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Online 1. Asymptotics of Gaussian processes and Markov chains [2018]
 Zhai, Alex, author.
 [Stanford, California] : [Stanford University], 2018.
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In this thesis, we present several results on the asymptotic behavior of Gaussian processes and Markov chains. In the first part, focused on Gaussian processes, we prove a central limit theorem for the sum of i.i.d. highdimensional random vectors. Surprisingly, not much is known about the optimal dependence of the convergence rate on the dimension of the vectors. Our main contribution is to prove a convergence rate in quadratic transportation distance that is close to optimal in both the dimension and the number of summands. We next prove a result (based on joint work with Jian Ding and Ronen Eldan) about general Gaussian processes: we show that if the maximum of a Gaussian process is strongly concentrated around its expectation (called "superconcentration"), then with high probability the process has many nearmaximal values with low pairwise correlations (called "multiple peaks"). Such phenomena naturally arise in the analysis of disordered systems in statistical physics, where the Gaussian process values correspond to energy levels. Our result adds to an overall picture of the behavior of superconcentrated Gaussian processes described by Chatterjee. The second part of the thesis contains results concerning asymptotic behavior of Markov chains. For random walk on a graph, we prove a sharpening of a relationship established by Ding, Lee, and Peres between the cover time and the Gaussian free field. In particular, our estimate implies that in families of graphs (of size growing to infinity) where the hitting time is asymptotically much smaller than the cover time, the cover time is exponentially concentrated around its expectation, and this expectation has a simple asymptotic formula in terms of the Gaussian free field. We also analyze the mixing time of a Markov chain, known as the product replacement walk, on ntuples of elements of some finite group. One step of the walk involves randomly choosing two of the elements a and b and multiplying a by either b or the inverse of b, with equal probability. The product replacement walk has been extensively studied in the context of random generation of group elements and is part of a larger class of Markov chains that includes random walks on matrix groups over finite fields and certain interacting particle system models. Based on joint work with Yuval Peres and Ryokichi Tanaka, we prove that the product replacement walk exhibits a cutoff phenomenon as n goes to infinity: the chain rapidly transitions from being unmixed to mixed after around 3/2 n log(n) steps.
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Online 2. The equivariant cobordism category [2018]
 Szűcs, Gergely, author.
 [Stanford, California] : [Stanford University], 2018.
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For a finite group G, we define an equivariant cobordism category C_d^G. Objects of the category are (d1)dimensional closed smooth Gmanifolds and morphisms are smooth ddimensional equivariant cobordisms. We identify the homotopy type of its classifying space (i.e. geometric realization of its simplicial nerve) as the fixed points of the infinite loop space of an equivariant spectrum.
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Online 3. Factorization theory of Thom spectra, twists, and duality [2018]
 Klang, Inbar, author.
 [Stanford, California] : [Stanford University], 2018.
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This thesis includes two related projects. The first project determines the factorization homology of Thom spectra of nfold loop maps, and uses this to study the topological Hochschild cohomology of such Thom spectra. Our description of the factorization homology of Thom spectra can be viewed as a twisted form of the nonabelian Poincare duality theorem of Segal, Salvatore, and Lurie, and permits calculations of factorization homology of cobordism spectra and certain EilenbergMacLane spectra. Our description of the Hochschild cohomology of these Thom spectra enables calculations and a description in terms of sections of a parametrized spectrum. This allows us to deduce a duality between topological Hochschild homology and topological Hochschild cohomology, and gives ring structures on a certain family of Thom spectra, which were not previously known to be ring spectra. The second project is joint work with Ralph Cohen in which we import the theory of ``CalabiYau" algebras and categories from symplectic topology and topological field theories to the setting of spectra. We define two types of CalabiYau structures in the setting of ring spectra: one that applies to compact algebras and one that applies to smooth algebras. We apply this theory to describe, prove, and explain a duality between the manifold string topology of Chas and Sullivan and the Lie group string topology of ChataurMenichi. Using results from the first project in this thesis, we prove that Thom ring spectra of (virtual) bundles over the loop space of a manifold have a CalabiYau structure. In the case when the manifold is a sphere, we use this structure to study Lagrangian immersions of the sphere into its cotangent bundle, recasting work of Abouzaid and Kragh.
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Online 4. The flexibility of caustics [electronic resource] [2018]
 AlvarezGavela, Daniel.
 2018.
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In this thesis we establish a full hprinciple for the simplification of singularities of Lagrangian and Legendrian fronts. More precisely, we show that if the obvious homotopy theoretic obstruction to the simplification of singularities vanishes, then the simplification can be achieved by means of an ambient Hamiltonian isotopy. The hprinciple is full in that it holds in C^0close, relative and parametric versions. Among several applications of the hprinciple we obtain a generalization of the Reidemeister theorem for Legendrian knots in the standard contact R^3, which allows for the simplification of the singularities of the front of a family of Legendrian knots parametrized by a space of arbitrary dimension. To prove our result we use two wellknown tools in the philosophy of the hprinciple: the holonomic approximation lemma and the wrinkled embeddings theorem. However, both of these tools need to be upgraded in order to be applicable to the situation at hand. For this purpose we refine the holonomic approximation lemma to a version in which cutoffs can be carefully controlled and we adapt the wrinkled embeddings theorem to the setting of Lagrangian and Legendrian embeddings.
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Online 5. Generalized DonaldsonThomas invariants via kirwan blowups [2018]
 Savvas, Michail, author.
 [Stanford, California] : [Stanford University], 2018.
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In this thesis, we develop a virtual cycle approach towards generalized DonaldsonThomas theory of CalabiYau threefolds. Starting with an Artin moduli stack parametrizing semistable sheaves or perfect complexes, we construct an associated DeligneMumford stack, called its Kirwan partial desingularization, with an induced semiperfect obstruction theory of virtual dimension zero, and define the generalized DonaldsonThomas invariant via Kirwan blowups as the degree of the corresponding virtual cycle. The key ingredients are a generalization of Kirwan's partial desingularization procedure and recent results from derived symplectic geometry regarding the local structure of stacks of sheaves and perfect complexes on CalabiYau threefolds. Examples of applications include Gieseker stability of coherent sheaves and Bridgeland and polynomial stability of perfect complexes. In the case of Gieseker semistable sheaves, this new DonaldsonThomas invariant is invariant under deformations of the complex structure of the CalabiYau threefold. More generally, deformation invariance is true under appropriate assumptions which are expected to hold in all cases.
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Online 6. Highergenus wallcrossing in LandauGinzburg theory [2018]
 Zhou, Yang, author.
