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Academic Commons Search Resultsen-usA GL(3) Kuznetsov Trace Formula and the Distribution of Fourier Coefficients of Maass Forms
https://academiccommons.columbia.edu/catalog/ac:202236
Guerreiro, João Leitãohttp://dx.doi.org/10.7916/D8GM87JPThu, 15 Sep 2016 18:03:01 +0000We study the problem of the distribution of certain GL(3) Maass forms, namely, we obtain a Weyl’s law type result that characterizes the distribution of their eigenvalues, and an orthogonality relation for the Fourier coefficients of these Maass forms. The approach relies on a Kuznetsov trace formula on GL(3) and on the inversion formula for the Lebedev-Whittaker transform. The family of Maass forms being studied has zero density in the set of all GL(3) Maass forms and contains all self-dual forms. The self-dual forms on GL(3) can also be realised as symmetric square lifts of GL(2) Maass forms by the work of Gelbart-Jacquet. Furthermore, we also establish an explicit inversion formula for the Lebedev-Whittaker transform, in the nonarchimedean case, with a view to applications.Mathematics, Weyl's problem, Eigenvalues, Mathematics, Trace formulasjlg2211MathematicsDissertationsA local relative trace formula for spherical varieties
https://academiccommons.columbia.edu/catalog/ac:202143
Filip, Ioanhttp://dx.doi.org/10.7916/D8HX1CWKFri, 09 Sep 2016 12:16:41 +0000Let F be a local non-Archimedean field of characteristic zero. We prove a Plancherel formula for the symmetric space GL(2,F)\GL(2,E), where E/F is an unramified quadratic extension. Our method relies on intrinsic geometric and combinatorial properties of spherical varieties and constitutes the local counterpart of the global computation of the Flicker-Rallis period as a residue of periods against Eisenstein series. We also give a novel derivation of the Plancherel formula for the strongly tempered variety T\PGL(2) over F (with maximal split torus T) using a canonical smooth asymptotics morphism and a contour shifting method. In this rank one local setting, our proof is similar to Langlands' proof over global fields describing the spectrum of a reductive group in terms of residues of Eisenstein series. Finally, using both L2-decompositions, we develop a local relative trace formula and outline a comparison result in the setting of the unitary rank one Gan-Gross-Prasad conjecture.Mathematics, Trace formulas, Combinatorial geometry, Geometry, Algebraic, Mathematicsif2179MathematicsDissertationsPeriodic symplectic cohomologies and obstructions to exact Lagrangian immersions
https://academiccommons.columbia.edu/catalog/ac:199078
Zhao, Jingyuhttp://dx.doi.org/10.7916/D8V69JMZMon, 09 May 2016 12:43:14 +0000Given a Liouville manifold, symplectic cohomology is defined as the Hamiltonian Floer homology for the symplectic action functional on the free loop space. In this thesis, we propose two versions of periodic S^1-equivariant homology or S^1-equivariant Tate homology for the natural S^1-action on the free loop space. The first version is called periodic symplectic cohomology. We prove that there is a localization theorem or a fix point property for periodic symplectic cohomology. The second version is called the completed periodic symplectic cohomology which satisfies Goodwillie's excision isomorphism.
Inspired by the work of Seidel and Solomon on the existence of dilations on symplectic cohomology, we formulate the notion of Liouville manifolds admitting higher dilations using Goodwillie's excision isomorphism on the completed periodic symplectic cohomology. As an application, we derive obstructions to existence of certain exact Lagrangian immersions in Liouville manifolds admitting higher dilations.Mathematics, Homology theory, Symplectic geometryjz2432MathematicsDissertationsNearly Overconvergent Forms and p-adic L-Functions for Symplectic Groups
https://academiccommons.columbia.edu/catalog/ac:198695
Liu, Zhenghttp://dx.doi.org/10.7916/D8ZC82VKThu, 05 May 2016 21:20:06 +0000We reformulate Shimura's theory of nearly holomorphic forms for Siegel modular forms using automorphic sheaves over Siegel varieties. This sheaf-theoretic reformulation allows us to define and study basic properties of nearly overconvergent Siegel modular forms as well as their p-adic families. Besides, it finds applications in the construction, via the doubling method, of p-adic partial standard L-functions associated to Siegel cuspidal Hecke eigensystems. We illustrate how the sheaf-theoretic definition of nearly holomorphic forms and Maass--Shimura differential operators helps with the choice of the archimedean sections for the Siegel Eisenstein series on the doubling group Sp(4n) and the study of the p-adic properties of their restrictions to Sp(2n)*Sp(2n). The selection of archimedean sections, together with p-adic interpolation considerations, then naturally gives the sections at the place p. We compute p-adic zeta integrals corresponding to those sections. Finally, we construct the p-adic standard L-functions associated to ordinary families of Siegel Hecke eigensystems and obtain their interpolation properties.Mathematicszl2283MathematicsDissertationsQuasi-local energy and isometric embedding
https://academiccommons.columbia.edu/catalog/ac:198605
Gimre, Karsten Trevorhttp://dx.doi.org/10.7916/D8765FB1Wed, 04 May 2016 21:30:05 +0000In this thesis, we consider the recent definition of gravitational energy at the quasi-local level provided by Mu-Tao Wang and Shing-Tung Yau. Their definition poses a variational question predicated on isometric embedding of Riemannian surfaces into the Minkowski space; as such, there is a naturally associated Euler-Lagrange equation, which is a fourth-order system of partial differential equations for the embedding functions. We prove a perturbation result for solutions of this Euler-Lagrange equation.Mathematics, Generalized spaces, Isometrics (Mathematics)ktg2117MathematicsDissertationsAn alternative proof of genericity for unitary group of three variables
https://academiccommons.columbia.edu/catalog/ac:198528
Wang, Chonglihttp://dx.doi.org/10.7916/D8C24WF7Wed, 04 May 2016 15:32:46 +0000In this thesis, we prove that local genericity implies globally genericity for the quasi-split unitary group U3 for a quadratic extension of number fields E/F. We follow [Fli1992] and [GJR2001] closely, using the relative trace formula approach. Our main result is the existence of smooth transfer for the relative trace formulae in [GJR2001], which is circumvented there. The basic idea is to compute the Mellin transform of Shalika germ functions and show that they are equal in the unitary case and the general linear case.Mathematics, Trace formulas, Unitary groups, Group theory, Mellin transform, Mathematicscw2639MathematicsDissertationsRelative Trace Formula for SO₂ × SO₃ and the Waldspurger Formula
https://academiccommons.columbia.edu/catalog/ac:198125
Krishna, Rahul Marathehttp://dx.doi.org/10.7916/D8FB52XBFri, 29 Apr 2016 21:15:19 +0000We provide a new relative trace formula approach to the theorem of Waldspurger on toric periods for GL₂, with possible applications to the global Gross-Prasad conjecture for orthogonal groups.Mathematics, Torus (Geometry), Trace formulas, Mathematics, Linear algebraic groupsrmk2138MathematicsDissertationsDualization and deformations of the Bar-Natan—Russell skein module
https://academiccommons.columbia.edu/catalog/ac:197984
Heyman, Andrea L.http://dx.doi.org/10.7916/D85D8RV2Tue, 26 Apr 2016 15:34:28 +0000This thesis studies the Bar-Natan skein module of the solid torus with a particular boundary curve system, and in particular a diagrammatic presentation of it due to Russell. This module has deep connections to topology and categorification: it is isomorphic to both the total homology of the (n,n)-Springer variety and the 0th Hochschild homology of the Khovanov arc ring H^n.
