectra of nonsemisimple twists of polynomial algebras Thesis

Ahlquist, Blair, 1979 (University of Oregon, September , 2010)[more][less]Ahlquist, Blair, 1979 20110504T01:19:26Z 20110504T01:19:26Z 201009 http://hdl.handle.net/1794/11144 vi, 48 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. We compare the relaxation times of two random walks  the simple random walk and the metropolis walk  on an arbitrary finite multigraph G. We apply this result to the random graph with n vertices, where each edge is included with probability p = [Special characters omitted.] where λ > 1 is a constant and also to the NewmanWatts small world model. We give a bound for the reconstruction problem for general trees and general 2 × 2 matrices in terms of the branching number of the tree and some function of the matrix. Specifically, if the transition probabilities between the two states in the state space are a and b , we show that we do not have reconstruction if Br( T ) [straight theta] < 1, where [Special characters omitted.] and Br( T ) is the branching number of the tree in question. This bound agrees with a result obtained by Martin for regular trees and is obtained by more elementary methods. We prove an inequality closely related to this problem. Committee in charge: David Levin, Chairperson, Mathematics; Christopher Sinclair, Member, Mathematics; Marcin Bownik, Member, Mathematics; Hao Wang, Member, Mathematics; Van Kolpin, Outside Member, Economics en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2010; Probability Graphs Random walks Reconstruction problem Metropolis walk Mixing time Probability on graphs: A comparison of sampling via random walks and a result for the reconstruction problem Thesis

Rupel, Dylan (University of Oregon, 2012)[more][less]Berenstein, Arkady Rupel, Dylan Rupel, Dylan 20121026T04:01:34Z 20121026T04:01:34Z 2012 http://hdl.handle.net/1794/12400 We de ne the quantum cluster character assigning an element of a quantum torus to each representation of a valued quiver (Q; d) and investigate its relationship to external and internal mutations of a quantum cluster algebra associated to (Q; d). We will see that the external mutations are related to re ection functors and internal mutations are related to tilting theory. Our main result will show the quantum cluster character gives a cluster monomial in this quantum cluster algebra whenever the representation is rigid, moreover we will see that each noninitial cluster variable can be obtained in this way from the quantum cluster character. en_US University of Oregon All Rights Reserved. Cluster Quantum Quiver Tilting Quantum Cluster Characters Electronic Thesis or Dissertation

Black, Samson, 1979 (University of Oregon, June , 2010)[more][less]Black, Samson, 1979 20101130T01:26:26Z 20101130T01:26:26Z 201006 http://hdl.handle.net/1794/10847 viii, 50 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. We study a certain quotient of the IwahoriHecke algebra of the symmetric group Sd , called the super TemperleyLieb algebra STLd. The Alexander polynomial of a braid can be computed via a certain specialization of the Markov trace which descends to STLd. Combining this point of view with Ocneanu's formula for the Markov trace and Young's seminormal form, we deduce a new statesum formula for the Alexander polynomial. We also give a direct combinatorial proof of this result. Committee in charge: Arkady Vaintrob, CoChairperson, Mathematics Jonathan Brundan, CoChairperson, Mathematics; Victor Ostrik, Member, Mathematics; Dev Sinha, Member, Mathematics; Paul van Donkelaar, Outside Member, Human Physiology en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2010; Hecke algebras Alexander polynomal Symmetric groups Markov trace Mathematics Theoretical mathematics Representations of Hecke algebras and the Alexander polynomial Thesis

Reynolds, Andrew (University of Oregon, August 18, 2015)[more][less]Brundan, Jon Reynolds, Andrew 20150818T22:59:49Z 20150818T22:59:49Z 20150818 http://hdl.handle.net/1794/19228 We study the representations of a certain specialization $\mathcal{OB}(\delta)$ of the oriented Brauer category in arbitrary characteristic $p$. We exhibit a triangular decomposition of $\mathcal{OB}(\delta)$, which we use to show its irreducible representations are labelled by the set of all $p$regular bipartitions. We then explain how its locally finite dimensional representations can be used to categorify the tensor product $V(\varpi_{m'}) \otimes V(\varpi_{m})$ of an integrable lowest weight and highest weight representation of the Lie algebra $\mathfrak{sl}_{\Bbbk}$. This is an example of a slight generalization of the notion of tensor product categorification in the sense of Losev and Webster and is the main result of this paper. We combine this result with the work of Davidson to describe the crystal structure on the set of irreducible representations. We use the crystal to compute the decomposition numbers of standard modules as well as the characters of simple modules assuming $p = 0$. We give another proof of the classification of irreducible modules over the walled Brauer algebra. We use this classification to prove that the irreducible $\mathcal{OB}(\delta)$modules are infinite dimensional unless $\delta = 0$, in which case they are all infinite dimensional except for the irreducible module labelled by the empty bipartition, which is one dimensional. en_US University of Oregon All Rights Reserved. Representations of the Oriented Brauer Category Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