 [Stanford, California] : [Stanford University], 2018.
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For a Fermat quasihomogeneous polynomial, we study the associated weighted FanJarvisRuanWitten theory with narrow insertions. We prove a wallcrossing formula in all genera via localization on a master space, which is constructed by introducing an additional tangent vector to the moduli problem. This is a LandauGinzburg theory analogue of the highergenus quasimap wallcrossing formula proved by CiocanFontanine and Kim. It generalizes the genus$0$ result by RossRuan and the genus$1$ result by GuoRoss. We apply similar techniques to prove an wallcrossing formula varying the weights of marked points in the hybridmodel. As an application, this removes the assumption on marked points in the wallcrossing formula of CladerJandaRuan.
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Online 7. NearlyKähler 6manifolds of cohomogeneity two : local theory [2018]
 Madnick, Jesse Ochs, author.
 [Stanford, California] : [Stanford University], 2018.
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We study nearlyKahler 6manifolds equipped with a cohomogeneitytwo Lie group action for which each principal orbit is coisotropic. If the metric is complete, this last condition is automatically satisfied. We will show that the acting Lie group must be 4dimensional and nonabelian. We partition the class of such nearlyKahler structures into three types (called I, II, III) and prove a local existence and generality result for each type. Metrics of Types I and II are shown to be incomplete. We also derive a quasilinear elliptic PDE system on a Riemann surface that nearlyKahler structures of Type I must satisfy. Finally, we remark on a relatively simple oneparameter family of Type III structures that turn out to be incomplete metrics cohomogeneityone under the action of a larger group.
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Online 8. Saddle connections on translation surfaces [2018]
 Dozier, Benjamin, author.
 [Stanford, California] : [Stanford University], 2018.
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In this thesis I prove several theorems on the distribution and number of saddle connections (and cylinders) on translation surfaces. The first main theorem says that saddle connections become equidistributed on the surface. To state this formally we fix a translation surface X, and consider the measures on X coming from averaging the uniform measures on all the saddle connections of length at most R. The theorem is that as R approaches infinity, the weak limit of these measures exists and is equal to the area measure on X coming from the flat metric. This implies that, on any rationalangled billiard table, the billiard trajectories that start and end at a corner of the table become equidistributed on the table. The main ingredients in the proof are new results on counting saddle connections whose angle lies in a given interval, and a theorem of KerckhoffMasurSmillie. The second main theorem concerns SiegelVeech constants, which govern counts of saddle connections averaged over different translation surfaces. We show that for any weakly convergent sequence of ergodic SL2(R)invariant probability measures on a stratum of unitarea translation surfaces, the corresponding SiegelVeech constants converge to the SiegelVeech constant of the limit measure. Combined with results of McMullen and EskinMirzakhaniMohammadi, this yields the (previously conjectured) convergence of sequences of SiegelVeech constants associated to Teichmuller curves in genus two. The key technical tool used in the proofs of both the main theorems is a recurrence result for arcs of circles in the moduli space of translation surfaces. This is proved using the "system of integral inequalities'' approach first used by EskinMasur for translation surfaces.
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Online 9. Schur indices and the padic langlands program [2018]
 Sherman, David Alfred, author.
 [Stanford, California] : [Stanford University], 2018.
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This dissertation uses the lens of the padic Langlands program to understand arithmetic questions about representations of some finite groups of Lie type. Any irreducible complex representation V of finite group G is realizable as a representation on a Kvector space for some number field K. Whether this is possible for a particular field K (necessarily containing all of the values of the character of V) is essentially controlled by a cohomological obstruction (belonging to the Brauer group of K), which is encoded in "local obstructions" at each place of K. To be specific, we consider cuspidal representations of the degreetwo general and special linear groups GL_2(F_p) and SL_2(F_p) over the field with p elements, p an odd prime, and focus on the obstructions at padic places of K. These obstructions have previously been shown (via grouptheoretic means) to vanish. In this dissertation, we present a new proof along the following lines: relate the original representation V to an (infinitedimensional) padic Banach space representation of the corresponding padic group GL_2(Q_p) or SL_2(Q_p), use the padic Langlands correspondence to further relate that to a padic Galois representation W (or a close cousin), and compute the obstruction using W. The padic Langlands correspondence was already known for the degreetwo general linear group over the padic numbers, but here we prove that it is suitably "natural" to transfer the Brauer obstruction from V to W (making our strategy possible). For the special linear group, on the other hand, there is no existing padic correspondence. Therefore, in this dissertation we construct a functor D_S, which we expect to realize the correspondence. This functor is a relative of the "Montreal functor" D that realizes the GL_2(Q_p) correspondence. Using the GL_2(Q_p) case as a guide, we then prove enough (though not all) of the expected properties of the SL_2(Q_p) correspondence, including its "naturality, " to carry out our above strategy.
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3781 2018 S  Inlibrary use 
 Li, Chao, author.
 [Stanford, California] : [Stanford University], 2018.
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In this thesis, we discuss metric properties of positive scalar curvature. Metrics with positive scalar curvature naturally arise from various geometric and physical problems. However, some basic questions of positive scalar curvature were unknown. Specifically, can one conclude that the scalar curvature of a metric is positive, just based on measurement by the metric, without taking any derivative? Such questions are usually answered via geometric comparison theorems. They are also built upon a good understanding of the singular set, along which a sequence of metrics with uniformly bounded curvature degenerate. The primary contributions of this thesis are twofold: Firstly, we study the effect of uniform Euclidean singularities on the Yamabe type of a closed, boundaryless manifold. We show that, in all dimensions, edge singularities with cone angles ≤ 2π along codimension2 submanifolds do not affect the Yamabe type. In three dimensions, we prove the same for more general singular sets, which are allowed to stratify along 1 skeletons, exhibiting edge singularities (angles ≤ 2π) and arbitrary L∞ isolated point singularities. Secondly, we establish a geometric comparison theorem for 3manifolds with positive scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov. For a large collections of polyhedra with interior nonnegative scalar curvature and mean convex faces, we prove that the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding Euclidean model, unless it is isometric to a flat polyhedron. From the viewpoint of metric geometry, our results show that R ≥ 0 is faithfully captured by polyhedra. They suggest the study of "R ≥ 0" with weak regularity assumptions, and the limit space of manifolds with scalar curvature lower bounds.
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Online 11. Tate duality in positive dimension and applications [2018]
 Rosengarten, Zev, author.
 [Stanford, California] : [Stanford University], 2018.
