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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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 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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 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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