Browsing Mathematics Theses and Dissertations by Title

Brown, Jonathan, 1975 (University of Oregon, June , 2009)[more][less]Brown, Jonathan, 1975 20100219T01:28:27Z 20100219T01:28:27Z 200906 http://hdl.handle.net/1794/10201 ix, 114 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. In this work we prove that the finite W algebras associated to nilpotent elements in the symplectic or orthogonal Lie algebras whose Jordan blocks are all the same size are quotients of twisted Yangians. We use this to classify the finite dimensional irreducible representations of these finite W algebras. Committee in charge: Jonathan Brundan, CoChairperson, Mathematics; Victor Ostrik, CoChairperson, Mathematics; Arkady Berenstein, Member, Mathematics; Hal Sadofsky, Member, Mathematics; Christopher Wilson, Outside Member, Computer & Information Science en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2009; Finite Walgebras Nilpotent Symplectic Quantum algebra Mathematics Walgebras Finite Walgebras of classical type Thesis

Thornton, Josiah (University of Oregon, 2012)[more][less]Ostrik, Victor Thornton, Josiah Thornton, Josiah 20121026T04:06:42Z 20121026T04:06:42Z 2012 http://hdl.handle.net/1794/12450 We give an exposition of neargroup categories and generalized neargroup categories. We show that both have a pseudounitary structure. We complete the classification of braided neargroup categories and discuss the inherent structures on both symmetric and modular generalized neargroup categories. en_US University of Oregon All Rights Reserved. Generalized NearGroup Categories Electronic Thesis or Dissertation

Heuser, Aaron, 1978 (University of Oregon, June , 2010)[more][less]Heuser, Aaron, 1978 20101203T22:34:13Z 20101203T22:34:13Z 201006 http://hdl.handle.net/1794/10870 x, 110 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. This dissertation examines the existence of the selfintersection local time for a superprocess over a stochastic flow in dimensions d ≤ 3, which through constructive methods, gives a Tanaka like representation. The superprocess over a stochastic flow is a superprocess with dependent spatial motion, and thus Dynkin's proof of existence, which requires multiplicity of the logLaplace functional, no longer applies. Skoulakis and Adler's method of calculating moments is extended to higher moments, from which existence follows. Committee in charge: Hao Wang, CoChairperson, Mathematics; David Levin, CoChairperson, Mathematics; Christopher Sinclair, Member, Mathematics; Huaxin Lin, Member, Mathematics; Van Kolpin, Outside Member, Economics en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2010; Selfintersection Tanaka representation Superprocess Stochastic flow Mathematics Theoretical mathematics Generalized selfintersection local time for a superprocess over a stochastic flow Thesis

Pelatt, Kristine (University of Oregon, 2012)[more][less]Sinha, Dev Pelatt, Kristine Pelatt, Kristine 20121026T04:03:49Z 20121026T04:03:49Z 2012 http://hdl.handle.net/1794/12423 We produce explicit geometric representatives of nontrivial homology classes in Emb(S1,Rd), the space of knots, when d is even. We generalize results of Cattaneo, CottaRamusino and Longoni to define cycles which live off of the vanishing line of a homology spectral sequence due to Sinha. We use con figuration space integrals to show our classes pair nontrivially with cohomology classes due to Longoni. We then give an alternate formula for the first differential in the homology spectral sequence due to Sinha. This differential connects the geometry of the cycles we define to the combinatorics of the spectral sequence. The new formula for the differential also simplifies calculations in the spectral sequence. en_US University of Oregon All Rights Reserved. embedding spaces spaces of knots Geometry and Combinatorics Pertaining to the Homology of Spaces of Knots Electronic Thesis or Dissertation

Collins, John, 1981 (University of Oregon, June , 2009)[more][less]Collins, John, 1981 20100225T23:49:36Z 20100225T23:49:36Z 200906 http://hdl.handle.net/1794/10218 vi, 85 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. We define and study a gluing procedure for Bridgeland stability conditions in the situation where a triangulated category has a semiorthogonal decomposition. As one application, we construct an open, contractible subset U in the stability manifold of the derived category [Special characters omitted.] of [Special characters omitted.] equivariant coherent sheaves on a smooth curve X , associated with a degree 2 map X [arrow right] Y , where Y is another curve. In the case where X is an elliptic curve we construct an open, connected subset in the stability manifold using exceptional collections containing the subset U . We also give a new proof of the constructibility of exceptional collections on [Special characters omitted.] . This dissertation contains previously unpublished coauthored material. Committee in charge: Alexander Polishchuk, Chairperson, Mathematics; Daniel Dugger, Member, Mathematics; Victor Ostrik, Member, Mathematics; Brad Shelton, Member, Mathematics; Michael Kellman, Outside Member, Chemistry en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2009; Stability conditions Equivariant sheaves Derived categories Elliptic curve Mathematics Gluing Bridgeland's stability conditions and Z2equivariant sheaves on curves Thesis