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In Part I, we generalize classical Tate duality (local duality, nineterm exact sequence, etc.) for finite discrete Galois modules (i.e., finite etale commutative group schemes) over global fields to all affine commutative group schemes of finite type (the "positivedimensional" case), building upon recent work of Cesnavicius generalizing Tate duality to all finite commutative group schemes (the "zerodimensional" case). We concentrate mainly on the more difficult function field setting, giving some remarks about the easier number field case along the way. In Part II, we give applications of this extension of Tate duality to the study of Picard groups, TateShafarevich sets, and Tamagawa numbers of linear algebraic groups over global function fields.
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Online 12. Adic moduli spaces [electronic resource] [2017]
 Warner, Evan B.
 2017.
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We prove a version of Artin's criteria for representability of moduli functors in the setting of nonarchimedean analytic geometry in characteristic zero, and deduce representability of the Picard functor under reasonable hypotheses.
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Online 13. Boundary fibration structures and quasihomogeneous geometries [electronic resource] [2017]
 Thorvaldsson, Sverrir.
 2017.
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In this thesis we extend work by Mazzeo on conformally compact manifolds to a class of manifolds with quasihomogeneous geometries, which we call kappamanifolds. Our results show that there are complete noncompact manifolds of negative curvature, that have 0 in the essential spectrum for the Hodge Laplacian on forms, and this applies in a range of degrees centered at the middle degree. As is typical for boundary fibration structures our methods give much more, namely we provide a general framework to study elliptic partial differential operators on kappamanifolds based on microlocal methods. We construct a calculus of pseudodifferential operators on the manifold, and give precise conditions for the existence of a parametrix for elliptic differential operators in this calculus. This work takes up the bulk of the thesis. We then apply this to the spectral theory of the Hodge Laplacian on a kappamanifold. This step requires detailed analysis of the Hodge Laplacian on a simpler model space, which in turn requires detailed study of a system of ordinary differential equations.
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Online 14. Combinatorial methods in Markov chain mixing [electronic resource] [2017]
 White, Graham Robert.
 2017.
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This thesis is concerned with various aspects of Markov chain mixing. The general problem is to examine the number of steps of a Markov chain required for the chain to be close to its stationary distribution. Chapter 2 discusses the problem of shuffling a large deck of cards, describing how to implement riffle shuffles using only operations which affect fewer cards at a time. It also gives upper bounds on the mixing times of several related shuffling schemes. Riffle shuffles are usually modelled by the GSR shuffle. Chapter 3 presents the results of several hundred physical riffle shuffles, and compares this data to the predictions of the GSR model. Similar analysis is done for mash shuffles. Chapter 4 describes how the convergence of a Markov chain may be affected by interweaving other operations. It gives examples where these modifications can drastically slow down or speed up convergence, and a conjecture regarding the efficient generation of random partitions. The random transposition walk on the symmetric group is wellunderstood. Chapter 5 extends the known strong stationary time results regarding this walk to a more general walk where at each step, a larger number of cards are chosen at random and shuffled amongst themselves. Chapter 6 introduces mutation times, a new combinatorial technique similar to the method of strong stationary times. Mutation times can give upper bounds on the mixing times of some Markov chains where strong stationary times may be difficult to construct. This chapter uses mutation times to analyse several models of wash shuffles, as well as some classical random walks on the symmetric group. Chapters 7 and 8 examine the convergence of statistics on Markov chains, rather than the convergence of the chains themselves. Chapter 7 describes how coupling may be used to obtain results about the convergence of statistics, and Chapter 8 uses strong stationary times. Both chapters contain a large number of examples. Finally, Chapter 9 presents a likelihood order for simple random walks on Coxeter groups, showing that the weak Bruhat order describes which elements are more or less likely than others after any number of steps.
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Online 15. Flexible Weinstein structures and applications to symplectic and contact topology [electronic resource] [2017]
 Lazarev, Oleg.
 2017.
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This thesis has three parts. In the first part, we introduce the notions of regular and flexible Lagrangian manifolds with Legendrian boundary in Weinstein domains. We show that flexible Lagrangians satisfy an existence and uniqueness hprinciple (up to ambient symplectomorphism) and give many examples of flexible Lagrangians in the standard symplectic ball. In the second part, we show that all flexible Weinstein fillings of a given contact manifold have isomorphic integral cohomology, generalizing similar results in the subcritical Weinstein case. We also prove relative analogs of our results for flexible Lagrangian fillings of Legendrians. As an application, we show that any closed exact, Maslov zero Lagrangian in a cotangent bundle that intersects a cotangent fiber exactly once has the same cohomology as the zerosection. In the third part, we construct many new exotic symplectic and contact structures. For instance, we show that many closed nmanifolds of dimension at least three can be realized as exact Lagrangian submanifolds of the cotangent bundle of the nsphere with possibly an exotic symplectic structure. We also show that in dimensions at least five any almost contact class that has an almost Weinstein filling has infinitely many different contact structures. We also construct the first known infinite family of almost symplectomorphic Weinstein domains whose contact boundaries are not contactomorphic.
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3781 2017 L  Inlibrary use 
 Gao, Jun.
 2017.
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KolmogrovPetrovskiiPiskunov (KPP) equations are a class of nonlinear parabolic equations which are used to model various biological, ecological, and physical phenomena. In particular it is used as a model for population dynamics. Originally it was studied by Kolmogrov, Petrovskii and Piskunov in 1937. This thesis investigates one type of integrodifferential equation: the nonlocal KPP equation, used in population dynamics. For the nonlocal KPP equation, we prove estimates regarding the front location and in particular introduce logarithmic correction. We also do a nonrigorous analysis to show the profile convergence of solution to the travelling wave.
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Online 17. Geometric variational problems [electronic resource] : regular and singular behavior [2017]
 Cheng, Da Rong.
 2017.
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This thesis is devoted to the study of two of the most fundamental geometric variational problems, namely the harmonic map problem and the minimal surface problem. Chapter 1 concerns the partial regularity of energyminimizing harmonic maps between Riemannian manifolds. In the case where both the domain and the target metrics are smooth, the regularity theory is rather welldeveloped, and we managed to extend part of this theory to the case where the domain metric is only bounded measurable. Specifically, we show that in this case the singular set of an energyminimizing map always has codimension strictly larger than two. In Chapter 2, we study the existence of codimensiontwo minimal submanifolds in a closed Riemannian manifold using the phasetransition approach, which is an alternative to the classical minmax theory and has enjoyed great success in the codimesionone case. To be precise, we show that one can obtain a codimensiontwo stationary rectifiable varifold as the energy concentration set of a sequence of suitably bounded critical points of the GinzburgLandau functional, which was originally a model for phase transition phenomena in superconducting materials. In Appendix A, we consider the fundamental solution of secondorder elliptic systems in divergence form, and prove that under mild assumptions on the coefficients, the fundamental solution can be bounded from above by the Green's function for the Laplacian. Such a growth estimate plays an important role in the analysis in Chapter 2, but is perhaps also interesting on its own.
