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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

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

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

Comes, Jonathan, 1981 (University of Oregon, June , 2010)[more][less]Comes, Jonathan, 1981 20101203T20:42:43Z 20101203T20:42:43Z 201006 http://hdl.handle.net/1794/10867 x, 81 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 give an exposition of Deligne's tensor category Rep(St) where t is not necessarily an integer. Thereafter, we give a complete description of the blocks in Rep(St) for arbitrary t. Finally, we use our result on blocks to decompose tensor products and classify tensor ideals in Rep(St). Committee in charge: Victor Ostrik, Chairperson, Mathematics; Daniel Dugger, Member, Mathematics; Jonathan Brundan, Member, Mathematics; Alexander Kleshchev, Member, Mathematics; Michael Kellman, Outside Member, Chemistry en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2010; Tensor category Symmetric groups Decomposed blocks Tensor products Tensor ideals Mathematics Theoretical mathematics Blocks in Deligne's category Rep(St) Thesis

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

Pearson, Kelly Jeanne, 1970 (2000)[more][less]Pearson, Kelly Jeanne, 1970 20080210T03:23:05Z 20080210T03:23:05Z 2000 0599841095 http://hdl.handle.net/1794/141 vii, 91 p. A print copy of this title is available through the UO Libraries under the call number: MATH QC20.7.H65 P43 2000 The OrlikSolomon algebra of a hyperplane arrangement first appeared from the Brieskorn and OrlikSolomon theorems as the cohomology of the complement of this arrangement (if the ground field is complex). Later, it was discovered that this algebra plays an important role in many other problems. In particular, define the cohomology of an OrlikSolomon algebra as that of the complex formed by its homogeneous components with the differential defined via multiplication by an element of degree one. Cohomology of the OrlikSolomon algebra is mostly studied in dimension one, and very little is known about the higher dimensions. We study this cohomology in higher dimensions. Adviser: Sergey Yuzvinsky. 3856498 bytes 122293 bytes 1483 bytes application/pdf text/plain text/plain en University of Oregon theses, Dept. of Mathematics, Ph. D., 2000 Cohomology operations Homology theory Cohomology of the OrlikSolomon algebras Thesis

Archey, Dawn Elizabeth, 1979 (University of Oregon, June , 2008)[more][less]Archey, Dawn Elizabeth, 1979 20081220T02:10:58Z 20081220T02:10:58Z 200806 http://hdl.handle.net/1794/8155 viii, 107 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. This dissertation consists of two related parts. In the first portion we use the tracial Rokhlin property for actions of a finite group G on stably finite simple unital C *algebras containing enough projections. The main results of this part of the dissertation are as follows. Let A be a stably finite simple unital C *algebra and suppose a is an action of a finite group G with the tracial Rokhlin property. Suppose A has real rank zero, stable rank one, and suppose the order on projections over A is determined by traces. Then the crossed product algebra C * ( G, A, Ã Ã Â±) also has these three properties. In the second portion of the dissertation we introduce an analogue of the tracial Rokhlin property for C *algebras which may not have any nontrivial projections called the projection free tracial Rokhlin property . Using this we show that under certain conditions if A is an infinite dimensional simple unital C *algebra with stable rank one and Ã Ã Â± is an action of a finite group G with the projection free tracial Rokhlin property, then C * ( G, A, Ã Ã Â±) also has stable rank one. Adviser: Phillips, N. Christopher en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph.D., 2008; Mathematics Cuntz subequivalence Stable rank one Tracial Rokhlin property Finite group actions Crossed product C*algebras Crossed product C*algebras by finite group actions with a generalized tracial Rokhlin property Thesis