Nash, David A., 1982 (University of Oregon, June , 2010)[more][less]Nash, David A., 1982 20101203T22:54:08Z 20101203T22:54:08Z 201006 http://hdl.handle.net/1794/10871 xii, 76 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 graded representation theory of the IwahoriHecke algebra, denoted by Hd , of the symmetric group over a field of characteristic zero at a root of unity. More specifically, we use graded Specht modules to calculate the graded decomposition numbers for Hd . The algorithm arrived at is the LascouxLeclercThibon algorithm in disguise. Thus we interpret the algorithm in terms of graded representation theory. We then use the algorithm to compute several examples and to obtain a closed form for the graded decomposition numbers in the case of twocolumn partitions. In this case, we also precisely describe the 'reduction modulo p' process, which relates the graded irreducible representations of Hd over [Special characters omitted.] at a p th root of unity to those of the group algebra of the symmetric group over a field of characteristic p. Committee in charge: Alexander Kleshchev, Chairperson, Mathematics; Jonathan Brundan, Member, Mathematics; Boris Botvinnik, Member, Mathematics; Victor Ostrik, Member, Mathematics; William Harbaugh, Outside Member, Economics en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2010; Symmetric groups Specht modules Irreducible representation Graded representation Hecke algebras Mathematics Theoretical mathematics Graded representation theory of Hecke algebras Thesis

Moseley, Daniel (University of Oregon, 2012)[more][less]Proudfoot, Nicholas Moseley, Daniel Moseley, Daniel 20121026T03:58:49Z 20121026T03:58:49Z 2012 http://hdl.handle.net/1794/12373 In this dissertation, we will look at two families of algebras with connections to hyperplane arrangements that admit actions of finite groups. One of the fundamental questions to ask is how these decompose into irreducible representations. For the first family of algebras, we will use equivariant cohomology techniques to reduce the computation to an easier one. For the second family, we will use two decompositions over the intersection lattice of the hyperplane arrangement to aid us in computation. en_US University of Oregon All Rights Reserved. Group Actions on Hyperplane Arrangements Electronic Thesis or Dissertation

Wilson, James B., 1980 (University of Oregon, June , 2008)[more][less]Wilson, James B., 1980 20090115T00:44:03Z 20090115T00:44:03Z 200806 http://hdl.handle.net/1794/8302 viii, 125 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. Finite p groups are studied using bilinear methods which lead to using nonassociative rings. There are three main results, two which apply only to p groups and the third which applies to all groups. First, for finite p groups P of class 2 and exponent p the following are invariants of fully refined central decompositions of P : the number of members in the decomposition, the multiset of orders of the members, and the multiset of orders of their centers. Unlike for direct product decompositions, Aut P is not always transitive on the set of fully refined central decompositions, and the number of orbits can in fact be any positive integer. The proofs use the standard semisimple and radical structure of Jordan algebras. These algebras also produce useful criteria for a p group to be centrally indecomposable. In the second result, an algorithm is given to find a fully refined central decomposition of a finite p group of class 2. The number of algebraic operations used by the algorithm is bounded by a polynomial in the log of the size of the group. The algorithm uses a Las Vegas probabilistic algorithm to compute the structure of a finite ring and the Las Vegas MeatAxe is also used. However, when p is small, the probabilistic methods can be replaced by deterministic polynomialtime algorithms. The final result is a polynomial time algorithm which, given a group of permutations, matrices, or a polycyclic presentation; returns a Remak decomposition of the group: a fully refined direct decomposition. The method uses group varieties to reduce to the case of p groups of class 2. Bilinear and ring theory methods are employed there to complete the process. Adviser: William M. Kantor en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2008; Computer science Mathematics pgroups Jordan algebras Group decompositions Central products Direct products Algorithms Group decompositions, Jordan algebras, and algorithms for pgroups Thesis

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

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

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

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