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We study two different geometrically flavored variational problems in mathematical physics: quasilocal mass in the initial data set approach to the general theory of relativity, and the theory of phase transitions. In the general relativity setting, we introduce a new moduli space of metrics on spheres and a new metric invariant on surfaces to help obtain a precise local understanding of the interaction of ambient scalar curvature and stable minimal surfaces in the context of threemanifolds with nonnegative scalar curvature; we use these tools to study the Bartnik and BrownYork notions quasilocal mass in general relativity. In the theory of phase transitions, we study the global behavior of twodimensional solutions, and relate their complexity at infinity to their variational instability.
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Online 19. Loop equations and string dualities in lattice gauge theories [electronic resource] [2017]
 Jafarov, Jafar.
 2017.
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The purpose of this dissertation is to explore loop equations and string dualities in lattice gauge theories. A lattice gauge theory involves a lattice, a compact Lie group, a matrix representation of the group and a parameter which is called an inverse coupling strength. The main objects of interest in the lattice gauge theories are the Wilson loop variables. A loop equation refers to expressing the expectation of a Wilson loop variable in terms of the expectations of Wilson loop sequences obtained from the loop by various loop operations. These loop equations can be used to establish the 1/N expansion of Wilson loop expectations in a strongly coupled regime. The coefficients of this expansion are represented as absolutely convergent sums over trajectories in a string theory on the lattice, establishing one kind of gaugestring duality. Finally, we will present several applications of this expansion such as the Wilson area law upper bound, the factorization property, and the correspondence of SO(N) and SU(N) Wilson loop expectations in the large N limit.
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Online 20. Modular forms in enumerative geometry [electronic resource] [2017]
 Greer, François.
 2017.
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Let X be an elliptically fibered CalabiYau threefold given by a very general Weierstrass equation over the projective plane. In this thesis, we answer the enumerative question of how many smooth rational curves lie on X over lines in the base plane, proving part of a conjecture by Huang, Katz, and Klemm. The key inputs are a modularity theorem of Kudla and Millson for locally symmetric spaces of orthogonal type and the deformation theory of ADE singularities.
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Online 21. Modular Koszul duality for Soergel bimodules [electronic resource] [2017]
 Makisumi, Shotaro.
 2017.
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We generalize the modular Koszul duality of AcharRiche to the setting of Soergel bimodules associated to any finite Coxeter system. In characteristic 0, our result together with Soergel's conjecture (proved by EliasWilliamson) imply that our Soergeltheoretic graded category O is Koszul selfdual, generalizing the result of BeilinsonGinzburgSoergel.
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In this thesis we study some questions related to Lfunctions over function fields. We obtain asymptotic formulas for the first four moments of the family of quadratic Dirichlet Lfunctions, when the size of the finite field is fixed and the genus goes to infinity. For the first moment we also compute an explicit lower order term of size approximately the cube root of the main term. In joint work with Hung Bui, we study lowlying zeros in the same family of Lfunctions. For the onelevel density of zeros, we detect a lower order term when the support of the Fourier transform of the test function is sufficiently restricted. We also compute the pair correlation of zeros and obtain a nonvanishing result and a lower bound for the proportion of simple zeros in the family.
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3781 2017 F  Inlibrary use 
 Ronchetti, Niccolo.
 2017.
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In this thesis we explore the structure of the derived Hecke algebra of a padic group, a graded associative algebra whose degree 0 subalgebra is the classical Hecke algebra. Working with Z/p^a coefficients, we will establish a Satake homomorphism relating the degree 1 component of this algebra, and the corresponding component for the algebra of a maximal torus. Generalizations and extensions of this construction are also discussed.
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Online 24. Pencils of fermat hypersurfaces and Galois cyclic covers of the projective line [electronic resource] [2017]
 Pan, Donghai.
 2017.
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Let $Z$ be a smooth intersection of two degree $p$ Fermat hypersurfaces in $\mathbb{P}^n$, and let $\pi:X\rightarrow \mathbb{P}^1$ be the pencil of hypersurfaces containing $Z$. We show that the irreducible $\mathbb{Q}$ sublocal systems of $\mathbf{R}^{n1} \pi_*\underline{\mathbb{Q} }_X$ arise from monodromy of the Galois cyclic covers of $\mathbb{P}^1$. This can be viewed as a higher degree analog of M.\ Reid's result on the correspondence between smooth intersection of two quadrics and hyperelliptic curves.
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3781 2017 P  Inlibrary use 
 Shabani, Beniada.
 2017.
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Advectionreactiondiffusion equations (ARD) are nonlinear elliptic or parabolic equations that are used to model a variety of systems in natural sciences and engineering, ranging from biological (population dynamics) to chemical and astrophysical systems (reactions in a fluid). They describe the evolution of a normalized quantity, such as a local density of a population or a temperature, as a result of three processes: transport inside the domain (advection), creation or depletion due to an energy source (reaction), and spatial movement (diffusion). These equations exhibit interesting behaviors such as growth/decay, spreading and mixing. Over the past decades there has been significant progress towards understanding and quantifying the behavior of ARD equations. A great focus was shown in equations of biological invasions, more specifically in the phenomenon of spreading, which happens as a result of the invasion of an unstable equilibrium state by a positive stable one. There are two main threads of research in this field: (1) frontlike spreading, which leads to the study of traveling fronts, their speeds of propagation and stability properties, and (2) multidirectional spreading, arising from initial data that have compact support or fast decay in every direction. In this thesis we study the problem of spreading rates for the Cauchy problem in multidimensional periodic FisherKPP equations. Localized initial data give rise to an invasion that will happen typically at different speeds and profiles in each unit direction, but independent on the size and distribution of the original mass. The main result will be on precise asymptotics for the location of level sets of solutions for these data. The trajectory of the thesis will be as follows: In the first part, after a brief introduction of the problem and past results, we study the linearized Dirichlet equation in a half space moving at a constant speed derived from the slowest traveling fronts. This provides intuition for the location of the fronts in the compactly supported case, as well as concrete bounds that can be compared to the solution of the original problem. In the second part we prove the main result by controlling the propagation of the FisherKPP solution using sub and supersolutions constructed from the linearized fronts.
