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Online 1. Bounds for sets with no nontrivial polynomial progressions [2019]
 Peluse, Sarah Anne, author.
 [Stanford, California] : [Stanford University], 2019.
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 Book — 1 online resource.
 Summary

In this thesis, we prove quantitative bounds in the polynomial Szemer\'edi theorem in several situations in which no bounds were previously known. In Chapter 2, we prove that if P_1, P_2\in\mathbb{Z}[y] are affinelinearly independent, then any subset of \mathbb{F}_q with no nontrivial polynomial progressions of the form x, x+P_1(y), x+P_2(y) must have size \ll_{P_1, P_2}q^{23/24}, provided the characteristic of \mathbb{F}_q is large enough. In Chapter 3, we prove that if P_1, \dots, P_m\in\mathbb{Z}[y] are affinelinearly independent, then any subset of \mathbb{F}_q with no nontrivial polynomial progressions of the form x, x+P_1(y), \dots, x+P_m(y) must have size \ll_{P_1, \dots, P_m}q^{1\gamma_{P_1, \dots, P_m}} for some \gamma_{P_1, \dots, P_m}> 0, again provided that the characteristic of \mathbb{F}_q is large enough. In Chapter 4, we prove that any subset of \{1, \dots, N\} with no nontrivial progressions of the form x, x+y, x+y^2 must have size \ll N/(\log\log{N})^{2^{157}}. In Chapter 5, we prove that if P_1, \dots, P_m\in\mathbb{Z}[y] have distinct degrees, then any subset of \{1, \dots, N\} with no nontrivial polynomial progressions of the form x, x+P_1(y), \dots, x+P_m(y) must have size \ll N/(\log\log{N})^{1\gamma_{P_1, \dots, P_m}} for some \gamma_{P_1, \dots, P_m}> 0. In the final chapter, Chapter 6, we move to the nonabelian setting and prove powersaving bounds for subsets of nonabelian finite simple groups with no nontrivial progressions of the form x, xy, xy^2.
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 Stanton, Caitlin King, author.
 [Stanford, California] : [Stanford University], 2019.
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 Book — 1 online resource.
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It is wellknown that there are 3264 conics in P^2 that are tangent to 5 general smooth quadrics. This result can be proved by taking a suitable space that parametrizes conics in P^2 and computing C^5, where C is the divisor corresponding to the condition of being tangent to a general conic. To answer similar enumerative questions about quadrics in P^n, we use the space of complete nquadrics, X_n. In this thesis we will give a brief overview of the equivalent ways of defining this space, determine ranks and generators of its Chow groups, and describe how one would use intersection theory on this space to compute the answers to enumerative problems involving hitting and tangency conditions.
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Online 3. Circle method and the subconvexity problem [2019]
 Raju, Chandra Sekhar, author.
 [Stanford, California] : [Stanford University], 2019.
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Munshi demonstrated the usefulness of the circle method in understanding the subconvexity problem, by exhibiting a subconvexity bound for GL3 automorphic forms twisted by a Dirichlet character. We take this idea further by exhibiting a subconvexity bound for GL2 X GL2 RankinSelberg Lfunctions using the circle method. We exhibit a subconvexity bound for the RankinSelberg L functions in the level aspect when both the automorphic forms are varying independently (i.e the arithmetic conductors don't have a large common factor). We beat the bestknown exponent for this problem using the circle method. We also use a different version of the circle method to exhibit a subconvexity bound in taspect.
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Online 4. Deformations of generalized Fuchsian representations [2019]
 Ungemach, Weston Joseph, author.
 [Stanford, California] : [Stanford University], 2019.
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 Summary

This thesis studies the deformation theory of surface group representations in to SL(4, R)  in particular, those near what we term the generalized Fuchsian locus. We show that deformations of these representations admit geodesic laminations on the boundary of the convex hull of the limit set in RP^3, which generalizes the classical theory of quasiFuchsian deformations. In this new setting, however, we show that there exist such deformations for which this lamination supports no transverse measure, which cannot happen for hyperbolic deformations.
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Online 5. Étale Steenrod operations and the ArtinTate pairing [2019]
 Feng, Tony, author.
 [Stanford, California] : [Stanford University], 2019.
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 Summary