Crossed product C*algebras of certain nonsimple C*algebras and the tracial quasiRokhlin propertyBuck, Julian Michael, 1982 (University of Oregon, June , 2010)[more][less]Buck, Julian Michael, 1982 20101130T23:48:55Z 20101130T23:48:55Z 201006 http://hdl.handle.net/1794/10849 viii, 113 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. This dissertation consists of four principal parts. In the first, we introduce the tracial quasiRokhlin property for an automorphism α of a C *algebra A (which is not assumed to be simple or to contain any projections). We then prove that under suitable assumptions on the algebra A , the associated crossed product C *algebra C *([Special characters omitted.] , A , α) is simple, and the restriction map between the tracial states of C *([Special characters omitted.] , A , α) and the αinvariant tracial states on A is bijective. In the second part, we introduce a comparison property for minimal dynamical systems (the dynamic comparison property) and demonstrate sufficient conditions on the dynamical system which ensure that it holds. The third part ties these concepts together by demonstrating that given a minimal dynamical system ( X, h ) and a suitable simple C *algebra A , a large class of automorphisms β of the algebra C ( X, A ) have the tracial quasiRokhlin property, with the dynamic comparison property playing a key role. Finally, we study the structure of the crossed product C *algebra B = C *([Special characters omitted.] , C ( X , A ), β) by introducing a subalgebra B { y } of B , which is shown to be large in a sense that allows properties B { y } of to pass to B . Several conjectures about the deeper structural properties of B { y } and B are stated and discussed. Committee in charge: Christopher Phillips, Chairperson, Mathematics; Daniel Dugger, Member, Mathematics; Huaxin Lin, Member, Mathematics; Marcin Bownik, Member, Mathematics; Van Kolpin, Outside Member, Economics en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2010; Dynamical systems Minimal homeomorphisms Crossed product algebras Tracial property Automorphisms C*algebras QuasiRokhlin property Mathematics Theoretical mathematics Crossed product C*algebras of certain nonsimple C*algebras and the tracial quasiRokhlin property Thesis

Sun, Wei, 1979 (University of Oregon, June , 2010)[more][less]Sun, Wei, 1979 20101222T01:32:21Z 20101222T01:32:21Z 201006 http://hdl.handle.net/1794/10912 vii, 124 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 is a study of the relationship between minimal dynamical systems on the product of the Cantor set ( X ) and torus ([Special characters omitted]) and their corresponding crossed product C *algebras. For the case when the cocyles are rotations, we studied the structure of the crossed product C *algebra A by looking at a large subalgebra A x . It is proved that, as long as the cocyles are rotations, the tracial rank of the crossed product C *algebra is always no more than one, which then indicates that it falls into the category of classifiable C *algebras. In order to determine whether the corresponding crossed product C *algebras of two such minimal dynamical systems are isomorphic or not, we just need to look at the Elliott invariants of these C *algebras. If a certain rigidity condition is satisfied, it is shown that the crossed product C *algebra has tracial rank zero. Under this assumption, it is proved that for two such dynamical systems, if A and B are the corresponding crossed product C *algebras, and we have an isomorphism between K i ( A ) and K i ( B ) which maps K i (C(X ×[Special characters omitted])) to K i (C( X ×[Special characters omitted])), then these two dynamical systems are approximately K conjugate. The proof also indicates that C *strongly flip conjugacy implies approximate K conjugacy in this case. We also studied the case when the cocyles are Furstenberg transformations, and some results on weakly approximate conjugacy and the K theory of corresponding crossed product C *algebras are obtained. Committee in charge: Huaxin Lin, Chairperson, Mathematics Daniel Dugger, Member, Mathematics; Christopher Phillips, Member, Mathematics; Arkady Vaintrob, Member, Mathematics; LiShan Chou, Outside Member, Human Physiology en_US University of Oregon University of Oregon theses, Dept. of Mathematics, Ph. D., 2010; Tracial rank Approximate conjugacy C*algebras Minimal dynamical systems Cantor set Torus Mathematics Theoretical mathematics Crossed product C*algebras of minimal dynamical systems on the product of the Cantor set and the torus 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

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
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