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Online 26. Sieves and iteration rules [electronic resource] [2017]
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A sieve is a type of inclusionexclusion inequality that comes up when bounding the size of number theoretic sets of interest. We explore iterative rules for improving sievetheoretic bounds when the sifting dimension is slightly greater than one. We also prove some asymptotic results for a model problem introduced by Selberg, in which we pretend that all primes have the same size, and prove the NPcompleteness of a decision variant of standard sievetheoretic questions.
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Online 27. Two results on period maps [electronic resource] [2017]
 Lawrence, Brian.
 2017.
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The content of this thesis is two separate results, loosely connected by the common theme of "period maps." The first is a result on real hyperelliptic curves with an application to extremal polynomials. The second is an application of padic Hodge theory to two Diophantine problems, the Sunit equation and the MordellFaltings theorem.
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Online 28. Covariance estimation and graphical models for infinite collections of random variables [electronic resource] [2016]
 Montague, David.
 2016.
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Estimating correlations for a large collection of random variables is an important part of an enormous number of modern applications of data science, statistics, and mathematics. In particular, it is often necessary to work with either an infinite collection of variables (such as in a spatialtemporal field) or to approximate a large highdimensional finite stochastic system with an infinitedimensional system. Two of the most common techniques used for high dimensional covariance estimation are Markov random fields, also known as graphical models, and positive definite functions, also known as covariance functions, but neither of these approaches has been adequately explored for use with infinite collections of random variables. In particular, the theory of Markov random fields is highly dependent on the finiteness of the graphs used to represent the random variables, and while covariance functions can be used for highdimensional covariance estimation in certain contexts, they generalize poorly to nonspatiotemporal collections of random variables and have other drawbacks (e.g., they may induce poorly conditioned covariance matrices). This dissertation attempts to expand the mathematical theory of Markov random fields to enable infinite dimensional applications, and explores the relationship between graphical approaches to covariance estimation and positive definite functions.
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Online 29. Differential calculus on graphon space and statistical applications of graph limit theory [electronic resource] [2016]
 Diao, Peter Zhiyi.
 2016.
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In this thesis, we build on the beautiful work on dense graph limit theory in two directions. In the first, we develop a framework for differential calculus on the space of graph limits. In the process, we discover a structure theory for differentiable graphon parameters and find that homomorphism densities can be used to expand such parameters in Taylor series. The methods developed are novel, robust and can be generalized. In the second, we use dense graph limit theory to provide a new framework for the study of stability of graph partitioning methods. By formulating statistical consistency as a continuity result on the graphon space, we obtain robust consistency results independent of needing to assume a specific form of the data generating mechanism. We derive the consistency of commonly used clustering algorithms such as clustering based on local graph statistics as well as spectral clustering using the normalized Laplacian. In the final chapter, we indicate how this work can lead to the discovery of new necessary mathematical abstractions to serve as foundations for modern data analysis in areas such as network science or machine learning.
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Online 30. The drift and minorization method for reversible Markov chains [electronic resource] [2016]
 Jerison, Daniel.
 2016.
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This thesis proves upper bounds on the convergence time to stationarity for reversible discrete time Markov chains on general state spaces. The method of proof is the wellestablished drift and minorization approach, which imposes a regenerative structure on the Markov chain according to a particular recipe. The resulting bounds are computable in terms of the ingredients of the recipe. The convergence theorems in this thesis are developed using a new perspective on the interplay between regeneration and reversibility. They are more widely applicable than previous results and provide better numerical bounds on the time to stationarity in specific examples. Two Gibbs samplers from Bayesian statistics are treated in detail. When applied to these chains, the new bounds improve on earlier work but are still quite conservative compared with the true convergence rates. For certain classes of finite chains, the drift and minorization method can give precise bounds on the mixing time. A striking result of this type is a sharp upper bound on the cutoff window for birth and death chains. In addition, an inductive argument shows that the spectral gap of the random walk on the hypercube can be recovered using drift and minorization up to a constant factor of 2. The thesis also contains: (1) An exposition of the drift and minorization method, explaining both the renewal theory approach of Meyn and Tweedie and the coupling approach of Rosenthal, and showing how the bounds improve when the chain is reversible or stochastically monotone. (2) A development of the properties of different types of regeneration times for general state space Markov chains. Defining a randomized stopping time usually requires enlarging the sample space to incorporate independent randomness. In full generality, this operation raises subtle questions of measurability, which are resolved using a new "compatibility condition." (3) A brief consideration of quantile estimation in Markov chain Monte Carlo, focusing on concentration inequalities that provide finitesample guarantees at specified confidence levels. The goal is to prove inequalities that rely as little as possible on theoretical convergence bounds (of the sort proved elsewhere in this thesis) and as much as possible on the empirical sample.
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Online 31. Geometric deformations of orthogonal and symplectic galois representations [electronic resource] [2016]
 Booher, Jeremy.
 2016.
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For a representation of the absolute Galois group of the rationals over a finite field of characteristic p, we would like to know if there exists a lift to characteristic zero with nice properties. In particular, we would like it to be geometric in the sense of the FontaineMazur conjecture: ramified at finitely many primes and potentially semistable at p. For twodimensional representations, Ramakrishna proved that under technical assumptions odd representations admit geometric lifts. We generalize this to higher dimensional orthogonal and symplectic representations. The key ingredient is a smooth local deformation condition obtained by analysing unipotent orbits and their centralizers in the relative situation, not just over fields.
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Online 32. GopakumarVafa conjecture for genus 0 real GromovWitten invariants [electronic resource] [2016]
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The GopakumarVafa conjecture arising from string theory states that BPS counts of a CalabiYau 3fold (obtained from GromovWitten invariants using a power series formula) are integers. A symplectic version of the conjecture was proved by IonelParker. In this thesis we prove a generalization of the symplectic GopakumarVafa conjecture to the case of real GromovWitten invariants. Namely, real genus 0 BPS states of a CalabiYau 3fold with an antisymplectic involution (which are obtained from real genus 0 GromovWitten invariants) are integers. The proof combines topological methods from IonelParker's proof of the original conjecture and KatzLiu's computation of local BPS counts.
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Online 33. New constructions and computations in rigid and flexible symplectic geometry and applications to several complex variables [electronic resource] [2016]
 Siegel, Kyler Bryce.
 2016.