We prove a 1966 conjecture of Tate concerning the ArtinTate pairing on the Brauer group of a surface over a finite field, which is the analogue of the CasselsTate pairing. Tate asked if this pairing is always alternating and we find an affirmative answer, which is somewhat surprising in view of the work of PoonenStoll on the CasselsTate pairing. Our method is based on studying a connection between the ArtinTate pairing and (generalizations of) Steenrod operations in etale cohomology. Inspired by an analogy to the algebraic topology of manifolds, we develop tools allowing us to calculate the relevant etale Steenrod operations in terms of characteristic classes.
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Online 6. Localization and free energy asymptotics in disordered statistical mechanics and random growth models [2019]
 Bates, Erik Walter, author.
 [Stanford, California] : [Stanford University], 2019.
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 Summary

This dissertation develops, for several families of statistical mechanical and random growth models, techniques for analyzing infinitevolume asymptotics. In the statistical mechanical setting, we focus on the lowtemperature phases of spin glasses and directed polymers, wherein the ensembles exhibit localization which is physically phenomenological. We quantify this behavior in several ways and establish connections to properties of the limiting free energy. We also consider two popular zerotemperature polymer models, namely first and lastpassage percolation. For these random growth models, we investigate the order of fluctuations in their growth rates, which are analogous to free energy.
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Online 7. Modern methods in extremal combinatorics [2019]
 Sauermann, Lisa, author.
 [Stanford, California] : [Stanford University], 2019.
 Description
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In this thesis, we apply modern probabilistic and algebraic techniques to different problems in extremal combinatorics. One of the most recent algebraic techniques is the new polynomial method which Croot, Lev and Pach introduced in 2016. This method has lead to the spectacular breakthrough of Ellenberg and Gijswijt on the capset problem, and has had many more applications in additive number theory and extremal combinatorics. In Chapter 2, we use various tools that resulted from the CrootLevPach polynomial method, combined with probabilistic and combinatorial arguments, to prove new upper bounds on the ErdosGinzburgZiv constant of F_p^n for a fixed prime p \geq 5 and large n. Chapter 3 also relies on developments arising from the CrootLevPach polynomial method as well as new combinatorial ideas. We prove a polynomial bound relating the parameters in Green's arithmetic kcycle removal lemma in F_p^n for all k \geq 3. The special case of k = 3 was previously proved by Fox and Lovasz and is used as the base case of an induction on k in our proof for all k \geq 3. In Chapter 4, we use methods from algebraic geometry (and basic differential topology) to prove an asymptotically tight lower bound for the number of graphs of a certain form where the edges are defined algebraically by the signs of a finite list of polynomials. We present many applications of this result, in particular to counting intersection graphs and containment orders for various families of geometric objects (e.g. segments of disks in the plane). Using probabilistic methods, we prove the socalled Edgestatistics conjecture of Alon, Hefetz, Krivelevich and Tyomkyn in Chapter 5. In a certain range of the parameters, this conjecture already follows from a result of Kwan, Sudakov and Tran. We solve the other cases, and thereby establish the full conjecture. Finally, in Chapter 6 we prove a conjecture of Erdos, Faudree, Rousseau and Schelp from 1990 concerning subgraphs of minimum degree k.
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Online 8. Moduli spaces of bundles via motivic probabilities [2019]
 Fayyazuddin Ljungberg, Benjamin Ake, author.
 [Stanford, California] : [Stanford University], 2019.
 Description
 Book — 1 online resource.
 Summary