We can also view the Bar-Natan--Russell skein module from a representation-theoretic viewpoint as an extension of the Frenkel--Khovanov graphical description of the Lusztig dual canonical basis of the nth tensor power of the fundamental U_q(sl_2)-representation. One of our primary results is to extend a dualization construction of Khovanov using Jones--Wenzl projectors from the Lusztig basis to the Russell basis.
We also construct and explore several deformations of the Russell skein module. One deformation is a quantum deformation that arises from embedding the Russell skein module in a space that obeys Kauffman--Lins diagrammatic relations. Our quantum version recovers the original Russell space when q is specialized to -1 and carries a natural braid group action that recovers the symmetric group action of Russell and Tymoczko. We also present an equivariant deformation that arises from replacing the TQFT algebra A used in the construction of the rings H^n by the equivariant homology of the two-sphere with the standard action of U(2) and taking the 0th Hochschild homology of the resulting deformed arc rings. We show that the equivariant deformation has the expected rank.
Finally, we consider the Khovanov two-functor F from the category of tangles. We show that it induces a surjection from the space of cobordisms of planar (2m, 2n)-tangles to the space of (H^m, H^n)-bimodule homomorphisms and give an explicit description of the kernel. We use our result to introduce a new quotient of the Russell skein module.Mathematics, Torus (Geometry), Mathematics, Duality theory (Mathematics), Quantum theory--Mathematicsalh2172MathematicsDissertationsConformally invariant random planar objects
https://academiccommons.columbia.edu/catalog/ac:197710
Benoist, Stephanehttp://dx.doi.org/10.7916/D80G3K4TTue, 19 Apr 2016 12:22:43 +0000This thesis explores different aspects of a surprising field of research: the conformally invariant scaling limits of planar statistical mechanics models.
The aspects developed here include the proof of convergence of certain interfaces in the critical Ising magnetization model (joint work with Hugo Duminil-Copin and Clement Hongler), a study of the near-critical behavior of the uniform spanning tree in the scaling limit (joint work with Laure Dumaz and Wendelin Werner), the construction of an interesting measure on continuous loops satisfying a certain stability property under deformation (joint work with Julien Dubedat) as well as some related algebraic considerations, and finally, notes on a paper of Sheffield, that studies a certain coupling of the scaling limits of discrete interfaces - SLE curves - together with random surfaces obtained from the Gaussian free field.Mathematics, Theoretical mathematics, Statistical mechanics, Ising model, Statistical mechanics--Mathematical models, Conformal invariantssb3193MathematicsDissertationsQuantum difference equations for quiver varieties
https://academiccommons.columbia.edu/catalog/ac:197653
Smirnov, Andreyhttp://dx.doi.org/10.7916/D8RN37T6Thu, 14 Apr 2016 12:21:44 +0000For an arbitrary Nakajima quiver variety X, we construct an analog of the quantum dynamical Weyl group acting in its equivariant K-theory. The correct generalization of the Weyl group here is the fundamental groupoid of a certain periodic locally finite hyperplane arrangement in Pic(X)⊗C. We identify the lattice part of this groupoid with the operators of quantum difference equation for X. The cases of quivers of finite and affine type are illustrated by explicit examples.Mathematics, Weyl groups, Quantum theory--Mathematics, K-theory, Difference equationsas4128MathematicsDissertationsDynamics, Graph Theory, and Barsotti-Tate Groups: Variations on a Theme of Mochizuki
https://academiccommons.columbia.edu/catalog/ac:197650
Krishnamoorthy, Rajuhttp://dx.doi.org/10.7916/D88K792NThu, 14 Apr 2016 12:21:26 +0000In this dissertation, we study etale correspondence of hyperbolic curves with unbounded dynamics. Mochizuki proved that over a field of characteristic 0, such curves are always Shimura curves. We explore variants of this question in positive characteristic, using graph theory, l-adic local systems, and Barsotti-Tate groups. Given a correspondence with unbounded dynamics, we construct an infinite graph with a large group of ”algebraic” automorphisms and roughly measures the ”generic dynamics” of the correspondence. We construct a specialization map to a graph representing the actual dynamics. Along the way, we formulate conjectures that etale correspondences with unbounded dynamics behave similarly to Hecke correspondences of Shimura curves. Using graph theory, we show that type (3,3) etale correspondences verify various parts of this philosophy. Key in the second half of this dissertation is a recent p-adic Langlands correspondence, due to Abe, which answers affirmatively the petites camarades conjecture of Deligne in the case of curves. This allows us the build a correspondence between rank 2 l-adic local systems with trivial determinant and Frobenius traces in Q and certain height 2, dimension 1 Barsotti-Tate groups. We formulate a conjecture on the fields of definitions of certain compatible systems of l-adic representations. Relatedly, we conjecture that the Barsotti-Tate groups over complete curves in positive characteristic may be ”algebraized” to abelian schemes.Mathematics, Exponential functions, Geometry, Hyperbolic, Shimura varieties, Mathematicsstk2117MathematicsDissertationsSimultaneous twists of elliptic curves and the Hasse principle for certain K3 surfaces
https://academiccommons.columbia.edu/catalog/ac:197554
Pal, Vivekhttp://dx.doi.org/10.7916/D81C1WVGMon, 11 Apr 2016 18:25:39 +0000In this thesis we unconditionally show that certain K3 surfaces satisfy the Hasse principle. Our method involves the 2-Selmer groups of simultaneous quadratic twists of two elliptic curves, only with places of good or additive reduction. More generally we prove that, given finitely many such elliptic curves defined over a number field (with rational 2-torsion and satisfying some mild conditions) there exists an explicit quadratic extension such that the quadratic twist of each elliptic curve has essential 2-Selmer rank one. Furthermore, given a 2-covering in each of the 2-Selmer groups, the quadratic extension above can be chosen so that the 2-Selmer group of the quadratic twist of each elliptic curve is generated by the given 2-covering and the image of the 2-torsion.