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This thesis is divided into three parts. In the first part, we give a complete characterization of those disk bundles over surfaces which embed as rationally convex strictly pseudoconvex domains in $\C^2$. We recall some classical obstructions and prove some deeper ones related to symplectic and contact topology. We explain the close connection to Lagrangian surfaces with isolated singularities and develop techniques for constructing such surfaces. Our proof also gives a complete characterization of Lagrangian surfaces with open Whitney umbrellas, answering a question first posed by Givental in 1986. In the second part, we introduce a class of Weinstein manifolds which are sublevel sets of flexible Weinstein manifolds but are not themselves flexible. These manifolds, called subflexible, exhibit rather subtle behavior with respect to both pseudoholomorphic curve invariants and symplectic flexibility. We construct a large class of examples and prove that every flexible Weinstein manifold can be Weinstein homotoped to have a nonflexible sublevel set. This resolves some recent open questions in symplectic flexibility. In the third part, we establish an infinitesimal version of fragility for squared Dehn twists around even dimensional Lagrangian spheres. The precise formulation involves twisting the Fukaya category by a closed twoform or bulk deforming it by a halfdimensional cycle. As our main application, we compute the twisted and bulk deformed symplectic cohomology of the subflexible Weinstein manifolds constructed in the second part.
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3781 2016 S  Inlibrary use 
 Skryzalin, Jacek.
 2016.
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Throughout the past decade, there have been many developments and innovations in topological data analysis and persistent homology. The theory of persistent homology is a powerful tool which assigns to any filtered space a persistence module, which is typically interpreted as a module over a ring of polynomials. Although persistent homology serves as a powerful method of analysis for filtered spaces, persistence modules can exhibit a large degree of complexity, especially if the dimension of the filtration is greater than one. Because of the high complexity of persistence modules, they cannot be used directly in the study of concrete data. Instead, one must utilize the structure of persistence modules in order to derive easily understood invariants which provide useful information about the underlying data. This thesis examines the algebraic geometric structure of the space of multidimensional persistence modules and uses the information gleaned to construct numeric invariants. Moreover, these numeric invariants are geometrically meaningful  if we had calculated these numeric invariants from a filtered space constructed from a (finite) data set, we expect these numeric invariants to encode information about the size, shape, and prominence of the topological features of the data set.
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Online 35. On the freeboundary mean curvature flow [electronic resource] [2016]
 Edelen, Nicholas.
 2016.
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We investigate the freeboundary mean curvature flow. This is an evolution of surfaces by ``steepest descent for area, '' while preserving the Neumanntype condition that all the surfaces meet some fixed barrier orthogonally. For example, a bubble in a sink. We first prove the HuiskenSinestrari convexity estimates for freeboundary mean curvature flow, which classifies ``typeII'' singularities. We then develop the notion of weak freeboundary mean curvature flow, extending Brakke's original definition, and proving a local regularity theorem. We also prove a geometric eigenvalue gap estimate, extending results of AshbaughBenguria and BenguriaLinde.
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Online 36. Quantum groups and the YangBaxter equation [electronic resource] [2016]
 Buciumas, Valentin.
 2016.
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In this thesis we will present a series of reconstruction results for Hopf algebras due to SaavedraRivano, Majid and Ulbrich. We will then apply them to solutions of the nonparametrized YangBaxter equation and recover the FRT construction, due to Faddeev, Reshetikhin and Takhtajan. The novel part is the application of the reconstruction theorem to a parametrized solution of the YangBaxter equation to develop a parametrized FRT construction. We apply this construction to the parametrized solution of the YangBaxter equation corresponding to the quantum group $U_q(\widehat{\mathfrak{sl}_2})$ and to another solution with parameter group $GL(2, \mathbb{C}) \times GL(1, \mathbb{C})$ that doesn't come from any known quantum group and build two new quantum groups. We study the representation theory of the newly built objects.
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Online 37. Rates of convergence of Markov chains to stationarity [electronic resource] : strong stationary times, coupling, Gelfand pairs and comparison theory [2016]
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The main purpose of this thesis is to find the mixing time for various random walks that the math community has been interested in. The methods used in this thesis are strong stationary times, introduced by Aldous and Diaconis, a coupling argument that was inspired by work of Aldous, bounding the eigenvalues of a walk via comparison theory and path arguments introduced by Diaconis and SaloffCoste and via Gelfand pair theory developped by Diaconis and Shahshahani, but also finding the eigenvalues of random walks using Fourier Transform arguments as introduced by Diaconis and Shahshahani. There are many norms that can use to study the mixing time of a random walk on a finite group. Coupling is used to bound the $l^1$ norm, strong stationary times are used to bound the separation distance, while the eigenvalues of a reversible Markov chain are used to bound the $l^2$ norm. This thesis focuses on specific examples of Markov chains on finite spaces, some of which have been studied before but through a different norm.
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Online 38. Some finiteness results for groups of automorphisms of manifolds [electronic resource] [2016]
 Kupers, Alexander.
 2016.
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We prove that in dimensions not equal to 4,5,7 the homology and homotopy groups of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses homological stability, embedding calculus and the arithmeticity of mapping class groups. From this we deduce similar results for the homeomorphisms of R^n and various types of automorphisms of 2connected manifolds.
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Online 39. Statistics of random integral matrices [electronic resource] [2016]
 Boreico, Iurie.
 2016.
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This thesis consists of two separate projects. The first part investigates the asymptotic behavior of the number of integral mbyn matrices, with entries bounded by T, whose cokernel is isomorphic to a fixed abelian group G. We answer this question by building on work of Katznelson, who obtained asymptotics on the number of such matrices of given rank. In particular, we show that if G has torsion B for a finite abelian group B, then a positive proportion of matrices of rank r have cokernel isomorphic to G, and we compute this proportion explicitly (as an infinite product over primes). The corresponding problem for symmetric matrices is also discussed, with a different answer. Part two of this thesis deals with the infinitesimal frequency of a monic polynomial appearing as the characteristic polynomial of an nbyn matrix with coefficients in the padic integers. Relying on the concept of rational singularities, we prove that this frequency is described by a continuous density on the space of monic polynomials, and show that the normalized density function is multiplicative, thus reducing its computation to the case of a monic irreducible polynomial. In the monic irreducible case, we express the density function as a finite sum over modules in the ring of integers of a finite extension of Q_p, and compute it in the case of degree < =3. For the general case, we conjecture bounds on the size of this function, as well as conjectures on the underlying geometric structures. In the end, we study a modification of this question as n goes to infinity. As an aside, we also use CohenLenstra measures to compute the distribution of the Jordan blocks of matrices with coefficients in a fixed finite field as the size goes to infinity.
