Browsing Mathematics Theses and Dissertations by Title
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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

Muth, Robert (University of Oregon, October 27, 2016)[more][less]Kleshchev, Alexander Muth, Robert 20161027T18:33:59Z 20161027T18:33:59Z 20161027 http://hdl.handle.net/1794/20432 We study representations of KhovanovLaudaRouquier (KLR) algebras of affine Lie type. Associated to every convex preorder on the set of positive roots is a system of cuspidal modules for the KLR algebra. For a balanced order, we study imaginary semicuspidal modules by means of `imaginary SchurWeyl duality'. We then generalize this theory from balanced to arbitrary convex preorders for affine ADE types. Under the assumption that the characteristic of the ground field is greater than some explicit bound, we prove that KLR algebras are properly stratified. We introduce affine zigzag algebras and prove that these are Morita equivalent to arbitrary imaginary strata if the characteristic of the ground field is greater than the bound mentioned above. Finally, working in finite or affine affine type A, we show that skew Specht modules may be defined over the KLR algebra, and real cuspidal modules associated to a balanced convex preorder are skew Specht modules for certain explicit hook shapes. en_US University of Oregon All Rights Reserved. KLR algebras Representation theory Representations of KhovanovLaudaRouquier algebras of affine Lie type Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

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

Wade, Jeremy, 1981 (University of Oregon, June , 2009)[more][less]Wade, Jeremy, 1981 20100310T00:12:17Z 20100310T00:12:17Z 200906 http://hdl.handle.net/1794/10245 vii, 99 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. We investigate Cesàro summability of the Fourier orthogonal expansion of functions on B d × I m , where B d is the closed unit ball in [Special characters omitted] and I m is the m fold Cartesian product of the interval [1, 1], in terms of orthogonal polynomials with respect to the weight functions (1  z ) α (1 + z ) β (1  x 2 ) λ1/2 , with z ∈ I m and x ∈ B d . In addition, we study a discretized Fourier orthogonal expansion on the cylinder B 2 × [1, 1], which uses a finite number of Radon projections. The Lebesgue constant of this operator is obtained, and the proof utilizes generating functions for associated orthogonal series. Committee in charge: Yuan Xu, Chairperson, Mathematics; Huaxin Lin, Member, Mathematics Jonathan Brundan, Member, Mathematics; Marcin Bownik, Member, Mathematics; Jun Li, Outside Member, Computer & Information Science en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2009; Fourier orthogonal expansions Radon projections Cylindrical functions Cartesian products Mathematics Summability of Fourier orthogonal expansions and a discretized Fourier orthogonal expansion involving radon projections for functions on the cylinder Thesis

Jordan, Alex, 1979 (University of Oregon, June , 2008)[more][less]Jordan, Alex, 1979 20090113T00:17:10Z 20090113T00:17:10Z 200806 http://hdl.handle.net/1794/8283 vii, 41 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. We generalize a theorem of Zhu relating the trace of certain vertex algebra representations and modular invariants to the arena of vertex super algebras. The theorem explains why the space of simple characters for the NeveuSchwarz minimal models NS( p, q ) is modular invariant. It also expresses negative products in terms of positive products, which are easier to compute. As a consequence of the main theorem, the subleading coefficient of the singular vectors of NS( p, q ) is determined for p and q odd. An interesting family of q series identities is established. These consequences established here generalize results of Milas in this field. Adviser: Arkady Vaintrob en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2008; Vertex algebras NeveuSchwarz model Super algebras Zhu's theorem Mathematics A Super Version of Zhu's Theorem Thesis

Dilts, James (University of Oregon, August 18, 2015)[more][less]Isenberg, James Dilts, James 20150818T23:00:52Z 20150818T23:00:52Z 20150818 http://hdl.handle.net/1794/19237 In this dissertation, we prove a number of results regarding the conformal method of finding solutions to the Einstein constraint equations. These results include necessary and sufficient conditions for the Lichnerowicz equation to have solutions, global supersolutions which guarantee solutions to the conformal constraint equations for nearconstantmeancurvature (nearCMC) data as well as for farfromCMC data, a proof of the limit equation criterion in the nearCMC case, as well as a model problem on the relationship between the asymptotic constants of solutions and the ADM mass. We also prove a characterization of the Yamabe classes on asymptotically Euclidean manifolds and resolve the (conformally) prescribed scalar curvature problem on asymptotically Euclidean manifolds for the case of nonpositive scalar curvatures. This dissertation includes previously published coauthored material. en_US University of Oregon All Rights Reserved. differential geometry general relativity partial differential equations The Einstein Constraint Equations on Asymptotically Euclidean Manifolds Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

Welly, Adam (University of Oregon, October 27, 2016)[more][less]He, Weiyong Welly, Adam 20161027T18:40:22Z 20161027T18:40:22Z 20161027 http://hdl.handle.net/1794/20466 Let (M,g) be a quasiSasaki manifold with Reeb vector field xi. Our goal is to understand the structure of M when g is an Einstein metric. Assuming that the S^1 action induced by xi is locally free or assuming a certain nonnegativity condition on the transverse curvature, we prove some rigidity results on the structure of (M,g). Naturally associated to a quasiSasaki metric g is a transverse Kahler metric g^T. The transverse KahlerRicci flow of g^T is the normalized Ricci flow of the transverse metric. Exploiting the transverse Kahler geometry of (M,g), we can extend results in KahlerRicci flow to our transverse version. In particular, we show that a deep and beautiful theorem due to Perleman has its counterpart in the quasiSasaki setting. We also consider evolving a Sasaki metric g by Ricci flow. Unfortunately, if g(0) is Sasaki then g(t) is not Sasaki for t>0. However, in some instances g(t) is quasiSasaki. We examine this and give some qualitative results and examples in the special case that the initial metric is etaEinstein. en_US University of Oregon All Rights Reserved. differential geometry Einstein metric Kahler quasiSasaki Ricci flow Sasaki The Geometry of quasiSasaki Manifolds Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