Our approach to the Hasse Principle is outlined below and was introduced by Skorobogatov and Swinnerton-Dyer. We also generalize the result proved in their paper. If each elliptic curve has a distinct multiplicative place of bad reduction, then we find a quadratic extension such that the quadratic twist of each elliptic curve has essential 2-Selmer rank one. Furthermore, given a 2-covering in each of the 2-Selmer groups, the quadratic extension above can be chosen so that the 2-Selmer group of the quadratic twist of each elliptic curve is generated by the given 2-covering and the image of the 2-torsion. If we further assume the finiteness of the Shafarevich-Tate groups (of the twisted elliptic curves) then each elliptic curve has Mordell-Weil rank one. If K = Q, then under the above assumptions the analytic rank of each elliptic curves is one. Furthermore, with the assumption on the Shafarevich-Tate group (and K = Q), we describe a single quadratic twist such that each elliptic curve has analytic rank zero and Mordell-Weil rank zero, again under some mild assumptions.Mathematics, Curves, Elliptic, Mathematics, Equations, Geometry, Differentialvp2262MathematicsDissertationsKuranishi atlases and genus zero Gromov-Witten invariants
https://academiccommons.columbia.edu/catalog/ac:197454
Castellano, Roberthttp://dx.doi.org/10.7916/D89W0FF0Mon, 11 Apr 2016 18:24:16 +0000Kuranishi atlases were introduced by McDuff and Wehrheim as a means to build a virtual fundamental cycle on moduli spaces of J-holomorphic curves and resolve some of the challenges in this field. This thesis considers genus zero Gromov-Witten invariants on a general closed symplectic manifold. We complete the construction of these invariants using Kuranishi atlases. To do so, we show that Gromov-Witten moduli spaces admit a smooth enough Kuranishi atlas to define a virtual fundamental class in any virtual dimension. In the process, we prove a stronger gluing theorem. Once we have defined genus zero Gromov-Witten invariants, we show that they satisfy the Gromov-Witten axioms of Kontsevich and Manin, a series of main properties that these invariants are expected to satisfy. A key component of this is the introduction of the notion of a transverse subatlas, a useful tool for working with Kuranishi atlases.Mathematics, Invariants, Moduli theory, Pseudoholomorphic curves, Gromov-Witten invariantsrtc2119Mathematics, Mathematics (Barnard College)DissertationsViscosity Characterizations of Explosions and Arbitrage
https://academiccommons.columbia.edu/catalog/ac:197340
Wang, Yinghuihttp://dx.doi.org/10.7916/D8125SMHWed, 06 Apr 2016 18:13:37 +0000This thesis analyzes the viscosity characterizations of the explosion time distribution for diffusions and of the arbitrage function in an equity market model with uncertainty. In the first part, we show that the tail distribution U of the explosion time for a multidimensional diffusion -- and more generally, a suitable function 𝒰 of the Feynman-Kac type involving the explosion time -- is a viscosity solution of an associated parabolic partial differential equation (PDE), provided that the dispersion and drift coefficients of the diffusion are continuous. This generalizes a result of Karatzas and Ruf (2015), who characterize U as a classical solution of a Cauchy problem for the PDE in the one-dimensional case, under the stronger condition of local Hölder continuity on the coefficients. We also extend their result to 𝒰 in the one-dimensional case by establishing the joint continuity of 𝒰. Furthermore, we show that 𝒰 is dominated by any nonnegative classical supersolution of this Cauchy problem. Finally, we consider another notion of weak solvability, that of the distributional (sub/super)solution, and show that 𝒰 is no greater than any nonnegative distributional supersolution of the relevant PDE. In the second part, a more elaborate mathematical finance setting is taken. We show that, in an equity market model with Knightian uncertainty regarding the relative risk and covariance structure of its assets, the arbitrage function -- defined as the reciprocal of the highest return on investment that can be achieved relative to the market using nonanticipative strategies, and under any admissible market model configuration -- is a viscosity solution of an associated Hamilton-Jacobi-Bellman (HJB) equation under appropriate boundedness, continuity and Markovian assumptions on the uncertainty structure. This result generalizes that of Fernholz and Karatzas (2011), who characterized this arbitrage function as a classical solution of a Cauchy problem for this HJB equation under much stronger conditions than those needed here. Our approach and results also extend to the Markovian Market Weight model introduced in Fernholz and Karatzas (2010b).Mathematics, Finance, Arbitrage, Business mathematics, Differential equations, Partial, Viscosity solutions, Hamilton-Jacobi equationsyw2450MathematicsDissertationsDerived Categories of Moduli Spaces of Semistable Pairs over Curves
https://academiccommons.columbia.edu/catalog/ac:197334
Potashnik, Natashahttp://dx.doi.org/10.7916/D8H99542Mon, 04 Apr 2016 18:46:08 +0000The context of this thesis is derived categories in algebraic geometry and geo- metric quotients. Specifically, we prove the embedding of the derived category of a smooth curve of genus greater than one into the derived category of the moduli space of semistable pairs over the curve. We also describe closed cover conditions under which the composition of a pullback and a pushforward induces a fully faithful functor. To prove our main result, we give an exposition of how to think of general Geometric Invariant Theory quotients as quotients by the multiplicative group.Mathematics, Geometry, Algebraic, Moduli theory, Mathematics, Derived categories (Mathematics)np2411MathematicsDissertationsDynamics of Large Rank-Based Systems of Interacting Diffusions
https://academiccommons.columbia.edu/catalog/ac:195668
Bruggeman, Cameronhttp://dx.doi.org/10.7916/D80G3K1GThu, 10 Mar 2016 12:18:05 +0000We study systems of n dimensional diffusions whose drift and dispersion coefficients depend only on the relative ranking of the processes. We consider the question of how long it takes for a particle to go from one rank to another. It is argued that as n gets large, the distribution of particles satisfies a Porous Medium Equation. Using this, we derive a deterministic limit for the system of particles. This limit allows for direct calculation of the properties of the rank traversal time. The results are extended to the case of asymmetrically colliding particles.
These models are of interest in the study of financial markets and economic inequality. In particular, we derive limits for the performance of some Functionally Generated Portfolios originating from Stochastic Portfolio Theory.Mathematics, Statistics, Diffusion processes, Diffusion--Mathematical models, Dispersion--Mathematical models, Porous materials--Mathematical models, Portfolio management--Mathematical modelscpb2133MathematicsDissertationsVariation in the Large-Scale Organization of Gene Expression Levels in the Hippocampus Relates to Stable Epigenetic Variability in Behavior
https://academiccommons.columbia.edu/catalog/ac:195407
Alter, Mark D.; Rubin, Daniel B.; Ramsey, Keri; Halpern, Rebecca; Stephan, Dietrich A.; Abbott, Larry; Hen, Renehttp://dx.doi.org/10.7916/D8WW7HJ3Tue, 08 Mar 2016 13:41:03 +0000Background
Despite sharing the same genes, identical twins demonstrate substantial variability in behavioral traits and in their risk for disease. Epigenetic factors–DNA and chromatin modifications that affect levels of gene expression without affecting the DNA sequence–are thought to be important in establishing this variability. Epigenetically-mediated differences in the levels of gene expression that are associated with individual variability traditionally are thought to occur only in a gene-specific manner. We challenge this idea by exploring the large-scale organizational patterns of gene expression in an epigenetic model of behavioral variability.
Methodology/Findings
To study the effects of epigenetic influences on behavioral variability, we examine gene expression in genetically identical mice. Using a novel approach to microarray analysis, we show that variability in the large-scale organization of gene expression levels, rather than differences in the expression levels of specific genes, is associated with individual differences in behavior. Specifically, increased activity in the open field is associated with increased variance of log-transformed measures of gene expression in the hippocampus, a brain region involved in open field activity. Early life experience that increases adult activity in the open field also similarly modifies the variance of gene expression levels. The same association of the variance of gene expression levels with behavioral variability is found with levels of gene expression in the hippocampus of genetically heterogeneous outbred populations of mice, suggesting that variation in the large-scale organization of gene expression levels may also be relevant to phenotypic differences in outbred populations such as humans. We find that the increased variance in gene expression levels is attributable to an increasing separation of several large, log-normally distributed families of gene expression levels. We also show that the presence of these multiple log-normal distributions of gene expression levels is a universal characteristic of gene expression in eurkaryotes. We use data from the MicroArray Quality Control Project (MAQC) to demonstrate that our method is robust and that it reliably detects biological differences in the large-scale organization of gene expression levels.