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Online 40. The string topology of holomorphic curves in BU(n) [electronic resource] [2016]
 Nolen, Sam.
 2015.
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In this thesis we compute the string topology algebra structure on the homology of a stabilized version of the space of holomorphic maps from 1dimensional complex projective space to a complex Grassmannian. This generalizes a computation of Kallel and Salvatore.
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3781 2015 N  Inlibrary use 
 Tripathy, Arnav.
 2016.
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We show that under mild hypotheses on a proper algebraic space X, the functors of taking its symmetric powers and its etale realisation commute up to weak equivalence. We conclude an effective version of the DoldThom theorem for the etale site and discuss the stabilisation results for the natural morphisms of etale homotopy groups π_k Sym^n X → π_k Sym^{n+1} X in the context of the Weil conjectures.
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3781 2016 T  Inlibrary use 
 Kališnik Verovšek, Sara.
 2016.
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In the last two decades applied topologists have developed numerous methods for `measuring' and building combinatorial representations of the shape of the data. The most famous example of the former is persistent homology. This adaptation of classical homology assigns a barcode, i.e. a collection of intervals with endpoints on the real line, to a finite metric space. Unfortunately, barcodes are not welladapted for use by practitioners in machine learning tasks. In this dissertation, I identify classes of maxplus polynomials and tropical rational functions that can be used as coordinates on the space of barcodes. All of these are stable with respect to standard distance functions (bottleneck distance, Wasserstein distances) used on the barcode space. I demonstrate how these coordinates can be used by combining persistent homology with SVM to classify numbers from the MNIST dataset. In order to identify functions on the barcode space, I find generators for the semirings of tropical polynomials, maxplus polynomials and tropical rational functions invariant under the action of the symmetric group. The fundamental theorem of ordinary symmetric polynomials has an equivalent in the tropical and maxplus semirings. There are interesting differences if we consider the tropical polynomial semiring with nr variables that come in n blocks of r variables each and are permuted by the symmetric group. As opposed to the ordinary polynomial case, the semiring of rsymmetric tropical polynomials is not finitely generated, but the semiring of rsymmetric tropical rational functions is.
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Online 43. Two models on limit order trading [electronic resource] [2016]
 Ren, Weiluo.
 2016.
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In this thesis we study two limit order trading models based on the one of Avellaneda and Stoikov (2008). For the first one, we study AvellanedaStoikov model under the condition when the underlying price is mean reverting. Our main result is that when time is far from the terminal, the optimal price for bid and ask limit orders is constant, which means that it does not track the underlying price. Numerical simulations confirm this behavior. When the underlying price is mean reverting, then for times sufficiently far from terminal, it is more advantageous to focus on the mean price and ignore fluctuations around it. Mean reversion suggests that limit orders will be executed with some regularity, and this is why they are optimal. We also explore intermediate time regimes where limit order prices are influenced by the inventory of outstanding orders. The duration of this intermediate regime depends on the liquidity of the market as measured by specific parameters in the model. The second model we study is a price impact model where the underlying price is affected by the mean field terms representing existing limit orders in the market. The main result on this model is that there are two different regimes for the representative trader in the solution of MFG: In the beginning, he trades aggressively which results in a large price impact; after a short period of time, he trades, by expectation, approximately symmetric about the underlying price. As a function of time, the price impact decays exponentially in the solution of MFG. Moreover the expectations of optimal prices for the representative trader are constants symmetric about a level close to the terminal value of expected underlying price.
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3781 2016 R  Inlibrary use 
 Siu, Ho Chung.
 2016.
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Consider the modular surface $X = SL_2(\BZ) \backslash \BH$. One would like to know how the values of holomorphic modular forms/nonholomorphic Maass forms would distribute in the limit. Hejhal and Rackner \cite{HejRac} was the first people to study this problem. They conjectured that (suitably normalized) Maass forms suitably normalized would distribute like the Gaussian with mean 0 and variance $\frac{3}{\pi}$ in the limit. This is known as the random wave conjecture. They also gave a heuristic argument on why the conjecture should hold. This thesis tries to address two questions centered around the random wave conjecture and related problems, such as Quantum Unique Ergodicity. The first part of the thesis (Chapter 3) addresses the discrepancy of predictions by physicists and by mathematicians. For a fixed test function on $X$, Quantum Unique Ergodicity asserts that $$\mu_j (\psi) := \int_X \psi(z) phi_j(z) 2 \frac{dxdy}{y^2} \to \frac{1}{vol(X)}\int_X \psi(z) \frac{dxdy}{y^2}$$ as the Laplacian eigenvalue $\lambda_j$ of the HeckeMaass forms $\phi_j$ approaches infinity. One may ask about the distribution of the sequence $\mu_j(\psi)$ as $\lambda_j \to \infty$. The correspondence principle from quantum chaos suggests that the distribution would be Gaussian, whereas the connection of $\mu_j(\psi)$ to $L$functions suggests that the distribution is logGaussian. We give evidence to the latter, by showing a onesided central limit theorem for $\log mu_j(\psi) . In particular, this shows that the distribution cannot be Gaussian. The second part of the thesis (joint with Professor Soundararajan) studies the distribution of modular forms when restricted to reasonable thin sets, such as horocycles. We show that for modular forms $f(z)$, the restrictions of $ ^kf(z) to a fixed horocycle is uniformly bounded, independent of the modular form $f$. This addresses an open question in Sarnak's letter to Reznikov \cite{Sar}, and gives another form of nonscarring for the family of modular forms.
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Online 45. Weak universality of interacting particle systems [electronic resource] [2016]
 Tsai, LiCheng.
 2016.
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In this dissertation we consider random growth phenomenon related to the KardarParisiZhang (KPZ) equation. We focus particularly on the weak universality of interacting particle systems. That is, under suitable scalings, a certain class of onedimensional, weakly irreversible interacting particle systems converge to the KPZ equation. Our discussion emphasizes on both the inter and intromodel universality. For the former, we derive the exact microscopic HopfCole transformation for the 4parameter family of Higher Spin Exclusion Processes (HSEPs) introduced by Corwin and Petrov (2016). This is done by exploiting the close relationship between HopfCole transformation and duality. As the HSEPs sit above most of the known integrable models in the KPZ class, we thus obtain the exact microscopic HopfCole transformation for all lowerlevel models. To demonstrate the weak universality, we further consider a particular weak scaling of the HSEPs, and show the convergence to the KPZ equation. This expands the relatively small number of systems for which weak universality of the KPZ equation has been demonstrated. As for intromodel universality, we analyze a class of nonnearestneighbor exclusion processes and the corresponding growth models. Our approach is to exploit an approximate HopfCole transformation, to which end we identify the main nonlinearity and eliminate it by imposing a gradient type condition. For hopping range at most 3, using the generalized transformation, we prove the convergence of the exclusion process toward the KPZ equation.