We give formulas for the classes in the completed Grothendieck ring of varieties of the moduli stacks of principal bundles for the special linear or symplectic groups over a curve X in terms of the motivic zeta function of X. The answers agree with predictions of Behrend and Dhillon. The computations rely on interpreting certain classes in the Grothendieck ring in a probabilistic way, using work of Margaret Bilu and Sean Howe.
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Online 9. On applications of Szemerédi's regularity lemma [2019]
 Wei, Fan (Mathematician), author.
 [Stanford, California] : [Stanford University], 2019.
 Description
 Book — 1 online resource.
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Szemeredi's regularity lemma is an important and powerful tool in graph theory. One of its prevalent application is the graph removal lemma, which has numerous applications to extremal problems for graphs and hypergraphs, additive combinatorics, discrete geometry, and theoretical computer science. In this thesis, we initiate and prove a new removal lemma with respect to a stronger metric on graphs, which is a strengthening of the usual graph removal lemma. We also prove a new application of the regularity lemma. The Ramsey number r(H) of a graph H is the minimum integer n such that any twocoloring of the edges of the complete graph Kn contains a monochromatic copy of H. Determining the Ramsey number for all graphs is a famously difficult question. A more general question is to study the minimum total number of monochromatic copies of H over all twocolorings of Kn. When n = r(H), this quantity is referred to as the threshold Ramsey multiplicity of H. Addressing a problem of Harary and Prins forty years ago, we proved the first quantitative sharp bound for cycles and paths. This thesis describes the work for path and even cycles. Although the regularity lemma is powerful, a notable drawback is that applications of the regularity lemma will result in very weak quantitative estimates. One important application of the regularity lemma is the field of property testing, whose goal is to very quickly distinguish between objects that stratify a certain property from those that are epsilonfar from satisfying that property. Some of the most general results in this area give ``constant query complexity" algorithms, which means the amount of information it looks at is independent of the input size. These results are proved using regularity lemmas or graph limits. Unfortunately, typically the proofs come with no explicit bound for the query complexity, or enormous bounds, of towertype or worse, as a function of 1/epsilon, making them impractical. We show by entirely new methods that for permutations property testing, such general results still hold with query complexity only polynomial in 1/epsilon.
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Online 10. On manifold bundles over classifying spaces [2019]
 Reinhold, Jens, author.
 [Stanford, California] : [Stanford University], 2019.
 Description
 Book — 1 online resource.
 Summary

This thesis comprises the results of three independent research projects in highdimensional geometric topology. The first one exhibits, for various manifolds as fibers, smooth bundles over the classifying space of SU(2) not induced from an action. The second project is joint work with M. Krannich. It gives conditions on when a manifold is, up to bordism, the total space of a fiber bundle over a surface with highlyconnected and almostparallelizable fiber M, including a computation of the corresponding characteristic numbers. As a corollary, we determine an explicit basis for the second integral cohomology of BDiff(M) up to torsion in terms of generalized Miller—Morita—Mumford classes. Finally, the third project analyzes models for higher parametrized cobordism categories, with a special focus on the case where the parametrizing space is a circle.
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Online 11. On the arithmetic of weight two Eisenstein series [2019]
 Silliman, Jesse Kyle, author.
 [Stanford, California] : [Stanford University], 2019.
 Description
 Book — 1 online resource.
 Summary

This thesis contains two separate results on the arithmetic of Eisenstein series. The first is a new proof of theorems of Merel and Lecouturier on the relationship between Mazur's Eisenstein ideal and special values of Dirichlet Lfunctions. The second is a computation of the Galois representations and mixed Hodge structures associated to certain Eisenstein series on Hilbert modular varieties.
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Online 12. On the orderability up to conjugation of certain open contact manifolds [2019]
 De Groote, Cédric, author.
 [Stanford, California] : [Stanford University], 2019.
 Description
 Book — 1 online resource.
 Summary

In this thesis, we define and study the orderability up to conjugation problem for groups of compactly supported contactomorphisms of certain open contact manifolds. In particular, we show that the groups of compactly supported contactomorphisms of certain subsets of odddimensional spheres are orderable up to conjugation. This class of manifolds has recently showed its importance in the flexible side of contact topology. Along the way, we prove nonsqueezingtype results for prequantizations of hyperboloids in the product of an evendimensional Euclidean space with a circle, which may be of independent interest. The main tool is an equivariant version of contact homology, adapted to subdomains of the product of an evendimensional Euclidean space with a circle.
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Online 13. Asymptotics of Gaussian processes and Markov chains [2018]
 Zhai, Alex, author.
 [Stanford, California] : [Stanford University], 2018.
 Description
 Book — 1 online resource.
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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 14. The equivariant cobordism category [2018]
 Szűcs, Gergely, author.
 [Stanford, California] : [Stanford University], 2018.
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 Book — 1 online resource.
 Summary

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 15. Factorization theory of Thom spectra, twists, and duality [2018]
 Klang, Inbar, author.
 [Stanford, California] : [Stanford University], 2018.
 Description
 Book — 1 online resource.
 Summary

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 16. The flexibility of caustics [electronic resource] [2018]
 AlvarezGavela, Daniel.
 2018.
 Description
 Book — 1 online resource.
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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 17. 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 18. 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 19. 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 20. 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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