Conclusions
Our results contrast with the traditional belief that epigenetic effects on gene expression occur only at the level of specific genes and suggest instead that the large-scale organization of gene expression levels provides important insights into the relationship of gene expression with behavioral variability. Understanding the epigenetic, genetic, and environmental factors that regulate the large-scale organization of gene expression levels, and how changes in this large-scale organization influences brain development and behavior will be a major future challenge in the field of behavioral genomics.Genetics, Health sciences, Neurosciences, Hippocampus (Brain), Gene expression, Behavior genetics, DNA microarrays--Data processingdr2525, lfa2103, rh95Psychiatry, Mathematics, NeuroscienceArticlesGrowth, Productivity, and Policy Instruments During the Lost Decade: A Replication of Beason and Weinstein (1996) for the Period 1992-1998
https://academiccommons.columbia.edu/catalog/ac:191453
Pinkovskiy, Maxim L.http://dx.doi.org/10.7916/D8CC10B3Wed, 25 Nov 2015 14:57:06 +0000We replicate the analysis of the connection between Japanese sectoral productivity growth and industrial policy performed by Beason and Weinstein (1996) of national accounts data for the period 1992 to 1999. We show that despite some positive raw correlations between growth and industrial policy tools, there is no robust association between growth and industrial policy in the 1990s. This is consistent with the conclusions of Beason and Weinstein for the high-growth period (1960-1973). We also confirm the development of some trends evident in the previous data, such as the skewed distribution of policy instrument application to politically influential industries and the inconsistent application of different instruments to industries. Overall, the data is much more consistent with a theory of Japanese industrial policy as a product of political economy than industrial policy as a response to growth and productivity.Economics, Commerce-Business, Industrial productivity, Industrial policy, Economic development--Political aspects, Japanese--Economic conditionsEconomics, MathematicsArticlesA Minkowski-Type Inequality for Hypersurfaces in the Reissner-Nordstrom-Anti-deSitter Manifold
https://academiccommons.columbia.edu/catalog/ac:187980
Wang, Zhuhaihttp://dx.doi.org/10.7916/D86H4GGNWed, 13 May 2015 12:21:47 +0000We prove a sharp Minkowski-type inequality for hypersurfaces in the n-dimensional Reissner-Nordström-Anti-deSitter(AdS) manifold for n ≥ 3. This inequality generalizes the one for hypersurfaces in the uncharged AdS-Schwarzschild manifold proved in 5. With the Minkowski inequality, we prove a charged Gibbons-Penrose inequality for a large class of (n - 1)-dimensional spacelike surfaces in the Reissner-Nordström spacetime.Mathematicszw2175MathematicsDissertationsThe Parity of Analytic Ranks among Quadratic Twists of Elliptic Curves over Number Fields
https://academiccommons.columbia.edu/catalog/ac:186977
Balsam, Nava Kaylahttp://dx.doi.org/10.7916/D87P8XF4Thu, 07 May 2015 00:16:25 +0000The parity of the analytic rank of an elliptic curve is given by the root number in the functional equation L(E,s). Fixing an elliptic curve over any number eld and considering the family of its quadratic twists, it is natural to ask what the average analytic rank in this family is. A lower bound on this number is given by the average root number. In this paper, we investigate the root number in such families and derive an asymptotic formula for the proportion of curves in the family that have even rank. Our results are then used to support a conjecture about the average analytic rank in this family of elliptic curves.MathematicsMathematicsDissertationsA Proof of Looijenga's Conjecture via Integral-Affine Geometry
https://academiccommons.columbia.edu/catalog/ac:186926
Engel, Philiphttp://dx.doi.org/10.7916/D8028QGQFri, 24 Apr 2015 18:34:12 +0000A cusp singularity is a surface singularity whose minimal resolution is a reduced cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. In 1981, Looijenga proved that whenever a cusp singularity is smoothable, the minimal resolution of the dual cusp is an anticanonical divisor of some smooth rational surface. He conjectured the converse. This dissertation provides a proof of Looijenga's conjecture based on a combinatorial criterion for smoothability given by Friedman and Miranda in 1983, and explores the geometry of the space of smoothings. The key tool in the proof is the use of integral-affine surfaces, two-dimensional manifolds whose transition functions are valued in the integral-affine transformation group. Motivated by the proof and recent work in mirror symmetry, we make a conjecture regarding the structure of the smoothing components of a cusp singularity.MathematicsMathematicsDissertationsQuantum Algebras and Cyclic Quiver Varieties
https://academiccommons.columbia.edu/catalog/ac:186938
Negut, Andreihttp://dx.doi.org/10.7916/D8J38RGFFri, 24 Apr 2015 18:33:17 +0000The purpose of this thesis is to present certain viewpoints on the geometric representation theory of Nakajima cyclic quiver varieties, in relation to the Maulik-Okounkov stable basis. Our main technical tool is the shuffle algebra, which arises as the K-theoretic Hall algebra of the double cyclic quiver. We prove the isomorphism between the shuffle algebra and the quantum toroidal algebra U_[q,t](sl_n), and identify the quotients of Verma modules for the shuffle algebra with the K-theory groups of Nakajima cyclic quiver varieties, which were studied by Nakajima and Varagnolo-Vasserot.
The shuffle algebra viewpoint allows us to construct the universal R-matrix of the quantum toroidal algebra U_[q,t](sl_n), and to factor it in terms of pieces that arise from subalgebras isomorphic to quantum affine groups U_q(gl_m), for various m. This factorization generalizes constructions of Khoroshkin-Tolstoy to the toroidal case, and matches the factorization that Maulik-Okounkov produce via the stable basis in the K-theory of Nakajima quiver varieties. We connect the two pictures by computing formulas for the root generators of U_[q,t](sl_n) acting on the stable basis, which provide a wide extension of Murnaghan-Nakayama and Pieri type rules from combinatorics.Mathematicsan2534MathematicsDissertationsPartial differential equations and variational approaches to constant scalar curvature metrics in Kähler geometry
https://academiccommons.columbia.edu/catalog/ac:186467
Rubin, Daniel Ilanhttp://dx.doi.org/10.7916/D8HD7TMGThu, 23 Apr 2015 12:24:35 +0000In this thesis we investigate two approaches to the problem of existence of metrics of constant scalar curvature in a fixed Kähler class. In the first part, we
examine the equation for constant scalar curvature under the assumption of toric symmetry, thus reducing the problem to a fourth order nonlinear degenerate elliptic equation for a convex function defined in a polytope in ℝ^n. We obtain partial results on this equation using an associated Monge-Ampère equation to determine the boundary behavior of the solution. In the second part, we consider the asymptotics of certain energy functionals and their relation to stability and the existence of minimizers. We derive explicit formulas for their asymptotic slopes, which allows one to determine whether or not (X,L) is stable, and in some cases rule out the existence of a canonical metric.Mathematicsdr2525MathematicsDissertationsSingular Solutions to the Monge-Ampere Equation
https://academiccommons.columbia.edu/catalog/ac:186488
Mooney, Connor R.http://dx.doi.org/10.7916/D89K4955Fri, 10 Apr 2015 15:20:22 +0000This thesis contains the author's results on singular solutions to the Monge-Ampere equation \det D^2u = 1. We first prove that solutions are smooth away from a small closed singular set of Hausdorff (n-1)-dimensional measure zero. We also construct solutions with a singular set of Hausdorff dimension n-1, showing that this result is optimal. As a consequence we obtain unique continuation for the Monge-Ampere equation. Finally, we prove an interior W^{2,1} estimate for singular solutions, and we construct an example to show that this estimate is optimal.Mathematicscrm2181MathematicsDissertationsOn a Spectral Bound for Congruence Subgroup Families in SL(3,Z)
https://academiccommons.columbia.edu/catalog/ac:184064