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Online 46. Diffraction of elastic waves by edges [electronic resource] [2015]
 Katsnelson, Vitaly.
 2015.
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The purpose of this thesis is to investigate the diffraction of singularities of solutions to the linear elastic equation on manifolds with edge singularities. Such manifolds are modeled on the product of a smooth manifold and a cone over a compact fiber. For the fundamental solution, the initial pole generates a pressure wave (pwave), and a secondary, slower shear wave (swave). If the initial pole is appropriately situated near the edge, we show that when a pwave strikes the edge, the diffracted pwaves and swaves (i.e. loosely speaking, are not limits of prays which just miss the edge) generated from such an interaction are weaker in a Sobolev sense than the incident pwave. More generally, we show that subject to a "coinvolutivity" hypothesis, if a psingularity (or ssingularity) of any solution strikes the edge, the diffracted p and s wavefronts are smoother that the incident singularity.
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3781 2015 K  Inlibrary use 
 Chodosh, Otis.
 2015.
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We discuss the largescale geometry of asymptotically hyperbolic manifolds. Asymptotically hyperbolic manifolds arise naturally in general relativity. However, several fundamental questions about them remain unresolved, including the asymptotically hyperbolic Penrose inequality and the static uniqueness of the Schwarzschildantide Sitter metric. The main contributions of this thesis are twofold: Firstly, we introduce a new notion of renormalized volume for asymptotically hyperbolic manifolds and prove a sharp Penrosetype inequality where mass is replaced by renormalized volume. Secondly, we use the notion of renormalized volume to study isoperimetric regions in asymptotically hyperbolic manifolds. We prove that for initial data sets that are Schwarzschildantide Sitter at infinity and satisfy appropriate scalar curvature lower bounds, sufficiently large coordinate spheres are uniquely isoperimetric. This is relevant in the context of Bray's isoperimetric approach to the Penrose inequality. From a geometric viewpoint, our results show that the largescale geometry of asymptotically hyperbolic manifolds significantly differs from the more familiar asymptotically flat setting. The renormalized volume is a very different quantity from the ``mass, '' and our results suggest that it is a stronger quantity. As a consequence of this, we uncover a link between scalar curvature and the behavior of large isoperimetric regions, which is not present in the asymptotically flat setting. Additionally, we discuss isoperimetric regions in warped products and consequences for the renormalized volume of a more general class of metrics. Finally, we study rotational symmetry of expanding Ricci solitons, a problem that is formally similar to the static uniqueness question with negative cosmological constant.
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Online 48. Global analysis of linear and nonlinear wave equations on cosmological spacetimes [electronic resource] [2015]
 Hintz, Peter.
 2015.
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We develop a general framework for the global analysis of linear and nonlinear wave equations on geometric classes of Lorentzian manifolds, based on microlocal analysis on compactified spaces. The main examples of manifolds that fit into this framework are cosmological spacetimes such as de Sitter and Kerrde Sitter spaces, as well as Minkowski space, and perturbations of these spacetimes. In particular, we establish the global solvability of quasilinear wave equations on cosmological black hole spacetimes and obtain the asymptotic behavior of solutions using a novel approach to the global study of nonlinear hyperbolic equations. The framework directly applies to nonscalar problems as well, and we present linear and nonlinear results both for scalar equations and for equations on natural vector bundles. To a large extent, our work was motivated by the black hole stability problem for cosmological spacetimes, and we expect the resolution of this problem to be within reach with the methods presented here.
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Online 49. Lagrangian Tori in R4 and S2xS2 [electronic resource] [2015]
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We study problems of classification of Lagrangian embeddings of a torus in a symplectic 4manifold. First we complete the proof of a claim that all Lagrangian tori in R^4 are isotopic. Next we present a construction of Lagrangian tori and Klein bottles in monotone S^2xS^2. Finally we show that all monotone tori may be produced from such a construction, and outline a new approach to the problem of finding the Hamiltonian isotopy classes of monotone tori in S^2xS^2.
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Online 50. Mean field games with common noise [electronic resource] [2015]
 Ahuja, Saran.
 2015.
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Mean Field Games (MFG) are a limit of stochastic differential games with a large number of identical players. They were proposed and first studied by Lasry and Lions and independently by Caines, Huang, and Malhame in 2006. They have attracted a lot of interest in the past decades due to their application in many fields. By assuming independence among each agent, taking the limit as N goes to infinity reduces a problem to a fullycoupled system of forwardbackward partial differential equations (PDE). The backward one is a HamiltonJacobiBellman (HJB) equation for the value function of each player while the forward one is the FokkerPlanck (FP) equation for the evolution of the players distribution. This limiting system is more tractable and one can use its solution to approximate the Nash equilibrium strategy of Nplayer games. In this thesis, we consider the MFG model in the presence of common noise, relaxing the usual independence assumption of individual random noise. The presence of common noise clearly adds an extra layer of complexity to the problem as the distribution of players now evolves stochastically. Our first task is proving existence and uniqueness of a Nash equilibrium strategy for this game, showing wellposedness of MFG with common noise. We use a probabilistic approach, namely the Stochastic Maximum Principle (SMP), instead of a PDE approach. This approach gives us a forwardbackward stochastic differential equation (FBSDE) of McKeanVlasov type instead of coupled HJBFP equations. This was first done by Carmona and Delarue in the case of no common noise and we extend their results to MFG with common noise. We are able to extend their results under a linearconvexity framework and a weak monotonicity assumption on the cost functions. In addition to wellposedness results, we also prove the Markov property of McKeanVlasov FBSDE by proving the existence of a decoupling function. In the second part of this thesis, we consider MFG models when the common noise is small. For simplicity, we assume a quadratic running cost function while keeping a general terminal cost function satisfying the same assumptions as in the first part. Our goal is to give an approximation of Nash equilibrium of this game using the solution from the original MFG with no common noise, which could be described through a finitedimensional system of PDEs. We characterize the first order approximation terms as the solution to a linear FBSDE of meanfield type. We then show that the solution to this FBSDE is a centered Gaussian process with respect to the common noise. By assuming regularity of the decoupling function of the 0MFG problem, we can find an explicit solution showing that they are in the form of a stochastic integral with respect to the common noise with the integrands adapted to the information from the 0MFG only. We then are able to compute the covariance function explicitly.
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