Mathematics Theses and Dissertations
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Gardella, Eusebio (University of Oregon, August 18, 2015)[more][less]Phillips, N. Christopher Gardella, Eusebio 20150818T23:13:52Z 20150818T23:13:52Z 20150818 http://hdl.handle.net/1794/19345 This dissertation is concerned with representations of locally compact groups on different classes of Banach spaces. The first part of this work considers representations of compact groups by automorphisms of C*algebras, also known as group actions on C*algebras. The actions we study enjoy a freenesstype of property, namely finite Rokhlin dimension. We investigate the structure of their crossed products, mainly in relation to their classifiability, and compare the notion of finite Rokhlin dimension with other existing notions of noncommutative freeness. In the case of Rokhlin dimension zero, also known as the Rokhlin property, we prove a number of classification theorems for these actions. Also, in this case, much more can be said about the structure of the crossed products. In the last chapter of this part, we explore the extent to which actions with Rokhlin dimension one can be classified. Our results show that even for Z_2actions on O_2, their classification is not Borel, and hence it is intractable. The second part of the present dissertation focuses on isometric representations of groups on Lpspaces. For p=2, these are the unitary representations on Hilbert spaces. We study the Lpanalogs of the full and reduced group \ca s, particularly in connection to their rigidity. One of the main results of this work asserts that for p different from 2, the isometric isomorphism type of the reduced group Lpoperator algebra recovers the group. Our study of group algebras acting on Lpspaces has also led us to answer a 20yearold question of Le Merdy and Junge: for p different from 2, the class of Banach algebras that can be represented on an Lpspace is not closed under quotients. We moreover study representations of groupoids, which are a generalization of groups where multiplication is not always defined. The algebras associated to these objects provide new examples of Lpoperator algebras and recover some previously existing ones. Groupoid Lpoperator algebras are particularly tractable objects. For instance, while groupoid Lpoperator algebras can be classified by their K_0group (an ordered, countable abelian group), we show that UHFLpoperator algebras not arising from groupoids cannot be classified by countable structures. This dissertation includes unpublished coauthored material. en_US University of Oregon All Rights Reserved. C*algebras Classification Cossed product Group action Lpspace ppseudofunctions Compact Group Actions on C*algebras: Classification, NonClassifiability and Crossed Products and Rigidity Results for Lpoperator Algebras 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

Bibby, Christin (University of Oregon, August 18, 2015)[more][less]Proudfoot, Nicholas Bibby, Christin 20150818T23:04:51Z 20150818T23:04:51Z 20150818 http://hdl.handle.net/1794/19273 An abelian arrangement is a finite set of codimension one abelian subvarieties (possibly translated) in a complex abelian variety. We are interested in the topology of the complement of an arrangement. If the arrangement is unimodular, we provide a combinatorial presentation for a differential graded algebra (DGA) that is a model for the complement, in the sense of rational homotopy theory. Moreover, this DGA has a bigrading that allows us to compute the mixed Hodge numbers. If the arrangement is chordal, then this model is a Koszul algebra. In this case, studying its quadratic dual gives a combinatorial description of the Qnilpotent completion of the fundamental group and the minimal model of the complement of the arrangement. This dissertation includes previously unpublished coauthored material. en_US University of Oregon All Rights Reserved. hyperplane arrangements Abelian Arrangements Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

Loubert, Joseph (University of Oregon, August 18, 2015)[more][less]Kleshchev, Alexander Loubert, Joseph 20150818T23:02:53Z 20150818T23:02:53Z 20150818 http://hdl.handle.net/1794/19255 This thesis consists of two parts. In the first we prove that the KhovanovLaudaRouquier algebras $R_\alpha$ of finite type are (graded) affine cellular in the sense of Koenig and Xi. In fact, we establish a stronger property, namely that the affine cell ideals in $R_\alpha$ are generated by idempotents. This in particular implies the (known) result that the global dimension of $R_\alpha$ is finite. In the second part we use the presentation of the Specht modules given by KleshchevMathasRam to derive results about Specht modules. In particular, we determine all homomorphisms from an arbitrary Specht module to a fixed Specht module corresponding to any hook partition. Along the way, we give a complete description of the action of the standard KLR generators on the hook Specht module. This work generalizes a result of James. This dissertation includes previously published coauthored material. en_US University of Oregon All Rights Reserved. Affine Cellularity KLR Algebras Specht Modules Affine Cellularity of Finite Type KLR Algebras, and Homomorphisms Between Specht Modules for KLR Algebras in Affine Type A Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

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

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

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

Ro, Min (University of Oregon, August 18, 2015)[more][less]Lin, Huaxin Ro, Min 20150818T22:52:07Z 20150818T22:52:07Z 20150818 http://hdl.handle.net/1794/19199 In this dissertation, we explore the approximate diagonalization of unital homomorphisms between C*algebras. In particular, we prove that unital homomorphisms from commutative C*algebras into simple separable unital C*algebras with tracial rank at most one are approximately diagonalizable. This is equivalent to the approximate diagonalization of commuting sets of normal matrices. We also prove limited generalizations of this theorem. Namely, certain injective unital homomorphisms from commutative C*algebras into simple separable unital C*algebras with rational tracial rank at most one are shown to be approximately diagonalizable. Also unital injective homomorphisms from AHalgebras with unique tracial state into separable simple unital C*algebras of tracial rank at most one are proved to be approximately diagonalizable. Counterexamples are provided showing that these results cannot be extended in general. Finally, we prove that for unital homomorphisms between AFalgebras, approximate diagonalization is equivalent to a combinatorial problem involving sections of lattice points in cones. en_US University of Oregon All Rights Reserved. approximate diagonalization C*algebras Elliott classification Approximate Diagonalization of Homomorphisms Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

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