Heath, Timothy Christopherhttp://dx.doi.org/10.7916/D8XW4HNMTue, 24 Feb 2015 12:16:08 +0000Spectral bounds on Maass forms of congruence families in algebraic groups are important ingredients to proving almost prime results for these groups. Extending the work of Gamburd [Gamburd, 2002] and Magee [Magee, 2013], we produce a condition under which such a bound exists in congruence subgroup families of SL(3,Z), uniformly and even when these groups are thin, i.e. of infinite index. The condition is analogous to the cusp and collar lemmas in Gamburd's work and is expected to hold for families whose Hausdorff dimension of the limit set is large enough.MathematicsMathematicsDissertationsEquivariant Gromov-Witten Theory of GKM Orbifolds
https://academiccommons.columbia.edu/catalog/ac:180940
Zong, Zhengyuhttp://dx.doi.org/10.7916/D8513WZCThu, 04 Dec 2014 09:40:48 +0000In this paper, we study the all genus Gromov-Witten theory for any GKM orbifold X. We generalize the Givental formula which is studied in the smooth case in [41] [42] [43] to the orbifold case. Specifically, we recover the higher genus Gromov-Witten invariants of a GKM orbifold X by its genus zero data. When X is toric, the genus zero Gromov-Witten invariants of X can be explicitly computed by the mirror theorem studied in [22] and our main theorem gives a closed formula for the all genus Gromov-Witten invariants of X. When X is a toric Calabi-Yau 3-orbifold, our formula leads to a proof of the remodeling conjecture in [38]. The remodeling conjecture can be viewed as an all genus mirror symmetry for toric Calabi-Yau 3-orbifolds. In this case, we apply our formula to the A-model higher genus potential and prove the remodeling conjecture by matching it to the B-model higher genus potential.MathematicsMathematicsDissertationsWorld Science U
https://academiccommons.columbia.edu/catalog/ac:177748
Greene, Brian; Dreifus, Claudia; Matiz, A. Mauriciohttp://dx.doi.org/10.7916/D869725QWed, 01 Oct 2014 11:47:48 +0000Brian Greene discusses his latest project, World Science U, with writer Claudia Dreifus. Brian Greene is a professor of physics and mathematics at Columbia University, and is widely recognized for a number of groundbreaking discoveries in superstring theory, including the co-discoveries of mirror symmetry and topology change. His first book for general audiences, The Elegant Universe, was a finalist for the Pulitzer Prize, and has sold more than a million copies worldwide. His more recent books, The Fabric of the Cosmos and The Hidden Reality, were both New York Times bestsellers, and inspired the Washington Post to call him "the single best explainer of abstruse concepts in the world today." Greene's latest project, World Science U, brings science education online with innovative digital courses available to anyone with an interest in science. Greene makes frequent media appearances on programs such as Charlie Rose, The Colbert Report and David Letterman. He has hosted two NOVA specials, based on The Elegant Universe and The Fabric of the Cosmos, which were nominated for four Emmy Awards and won a George Foster Peabody Award. Professor Greene is co-director of Columbia's Institute for Strings, Cosmology, and Astroparticle Physics, and with producer Tracy Day, he is co-founder of the World Science Festival. Claudia Dreifus, a journalist and adjunct professor at Columbia's School of International and Public Affairs, is well known for her interviews in the New York Times with leading figures in world politics and science. Maurice Matiz, CCNMTL's acting executive director, moderates the discussion on World Science U, online learning, and the changing role of the university professor.Science educationbg111, cd2106, amm8Mathematics, International and Public Affairs, Columbia Center for New Media Teaching and LearningInterviews and roundtablesA complete classification of epistatic two-locus models
https://academiccommons.columbia.edu/catalog/ac:190069
Hallgrímsdóttir, Ingileif; Yuster, Debbie S.http://dx.doi.org/10.7916/D880512QTue, 09 Sep 2014 00:01:35 +0000Background: The study of epistasis is of great importance in statistical genetics in fields such as linkage and association analysis and QTL mapping. In an effort to classify the types of epistasis in the case of two biallelic loci Li and Reich listed and described all models in the simplest case of 0/1 penetrance values. However, they left open the problem of finding a classification of two-locus models with continuous penetrance values.
Results: We provide a complete classification of biallelic two-locus models. In addition to solving the classification problem for dichotomous trait disease models, our results apply to any instance where real numbers are assigned to genotypes, and provide a complete framework for studying epistasis in QTL data. Our approach is geometric and we show that there are 387 distinct types of two-locus models, which can be reduced to 69 when symmetry between loci and alleles is accounted for. The model types are defined by 86 circuits, which are linear combinations of genotype values, each of which measures a fundamental unit of interaction.
Conclusion: The circuits provide information on epistasis beyond that contained in the additive × additive, additive × dominance, and dominance × dominance interaction terms. We discuss the connection between our classification and standard epistatic models and demonstrate its utility by analyzing a previously published dataset.Genetics, Developmental biology, Evolution and development, Epistasis (Genetics), Genetics, Scientific literature--Mathematical models, ClassificationMathematicsArticlesMultiple Dirichlet Series for Affine Weyl Groups
https://academiccommons.columbia.edu/catalog/ac:176818
Whitehead, Ianhttp://dx.doi.org/10.7916/D8BK19HTMon, 07 Jul 2014 11:54:13 +0000Let W be the Weyl group of a simply-laced affine Kac-Moody Lie group, excepting type A affine root systems of even rank. We construct a multiple Dirichlet series Z(x_1, ... x_n+1 meromorphic in a half-space, satisfying a group W of functional equations. This series is analogous to the multiple Dirichlet series for classical Weyl groups constructed by Brubaker-Bump-Friedberg, Chinta-Gunnells, and others. It is completely characterized by four natural axioms concerning its coefficients, axioms which come from the geometry of parameter spaces of hyperelliptic curves. The series constructed this way is optimal for computing moments of character sums and L-functions, including the fourth moment of quadratic L-functions at the central point via affine D4 and the second moment weighted by the number of divisors of the conductor via affine A_3. We also give evidence to suggest that this series appears as a first Fourier-Whittaker coefficient in an Eisenstein series on the twofold metaplectic cover of the relevant Kac-Moody group. The construction is limited to the rational function field, but it also describes the p-part of the multiple Dirichlet series over an arbitrary global field.MathematicsMathematicsDissertationsDemazure-Lusztig Operators and Metaplectic Whittaker Functions on Covers of the General Linear Group
https://academiccommons.columbia.edu/catalog/ac:176190
Puskas, Annahttp://dx.doi.org/10.7916/D8J964J6Mon, 07 Jul 2014 11:51:49 +0000There are two different approaches to constructing Whittaker functions of metaplectic groups over non-archimedean local fields. One approach, due to Chinta and Offen for the general linear group and to McNamara in general, represents the spherical Whittaker function in terms of a sum over a Weyl group. The second approach, by Brubaker, Bump and Friedberg and separately by McNamara, expresses it as a sum over a highest weight crystal.
This work builds a direct, combinatorial connection between the two approaches. This is done by exploring both in terms of Demazure and Demazure-Lusztig operators associated to the Weyl group of an irreducible root system. The relevance of Demazure and Demazure-Lusztig operators is indicated by results in the non-metaplectic setting: the Demazure character formula, Tokuyama's theorem and the work of Brubaker, Bump and Licata in describing Iwahori-Whittaker functions.
The first set of results is joint work with Gautam Chinta and Paul E. Gunnells. We define metaplectic Demazure and Demazure-Lusztig operators for a root system of any type. We prove that they satisfy the same Braid relations and quadratic relations as their nonmetaplectic analogues. Then we prove two formulas for the long word in the Weyl group. One is a metaplectic generalization of Demazure's character formula, and the other connects the same expression to Demazure-Lusztig operators. Comparing the two results to McNamara's construction of metaplectic Whittaker functions results in a formula for the Whittaker functions in the spirit of the Demazure character formula.
