Mathematics Theses and Dissertations
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This collection contains some of the theses and dissertations produced by students in the University of Oregon Mathematics Graduate Program. Paper copies of these and other dissertations and theses are available through the UO Libraries.
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Chettih, Safia (University of Oregon, November 21, 2016)[more][less]Sinha, Dev Chettih, Safia 20161121T17:00:59Z 20161121T17:00:59Z 20161121 http://hdl.handle.net/1794/20728 We prove that the nonkequal configuration space of a graph has a discretized model, analogous to the discretized model for configurations on graphs. We apply discrete Morse theory to the latter to give an explicit combinatorial formula for the ranks of homology and cohomology of configurations of two points on a tree. We give explicit presentations for homology and cohomology classes as well as pairings for ordered and unordered configurations of two and three points on a few simple trees, and show that the first homology group of ordered and unordered configurations of two points in any tree is generated by the first homology groups of configurations of two points in three particular graphs, K_{1,3}, K_{1,4}, and the trivalent tree with 6 vertices and 2 vertices of degree 3, via graph embeddings. en_US University of Oregon All Rights Reserved. configuration space discrete Morse theory graph braid group nonkequal configuration Dancing in the Stars: Topology of Nonkequal Configuration Spaces of Graphs Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

Davidson, Nicholas (University of Oregon, November 21, 2016)[more][less]Brundan, Jonathan Davidson, Nicholas 20161121T16:58:19Z 20161121T16:58:19Z 20161121 http://hdl.handle.net/1794/20704 This dissertation uses techniques from the theory of categorical actions of KacMoody algebras to study the analog of the BGG category O for the queer Lie superalgebra. Chen recently reduced many questions about this category to its socalled types A, B, and C blocks. The type A blocks were completely described in joint work with Brundan in terms of the general linear Lie superalgebra. This dissertation proves that the type C blocks admit the structure of a tensor product categorification of the nfold tensor power of the natural sp_\inftymodule. Using this result, we relate the combinatorics for these blocks to Webster’s orthodox bases for the quantum group of type C_\infty, verifying the truth of a recent conjecture of ChengKwonWang. This dissertation contains coauthored material. en_US University of Oregon All Rights Reserved. Categorification KacMoody Representation Theory Superalgebra Supercategorification Supercategory Categorical Actions on Supercategory O Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

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

Hilburn, Justin (University of Oregon, October 27, 2016)[more][less]Proudfoot, Nicholas Hilburn, Justin 20161027T18:38:29Z 20161027T18:38:29Z 20161027 http://hdl.handle.net/1794/20456 In this thesis I show that indecomposable projective and tilting modules in hypertoric category O are obtained by applying a variant of the geometric Jacquet functor of Emerton, Nadler, and Vilonen to certain Gel'fandKapranovZelevinsky hypergeometric systems. This proves the abelian case of a conjecture of Bullimore, Gaiotto, Dimofte, and Hilburn on the behavior of generic Dirichlet boundary conditions in 3d N=4 SUSY gauge theories. en_US University of Oregon All Rights Reserved. 3d N=4 boundary condition category O hypertoric symplectic duality symplectic resolution GKZ Hypergeometric Systems and Projective Modules in Hypertoric Category O Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

Iverson, Joseph (University of Oregon, October 27, 2016)[more][less]Bownik, Marcin Iverson, Joseph 20161027T18:36:03Z 20161027T18:36:03Z 20161027 http://hdl.handle.net/1794/20443 Let $G$ be a second countable, locally compact group which is either compact or abelian, and let $\rho$ be a unitary representation of $G$ on a separable Hilbert space $\mathcal{H}_\rho$. We examine frames of the form $\{ \rho(x) f_j \colon x \in G, j \in I\}$ for families $\{f_j\}_{j \in I}$ in $\mathcal{H}_\rho$. In particular, we give necessary and sufficient conditions for the joint orbit of a family of vectors in $\mathcal{H}_\rho$ to form a continuous frame. We pay special attention to this problem in the setting of shift invariance. In other words, we fix a larger second countable locally compact group $\Gamma \supset G$ containing $G$ as a closed subgroup, and we let $\rho$ be the action of $G$ on $L^2(\Gamma)$ by left translation. In both the compact and the abelian settings, we introduce notions of Zak transforms on $L^2(\Gamma)$ which simplify the analysis of group frames. Meanwhile, we run a parallel program that uses the Zak transform to classify closed subspaces of $L^2(\Gamma)$ which are invariant under left translation by $G$. The two projects give compatible outcomes. This dissertation contains previously published material. en_US University of Oregon All Rights Reserved. compact group dual integrable group frame LCA group shiftinvariant Zak transform Frames Generated by Actions of Locally Compact Groups Electronic Thesis or Dissertation Ph.D. doctoral Department of Mathematics University of Oregon

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

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

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