Vicinsky, Deborah (University of Oregon, August 18, 2015)[more][less]Sadofsky, Hal Vicinsky, Deborah 20150818T23:06:22Z 20150818T23:06:22Z 20150818 http://hdl.handle.net/1794/19283 We construct categories of spectra for two model categories. The first is the category of small categories with the canonical model structure, and the second is the category of directed graphs with the BissonTsemo model structure. In both cases, the category of spectra is homotopically trivial. This implies that the Goodwillie derivatives of the identity functor in each category, if they exist, are weakly equivalent to the zero spectrum. Finally, we give an infinite family of model structures on the category of small categories. en_US University of Oregon All Rights Reserved. Algebraic topology Goodwillie calculus Homotopy theory Model categories The Homotopy Calculus of Categories and Graphs Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

Montgomery, Aaron (University of Oregon, October 3, 2013)[more][less]Levin, David Montgomery, Aaron 20131003T23:37:50Z 20131003T23:37:50Z 20131003 http://hdl.handle.net/1794/13335 We study a family of random walks defined on certain Euclidean lattices that are related to incidence matrices of balanced incomplete block designs. We estimate the return probability of these random walks and use it to determine the asymptotics of the number of balanced incomplete block design matrices. We also consider the problem of collisions of independent simple random walks on graphs. We prove some new results in the collision problem, improve some existing ones, and provide counterexamples to illustrate the complexity of the problem. en_US University of Oregon All Rights Reserved. balanced incomplete block designs collisions of random walks Markov chains Topics in Random Walks Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

Sun, Michael (University of Oregon, September 29, 2014)[more][less]Lin, Huaxin Sun, Michael 20140929T17:46:18Z 20140929T17:46:18Z 20140929 http://hdl.handle.net/1794/18368 In this dissertation we explore the question of existence of a property of group actions on C*algebras known as the tracial Rokhlin property. We prove existence of the property in a very general setting as well as specialise the question to specific situations of interest. For every countable discrete elementary amenable group G, we show that there always exists a Gaction ω with the tracial Rokhlin property on any unital simple nuclear tracially approximately divisible C*algebra A. For the ω we construct, we show that if A is unital simple and Zstable with rational tracial rank at most one and G belongs to the class of countable discrete groups generated by finite and abelian groups under increasing unions and subgroups, then the crossed product A ω G is also unital simple and Zstable with rational tracial rank at most one. We also specialise the question to UHF algebras. We show that for any countable discrete maximally almost periodic group G and any UHF algebra A, there exists a strongly outer product type action α of G on A. We also show the existence of countable discrete almost abelian group actions with the "pointwise" Rokhlin property on the universal UHF algebra. Consequently we get many examples of unital separable simple nuclear C*algebras with tracial rank zero and a unique tracial state appearing as crossed products. en_US University of Oregon All Rights Reserved. C*algebras classification crossed product existence group actions tracial Rokhlin property The Tracial Rokhlin Property for Countable Discrete Amenable Group Actions on Nuclear Tracially Approximately Divisible C*Algebras Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

Bell, Thomas (University of Oregon, October 3, 2013)[more][less]Lu, Peng Bell, Thomas 20131003T23:31:15Z 20131003T23:31:15Z 20131003 http://hdl.handle.net/1794/13231 In the first chapter we consider the question of uniqueness of conformal Ricci flow. We use an energy functional associated with this flow along closed manifolds with a metric of constant negative scalar curvature. Given initial conditions we use this functional to demonstrate the uniqueness of the solution to both the metric and the pressure function along conformal Ricci flow. In the next chapter we study backward Ricci flow of locally homogeneous geometries of 4manifolds which admit compact quotients. We describe the longterm behavior of each class and show that many of the classes exhibit the same behavior near the singular time. In most cases, these manifolds converge to a subRiemannian geometry after suitable rescaling. en_US University of Oregon All Rights Reserved. Conformal Differential Flow Geometry Homogeneous Ricci Uniqueness of Conformal Ricci Flow and Backward Ricci Flow on Homogeneous 4Manifolds Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

Arbo, Matthew (University of Oregon, February 23, 2016)[more][less]Proudfoot, Nicholas Arbo, Matthew 20160224T00:17:14Z 20160224T00:17:14Z 20160223 http://hdl.handle.net/1794/19686 Hypertoric varieties are a class of conical symplectic resolutions which are very computable. In the current literature, they are only defined constructively, using hyperplane arrangements. We provide an abstract definition of a hypertoric variety and a new construction using zonotopal tilings and relate the zonotopal construction to the hyperplane construction. en_US University of Oregon All Rights Reserved. Zonotopes and Hypertoric Varieties Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon
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