The second set of results relates to Tokuyama's theorem about the crystal description of type A characters. We prove a metaplectic generalization of this theorem. This establishes a combinatorial link between the two approaches to constructing Whittaker functions for metaplectic covers of any degree. The metaplectic version of Tokuyama's theorem is proved as a special case of a stronger result: a crystal description of polynomials produced by sums of Demazure-Lusztig operators acting on a monomial. These results make use of the Demazure and Demazure-Lusztig formulas above, and the branching structure of highest weight crystals of type A. The polynomials produced by sums of Demazure-Lusztig operators acting on a monomial are related to Iwahori fixed Whittaker functions in the nonmetaplectic setting.MathematicsMathematicsDissertationsTowards a definition of Shimura curves in positive characteristics
https://academiccommons.columbia.edu/catalog/ac:176077
Xia, Jiehttp://dx.doi.org/10.7916/D8ZP448CMon, 07 Jul 2014 11:47:28 +0000In the thesis, we present some answers to the question
What is an appropriate definition of Shimura curves in positive characteristics ?
The answer is obvious for Shimura curves of PEL type due to the moduli interpretation. Thus what is more interesting is the answer on Shimura curves of Hodge type.
Inspired by an example constructed by David Mumford, we find conditions on a proper smooth curve over a field of positive characteristic which guarantee that it lifts to a Shimura curve of Hodge type over the complex numbers. These conditions are in terms of geometry mod p, such as Barsotti-Tate groups, Dieudonne isocrystals, crystalline Hodge cycles and l-adic monodromy. Thus one can take them as definitions of Shimura curves in positive characteristics. More generally, We define ``weak" Shimura curves in characteristic p.
Along the way, we prove if a Barsotti-Tate group is versally deformed over a proper curve over an algebraically closed field of positive characteristic, then it admits a unique deformation to the corresponding Witt ring. This deformation result serves as one of the key ingredients in the proofs.Mathematicsjx2149MathematicsDissertationsOn a triply-graded generalization of Khovanov homology
https://academiccommons.columbia.edu/catalog/ac:175996
Putyra, Krzysztofhttp://dx.doi.org/10.7916/D86971RXMon, 07 Jul 2014 11:44:36 +0000In this thesis we study a certain generalization of Khovanov homology that unifies both the original theory due to M. Khovanov, referred to as the even Khovanov homology, and the odd Khovanov homology introduced by P. Ozsv´ath, Z. Szab´o, and J. Rasmussen.
The generalized Khovanov complex is a variant of the formal Khovanov bracket introduced by Bar Natan, constructed in a certain 2-categorical extension of cobordisms, in which the disjoint union is a cubical 2-functor, but not a strict one. This allows us to twist the usual relations between cobordisms with signs or, more generally, other invertible scalars. We prove the homotopy type of the complex is a link invariant, and we show how both even and odd Khovanov homology can be recovered. Then we analyze other link homology theories arising from this construction such as a unified theory over the ring Z_p :=Z[p]/(p²−1), and a variant of the algebra of dotted cobordisms, defined over k := Z[X,Y,Z^±1]/(X² = Y² = 1).
The generalized chain complex is bigraded, but the new grading does not make it a stronger invariant. However, it controls up to some extend signs in the complex, the property we use to prove several properties of the generalized Khovanov complex such as multiplicativity with respect to disjoint unions and connected sums of links, and the duality between complexes for a link and its mirror image. In particular, it follows the odd Khovanov homology of anticheiral links is self-dual. Finally, we explore Bockstein-type homological operations, proving the unified theory is a finer invariant than the even and odd Khovanov homology taken together.Theoretical mathematics, MathematicsMathematicsDissertationsRational normal curves on complete intersections
https://academiccommons.columbia.edu/catalog/ac:175993
Pan, Xuanyuhttp://dx.doi.org/10.7916/D8KK98X0Mon, 07 Jul 2014 11:42:57 +0000We prove that the moduli space of rational normal curves on a low degree complete intersection passing several suitable points is a complete intersection.MathematicsMathematicsDissertationsA Spacetime Alexandrov Theorem
https://academiccommons.columbia.edu/catalog/ac:175978
Wang, Ye-Kaihttp://dx.doi.org/10.7916/D8MG7MN2Mon, 07 Jul 2014 11:41:44 +0000Let Σ be an embedded spacelike codimension-2 submanifold in a spherically symmetric spacetime satisfying null convergence condition. Suppose Σ has constant null mean curvature and zero torsion. We prove that Σ must lie in a standard null cone. This generalizes the classical Alexandrov theorem which classifies embedded constant mean curvature hypersurfaces in Euclidean space. The proof follows the idea of Ros and Brendle. We first derive a spacetime Minkowski formula for spacelike codimension-2 submanifolds using conformal Killing-Yano 2-forms. The Minkowski formula is then combined with a Heintze-Karcher type geometric inequality to prove the main theorem. We also obtain several rigidity results for codimension-2 submanifolds in spherically symmetric spacetimes.MathematicsMathematicsDissertationsSelf-duality and singularities in the Yang-Mills flow
https://academiccommons.columbia.edu/catalog/ac:175717
Waldron, Alexhttp://dx.doi.org/10.7916/D81V5C3RMon, 07 Jul 2014 11:38:13 +0000We investigate the long-time behavior and smooth convergence properties of the Yang-Mills flow in dimension four. Two chapters are devoted to equivariant solutions and their precise blowup asymptotics at infinite time. The last chapter contains general results. We show that a singularity of pure + or - charge cannot form within finite time, in contrast to the analogous situation of harmonic maps between Riemann surfaces. This implies long-time existence given low initial self-dual energy. In this case we study convergence of the flow at infinite time: if a global weak Uhlenbeck limit is anti-self-dual and has vanishing self-dual second cohomology, then the limit exists smoothly and exponential convergence holds. We also recover the classical grafting theorem, and derive asymptotic stability of this class of instantons in the appropriate sense.MathematicsMathematicsDissertationsPro-p-Iwahori-Hecke Algebras in the mod-p Local Langlands Program
https://academiccommons.columbia.edu/catalog/ac:175711
Koziol, Karolhttp://dx.doi.org/10.7916/D89C6VKVMon, 07 Jul 2014 11:38:05 +0000Let p be a prime number, and F a nonarchimedean local field of residual characteristic p. This thesis is dedicated to the study of the pro-p-Iwahori-Hecke algebra H_{F_p}(G, I(1)) in the mod-p Local Langlands Program, where G is the group of F-points of a connected, reductive group, and I(1) is a pro-p-Iwahori subgroup of G.
When G = U(2,1)(E/F) is an unramified unitary group in three variables, we first describe the structure and simple modules of the algebra H_{F_p}(G, I(1)). We then adapt methods of Schneider-Stuhler and Paskunas to construct, for each supersingular H_{F_p}(G, I(1))-module, a supersingular representation of G. These are exactly the representations which are expected to correspond to irreducible Galois parameters.
When G = U(1,1)(Q_{p^2} /Q_p) is an unramified unitary group in two variables, we use the pro-p-Iwahori-Hecke algebra H_{F_p}(G_S , I_S(1)) of the derived subgroup G_S to classify the supersingular representations of G. Combining this with previous results, we obtain a classification of all irreducible representations of G, and then construct a correspondence between representations of G and Galois parameters.