Kronholm, William C., 1980 (University of Oregon, June , 2008)[more][less]Kronholm, William C., 1980 20090113T00:36:10Z 20090113T00:36:10Z 200806 http://hdl.handle.net/1794/8284 x, 72 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. The theory of equivariant homology and cohomology was first created by Bredon in his 1967 paper and has since been developed and generalized by May, Lewis, Costenoble, and a host of others. However, there has been a notable lack of computations done. In this paper, a version of the Serre spectral sequence of a fibration is developed for RO ( G )graded equivariant cohomology of G spaces for finite groups G . This spectral sequence is then used to compute cohomology of projective bundles and certain loop spaces. In addition, the cohomology of Rep( G )complexes, with appropriate coefficients, is shown to always be free. As an application, the cohomology of real projective spaces and some Grassmann manifolds are computed, with an eye towards developing a theory of equivariant characteristic classes. Adviser: Daniel Dugger en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2008; Algebraic topology Equivariant topology Spectral sequence Serre spectral sequence Mathematics The RO(G)graded Serre Spectral Sequence Thesis

Foster, John (University of Oregon, October 3, 2013)[more][less]Berenstein, Arkady Foster, John 20131003T23:33:29Z 20131003T23:33:29Z 20131003 http://hdl.handle.net/1794/13269 We exhibit a correspondence between subcategories of modules over an algebra and subbimodules of the dual of that algebra. We then prove that the semisimplicity of certain such categories is equivalent to the existence of a PeterWeyl decomposition of the corresponding subbimodule. Finally, we use this technique to establish the semisimplicity of certain finitedimensional representations of the quantum double $D(U_q(sl_2))$ for generic $q$. en_US University of Oregon All Rights Reserved. Semisimplicity of Certain Representation Categories Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

Shum, Christopher (University of Oregon, October 3, 2013)[more][less]Sinclair, Christopher Shum, Christopher 20131003T23:35:27Z 20131003T23:35:27Z 20131003 http://hdl.handle.net/1794/13302 For beta > 0, the betaensemble corresponds to the joint probability density on the real line proportional to prod_{n > m}^N abs{x_n  x_m}^beta prod_{n = 1}^N w(x_n) where w is the weight of the system. It has the application of being the Boltzmann factor for the configuration of N chargeone particles interacting logarithmically on an infinite wire inside an external field Q = log w at inverse temperature beta. Similarly, the circular betaensemble has joint probability density proportional to prod_{n > m}^N abs{e^{itheta_n}  e^{itheta_m}}^beta prod_{n = 1}^N w(x_n) quad for theta_n in [ pi, pi) and can be interpreted as N chargeone particles on the unit circle interacting logarithmically with no external field. When beta = 1, 2, and 4, both ensembles are said to be solvable in that their correlation functions can be expressed in a form which allows for asymptotic calculations. It is not known, however, whether the general betaensemble is solvable. We present four families of particle models which are solvable point processes related to the betaensemble. Two of the examples interpolate between the circular betaensembles for beta = 1, 2, and 4. These give alternate ways of connecting the classical betaensembles besides simply changing the values of beta. The other two examples are "mirrored" particle models, where each particle has a paired particle reflected about some point or axis of symmetry. en_US University of Oregon All Rights Reserved. Beta Ensemble Random Matrix Theory Solvable Particle Models Related to the BetaEnsemble Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

Leeman, Aaron, 1974 (University of Oregon, June , 2009)[more][less]Leeman, Aaron, 1974 20100301T23:23:03Z 20100301T23:23:03Z 200906 http://hdl.handle.net/1794/10227 vii, 34 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. We study the Bousfield localization functors known as [Special characters omitted], as described in [MahS]. In particular we would like to understand how they interact with suspension and how they stabilize. We prove that suitably connected [Special characters omitted]acyclic spaces have suspensions which are built out of a particular type n space, which is an unstable analog of the fact that [Special characters omitted]acyclic spectra are built out of a particular type n spectrum. This theorem follows DrorFarjoun's proof in the case n = 1 with suitable alterations. We also show that [Special characters omitted] applied to a space stabilizes in a suitable way to [Special characters omitted] applied to the corresponding suspension spectrum. Committee in charge: Hal Sadofsky, Chairperson, Mathematics; Arkady Berenstein, Member, Mathematics; Daniel Dugger, Member, Mathematics; Dev Sinha, Member, Mathematics; William Rossi, Outside Member, English en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2009; Chromatic functors Bousfield functors Acyclic spaces Suspension spectrum Algebraic topology Mathematics Stabilization of chromatic functors Thesis