Mathematics Theses and Dissertations
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Schultz, Patrick (University of Oregon, September 29, 2014)[more][less]Dugger, Daniel Schultz, Patrick 20140929T17:53:19Z 20140929 http://hdl.handle.net/1794/18429 We present a generalized framework for the theory of algebraic weak factorization systems, building on work by Richard Garner and Emily Riehl. We define cyclic 2fold double categories, and bimonads (or bialgebras) and lax/colax bimonad morphisms inside cyclic 2fold double categories. After constructing a cyclic 2fold double category <bold>FF</bold>(D) of functorial factorization systems in any sufficiently nice 2category D, we show that bimonads and lax/colax bimonad morphsims in <bold>FF</bold>(Cat) agree with previous definitions of algebraic weak factorization systems and lax/colax morphisms. We provide a proof of one of the core technical theorems from previous work on algebraic weak factorization systems in our generalized framework. Finally, we show that this framework can be further generalized to cyclic 2fold double multicategories, incorporating Quillen functors of several variables. en_US University of Oregon All Rights Reserved. Category Theory Double Categories Model Categories Algebraic Weak Factorization Systems in Double Categories Electronic Thesis or Dissertation 20160929 Ph.D. doctoral Department of Mathematics University of Oregon

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

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

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

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

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

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

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

Platt, David (University of Oregon, October 3, 2013)[more][less]Polishchuk, Alexander Platt, David 20131003T23:32:01Z 20131003T23:32:01Z 20131003 http://hdl.handle.net/1794/13244 We give a formula for the Chern character on the DG category of global matrix factorizations on a smooth scheme $X$ with superpotential $w\in \Gamma(\O_X)$. Our formula takes values in a Cech model for Hochschild homology. Our methods may also be adapted to get an explicit formula for the Chern character for perfect complexes of sheaves on $X$ taking values in right derived global sections of the DeRham algebra. Along the way we prove that the DG version of the Chern Character coincides with the classical one for perfect complexes. en_US University of Oregon All Rights Reserved. Chern Character Matrix Factorizations Noncommutative Geometry Chern Character for Global Matrix Factorizations 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

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

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

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

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

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

Fisette, Robert (University of Oregon, 2012)[more][less]Polishchuk, Alexander Fisette, Robert Fisette, Robert 20121026T03:58:22Z 20121026T03:58:22Z 2012 http://hdl.handle.net/1794/12368 We choose a generator G of the derived category of coherent sheaves on a smooth curve X of genus g which corresponds to a choice of g distinguished points P1, . . . , Pg on X. We compute the Hochschild cohomology of the algebra B = Ext (G,G) in certain internal degrees relevant to extending the associative algebra structure on B to an A1structure, which demonstrates that A1structures on B are finitely determined for curves of arbitrary genus. When the curve is taken over C and g = 1, we amend an explicit A1structure on B computed by Polishchuk so that the higher products m6 and m8 become Hochschild cocycles. We use the cohomology classes of m6 and m8 to recover the jinvariant of the curve. When g 2, we use Massey products in Db(X) to show that in the A1structure on B, m3 is homotopic to 0 if and only if X is hyperelliptic and P1, . . . , Pg are chosen to be Weierstrass points. iv en_US University of Oregon All Rights Reserved. Ainfinity Curve Elliptic curve Hochschild cohomology jinvariant The Ainfinity Algebra of a Curve and the Jinvariant Electronic Thesis or Dissertation

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

Conner, Andrew Brondos, 1981 (University of Oregon, June , 2011)[more][less]Conner, Andrew Brondos, 1981 20110912T19:31:59Z 20130604T17:45:42Z 201106 http://hdl.handle.net/1794/11559 x, 68 p. : ill. (some col.) Motivated by the Adams spectral sequence for computing stable homotopy groups, Priddy defined a class of algebras called Koszul algebras with nice homological properties. Many important algebras arising naturally in mathematics are Koszul, and the Koszul property is often tied to important structure in the settings which produced the algebras. However, the strong defining conditions for a Koszul algebra imply that such algebras must be quadratic. A very natural generalization of Koszul algebras called K 2 algebras was recently introduced by Cassidy and Shelton. Unlike other generalizations of the Koszul property, the class of K 2 algebras is closed under many standard operations in ring theory. The class of K 2 algebras includes ArtinSchelter regular algebras of global dimension 4 on three linear generators as well as graded complete intersections. Our work comprises two distinct projects. Each project was motivated by an aspect of the theory of Koszul algebras which we regard as sufficiently powerful or fundamental to warrant an interpretation for K 2 algebras. A very useful theorem due to Backelin and Fröberg states that if A is a Koszul algebra and I is a quadratic ideal of A which is Koszul as a left A module, then the factor algebra A/I is a Koszul algebra. We prove that if A is Koszul algebra and A I is a K 2 module, then A/I is a K 2 algebra provided A/I acts trivially on Ext A ( A/I,k ). As an application of our theorem, we show that the class of sequentially CohenMacaulay StanleyReisner rings are K 2 algebras and we give examples that suggest the class of K 2 StanleyReisner rings is actually much larger. Another important recent development in ring theory is the use of A ∞ algebras. One can characterize Koszul algebras as those graded algebras whose Yoneda algebra admits only trivial A ∞ structure. We show that, in contrast to the situation for Koszul algebras, vanishing of higher A ∞ structure on the Yoneda algebra of a K 2 algebra need not be determined in any obvious way by the degrees of defining relations. We also demonstrate that obvious patterns of vanishing among higher multiplications cannot detect the K 2 property. This dissertation includes previously unpublished coauthored material. Committee in charge: Dr. Brad Shelton, Chair; Dr. Victor Ostrik, Member; Dr. Nicholas Proudfoot, Member; Dr. Arkady Vaintrob, Member; Dr. David Boush, Outside Member en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2011; Ainfinity Face ring K2 Koszul algebras StanleyReisner Yoneda algebra Ring theory Mathematics A(infinity)structures, generalized Koszul properties, and combinatorial topology 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

Liang, Hutian (University of Oregon, June , 2010)[more][less]Liang, Hutian 20110114T18:44:40Z 20110114T18:44:40Z 201006 http://hdl.handle.net/1794/10938 viii, 133 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 dissertation, we will study the crossed product C*algebras obtained from free and minimal [Special characters omitted.] actions on compact metric spaces with finite covering dimension. We first define stable recursive subhomogeneous algebras (SRSHAs), which differ from recursive subhomogeneous algebras introduced by N. C. Phillips in that the irreducible representations of SRSHAs are infinite dimensional instead of finite dimensional. We show that simple inductive limits of SRSHAs with no dimension growth in which the connecting maps are injective and nonvanishing have topological stable rank one. We then construct C*subalgebras of the crossed product that are analogous to the C*subalgebras in the studies of free minimal [Special characters omitted.] actions on compact metric spaces with finite covering dimension. Finally, we prove that these C*algebras are in fact simple inductive limits of SRSHAs in which the connecting maps are injective and nonvanishing. Thus these C*subalgebras have topological stable rank one. Committee in charge: Christopher Phillips, Chairperson, Mathematics; Boris Botvinnik, Member, Mathematics; Huaxin Lin, Member, Mathematics; Yuan Xu, Member, Mathematics; Dietrich Belitz, Outside Member, Physics en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2010; Free minimal action Metric space Subhomogeneous algebras Infinite dimensions Topological stability Mathematics Theoretical mathematics The crossed product of C(X) by a free minimal action of R Thesis