Finally, when G = GL_n(F) and G_S = SL_n(F), we show how to relate the two algebras H_{F_p}(G, I(1)) and H_{F_p}(G_S, I_S(1)). Using this interplay, we prove a numerical correspondence between L-packets of supersingular H_{F_p}(G_S , I_S(1))-modules and irreducible projective n-dimensional Galois representations, and prove that this correspondence is induced by a functor when F = Q_p.MathematicsMathematicsDissertationsConstant Scalar Curvature of Toric Fibrations
https://academiccommons.columbia.edu/catalog/ac:175513
Nyberg, Thomashttp://dx.doi.org/10.7916/D8TH8JVHMon, 07 Jul 2014 11:34:39 +0000We study the conditions under which a fibration of toric varieties, fibered over a flag variety, admits a constant scalar curvature Kähler metric. We first provide an introduction to toric varieties and toric fibrations and derive the scalar curvature equation. Next we derive interior a priori estimates of all orders and a global L^∞-estimate for the scalar curvature equation. Finally we extend the theory of K-Stability to this setting and construct test-configurations for these spaces.Mathematicstwn2103MathematicsDissertationsCanonical Metrics in Sasakian Geometry
https://academiccommons.columbia.edu/catalog/ac:175504
Collins, Tristanhttp://dx.doi.org/10.7916/D86Q1VCSMon, 07 Jul 2014 11:34:18 +0000The aim of this thesis is to study the existence problem for canonical Sasakian metrics, primarily Sasaki-Einstein metrics. We are interested in providing both necessary conditions, as well as sufficient conditions for the existence of such metrics.
We establish several sufficient conditions for the existence of Sasaki-Einstein metrics by studying the Sasaki-Ricci flow. In the process, we extend some fundamental results from the study of the Kahler-Ricci flow to the Sasakian setting. This includes finding Sasakian analogues of Perelman's energy and entropy functionals which are monotonic along the Sasaki-Ricci flow. Using these functionals we extend Perelman's deep estimates for the Kahler-Ricci flow to the Sasaki-Ricci flow. Namely, we prove uniform scalar curvature, diameter and non-collapsing estimates along the Sasaki-Ricci flow. We show that these estimates imply a uniform transverse Sobolev inequality. Furthermore, we introduce the sheaf of transverse foliate vector fields, and show that it has a natural, transverse complex structure. We show that the convergence of the flow is intimately related to the space of global transversely holomorphic sections of this sheaf.
We introduce an algebraic obstruction to the existence of constant scalar curvature Sasakian metrics, extending the notion of K-stability for projective varieties.
Finally, we show that, for regular Sasakian manifolds whose quotients are Kahler-Einstein Fano manifolds, the Sasaki-Ricci flow, or equivalently, the Kahler-Ricci flow, converges exponentially fast to a (transversely) Kahler-Einstein metric.Mathematicstcc2119MathematicsDissertationsThe arithmetic and geometry of genus four curves
https://academiccommons.columbia.edu/catalog/ac:175469
Xue, Hanghttp://dx.doi.org/10.7916/D87P8WHMMon, 07 Jul 2014 11:33:44 +0000We construct a point in the Jacobian of a non-hyperelliptic genus four curve which is defined over a quadratic extension of the base field. We attempt to answer two questions:
1. Is this point torsion?
2. If not, does it generate the Mordell--Weil group of the Jacobian?
We show that this point generates the Mordell--Weil group of the Jacobian of the universal genus four curve. We construct some families of genus four curves over the function field of $\bP^1$ over a finite field and prove that half of the Jacobians in this family are generated by this point via the other half are not. We then turn to the case where the base field is a number field or a function field. We compute the Neron--Tate height of this point in terms of the self-intersection of the relative dualizing sheaf of (the stable model of) the curve and some local invariants depending on the completion of
the curve at the places where this curve has bad or smooth hyperelliptic reduction. In the case where the reduction satisfies some certain conditions, we compute these local invariants explicitly.Mathematicshx2119MathematicsDissertationsBordered Heegaard Floer Homology and Graph Manifolds
https://academiccommons.columbia.edu/catalog/ac:175430
Hanselman, Jonathanhttp://dx.doi.org/10.7916/D8NZ85TFMon, 07 Jul 2014 11:32:47 +0000We use the techniques of bordered Heegaard Floer homology to investigate the Heegaard Floer homology of graph manifolds. Bordered Heegaard Floer homology allows us to split a graph manifold into pieces and perform computations for each piece separately. The resulting invariants can then be combined by a simple algebraic procedure to recover HFhat. Graph manifolds by definition decompose into pieces which are S¹-bundles over surfaces. This decomposition makes them particularly well suited to the divide-and-conquer techniques of bordered Heegaard Floer homology. In fact, the problem reduces to computing bordered Heegaard Floer invariants of just two pieces. The first invariant is the type D trimodule associated to the trivial S¹-bundle over the pair of pantsMathematicsMathematicsDissertationsThree-Manifold Mutations Detected by Heegaard Floer Homology
https://academiccommons.columbia.edu/catalog/ac:175403
Clarkson, Corrinhttp://dx.doi.org/10.7916/D8GF0RNGMon, 07 Jul 2014 11:31:33 +0000Given a self-diffeomorphism h of a closed, orientable surface S with genus greater than one and an embedding f of S into a three-manifold M, we construct a mutant manifold by cutting M along f(S) and regluing by h. We will consider whether there exist nontrivial gluings such that for any embedding, the manifold M and its mutant have isomorphic Heegaard Floer homology. In particular, we will demonstrate that if h is not isotopic to the identity map, then there exists an embedding of S into a three-manifold M such that the rank of the non-torsion summands of HF-hat of M differs from that of its mutant. We will also show that if the gluing map is isotopic to neither the identity nor the genus-two hyperelliptic involution, then there exists an embedding of S into a three-manifold M such that the total rank of HF-hat of M differs from that of its mutant.MathematicsMathematicsDissertationsDiscriminative topological features reveal biological network mechanisms
https://academiccommons.columbia.edu/catalog/ac:174771
Middendorf, Manuel; Ziv, Etay; Adams, Carter; Hom, Jennifer C.; Koytcheff, Robin; Levovitz, Chaya; Woods, Gregory; Chen, Linda; Wiggins, Chris H.http://dx.doi.org/10.7916/D8VD6WKKFri, 06 Jun 2014 13:31:01 +0000Background: Recent genomic and bioinformatic advances have motivated the development of numerous network models intending to describe graphs of biological, technological, and sociological origin. In most cases the success of a model has been evaluated by how well it reproduces a few key features of the real-world data, such as degree distributions, mean geodesic lengths, and clustering coefficients. Often pairs of models can reproduce these features with indistinguishable fidelity despite being generated by vastly different mechanisms. In such cases, these few target features are insufficient to distinguish which of the different models best describes real world networks of interest; moreover, it is not clear a priori that any of the presently-existing algorithms for network generation offers a predictive description of the networks inspiring them. Results: We present a method to assess systematically which of a set of proposed network generation algorithms gives the most accurate description of a given biological network. To derive discriminative classifiers, we construct a mapping from the set of all graphs to a high-dimensional (in principle infinite-dimensional) "word space". This map defines an input space for classification schemes which allow us to state unambiguously which models are most descriptive of a given network of interest. Our training sets include networks generated from 17 models either drawn from the literature or introduced in this work. We show that different duplication-mutation schemes best describe the E. coli genetic network, the S. cerevisiae protein interaction network, and the C. elegans neuronal network, out of a set of network models including a linear preferential attachment model and a small-world model. Conclusions: Our method is a first step towards systematizing network models and assessing their predictability, and we anticipate its usefulness for a number of communities.Bioinformaticsjch149, chw2Physics, Mathematics, Applied Physics and Applied MathematicsArticlesGeneralized Volatility-Stabilized Processes
https://academiccommons.columbia.edu/catalog/ac:165162
Pickova, Radkahttp://hdl.handle.net/10022/AC:P:21616Fri, 13 Sep 2013 15:07:49 +0000In this thesis, we consider systems of interacting diffusion processes which we call Generalized Volatility-Stabilized processes, as they extend the Volatility-Stabilized Market models introduced in Fernholz and Karatzas (2005). First, we show how to construct a weak solution of the underlying system of stochastic differential equations. In particular, we express the solution in terms of time-changed squared-Bessel processes and argue that this solution is unique in distribution. In addition, we also discuss sufficient conditions under which this solution does not explode in finite time, and provide sufficient conditions for pathwise uniqueness and for existence of a strong solution.
Secondly, we discuss the significance of these processes in the context of Stochastic Portfolio Theory. We describe specific market models which assume that the dynamics of the stocks' capitalizations is the same as that of the Generalized Volatility-Stabilized processes, and we argue that strong relative arbitrage opportunities may exist in these markets, specifically, we provide multiple examples of portfolios that outperform the market portfolio. Moreover, we examine the properties of market weights as well as the diversity weighted portfolio in these models.
Thirdly, we provide some asymptotic results for these processes which allows us to describe different properties of the corresponding market models based on these processes.Statisticsrp2424Statistics, MathematicsDissertationsProperties of Hamiltonian Torus Actions on Closed Symplectic Manifolds
https://academiccommons.columbia.edu/catalog/ac:161552
Fanoe, Andrew L.http://hdl.handle.net/10022/AC:P:20455Fri, 24 May 2013 09:40:41 +0000In this thesis, we will study the properties of certain Hamiltonian torus actions on closed symplectic manifolds. First, we will consider counting Hamiltonian torus actions on closed, symplectic manifolds M with 2-dimensional second cohomology. In particular, all such manifolds are bundles with fiber and base equal to projective spaces. We use cohomological techniques to show that there is a unique toric structure if the fiber has a smaller dimension than the base. Furthermore, if the fiber and base are both at least 2-dimensional projective spaces, we show that there is a finite number of toric structures on M that are compatible with some symplectic structure on M. Additionally, we show there is uniqueness in certain other cases, such as the case where M is a monotone symplectic manifold. Finally, we will be interested in the existence of symplectic, non-Hamiltonian circle actions on closed symplectic 6-manifolds. In particular, we will use J-holomorphic curve techniques to show that there are no such actions that satisfy certain fixed point conditions. This lends support to the conjecture that there are no such actions with a non-empty set of isolated fixed points.Mathematicsalf2140MathematicsDissertationsSingular theta lifts and near-central special values of Rankin-Selberg L-functions
https://academiccommons.columbia.edu/catalog/ac:161464
Garcia, Luis Emiliohttp://hdl.handle.net/10022/AC:P:20421Thu, 23 May 2013 11:35:19 +0000In this thesis we study integrals of a product of two automorphic forms of weight 2 on a Shimura curve over Q against a function on the curve with logarithmic singularities at CM points obtained as a Borcherds lift. We prove a formula relating periods of this type to a near-central special value of a Rankin-Selberg L-function. The results provide evidence for Beilinson's conjectures on special values of L-functions.Mathematicslg2440MathematicsDissertationsDel Pezzo surfaces with irregularity and intersection numbers on quotients in geometric invariant theory
https://academiccommons.columbia.edu/catalog/ac:161405
Maddock, Zachary Alexanderhttp://hdl.handle.net/10022/AC:P:20379Mon, 20 May 2013 15:28:03 +0000This thesis comprises two parts covering distinct topics in algebraic geometry. In Part I, we construct the first examples of regular del Pezzo surfaces for which the first cohomology group of the structure sheaf is nonzero. Such surfaces, which only exist over imperfect fields, arise as generic fibres of fibrations of singular del Pezzo surfaces in positive characteristic whose total spaces are smooth, and their study is motivated by the minimal model program. We also find a restriction on the integer pairs that are possible as the irregularity (that is, the dimension of the first cohomology group of the structure sheaf) and anti-canonical degree of regular del Pezzo surfaces with positive irregularity. In Part II, we consider a connected reductive group acting linearly on a projective variety over an arbitrary field. We prove a formula that compares intersection numbers on the geometric invariant theory quotient of the variety by the reductive group with intersection numbers on the geometric invariant theory quotient of the variety by a maximal torus, in the case where all semi-stable points are properly stable. These latter intersection numbers involve the top equivariant Chern class of the maximal torus representation given by the quotient of the adjoint representation on the Lie algebra of the reductive group by that of the maximal torus. We provide a purely algebraic proof of the formula when the root system decomposes into irreducible root systems of type A. We are able to remove this restriction on root systems by applying a related result of Shaun Martin from symplectic geometry.Mathematicszam2104MathematicsDissertationsA geometric construction of a Calabi quasimorphism on projective space
https://academiccommons.columbia.edu/catalog/ac:188472
Carneiro, Andre R.http://dx.doi.org/10.7916/D8N29W9TFri, 17 May 2013 10:14:46 +0000We use the rotation numbers defined by Théret in [T] to construct a quasimorphism on the universal cover of the Hamiltonian group of CP^n. We also show that this quasimorphism agrees with the Calabi invariant for isotopies that are supported in displaceable subsets of CP^n.Mathematicsarc2142Mathematics, Mathematics (Barnard College)DissertationsPurity of the stratification by Newton polygons and Frobenius-periodic vector bundles
https://academiccommons.columbia.edu/catalog/ac:161158
Yang, Yanhonghttp://hdl.handle.net/10022/AC:P:20334Wed, 15 May 2013 10:21:52 +0000This thesis includes two parts. In the first part, we show a purity theorem for stratifications by Newton polygons coming from crystalline cohomology, which says that the family of Newton polygons over a noetherian scheme have a common break point if this is true outside a subscheme of codimension bigger than 1. The proof is similar to the proof of [dJO99, Theorem 4.1]. In the second part, we prove that for every ordinary genus-2 curve X over a finite field k of characteristic 2 with automorphism group Z/2Z × S_3, there exist SL(2,k[[s]])-representations of π_1(X) such that the image of π_1(X^-) is infinite. This result produces a family of examples similar to Laszlo's counterexample [Las01] to a question regarding the finiteness of the geometric monodromy of representations of the fundamental group [dJ01].Mathematicsyy2244MathematicsDissertationsLocal Regularity of the Complex Monge-Ampere Equation
https://academiccommons.columbia.edu/catalog/ac:161155
Wang, Yuhttp://hdl.handle.net/10022/AC:P:20333Wed, 15 May 2013 10:11:41 +0000In this thesis, we present a self-contained account of the current development in the local regularity theory of the complex Monge-Ampere equation through the modern fully-nonlinear PDE point of view. We have apply the modern elliptic techniques to establish new local regularity results. These includes: regularity of small perturbed solutions, Holder regularity of the Hessian of the W^{2,p} solutions and a Liouville-type theorem.Mathematicsyw2340MathematicsDissertationsOdd symmetric functions and categorification
https://academiccommons.columbia.edu/catalog/ac:161123
Ellis, Alexander Palenhttp://hdl.handle.net/10022/AC:P:20307Tue, 14 May 2013 15:16:04 +0000We introduce q- and signed analogues of several constructions in and around the theory of symmetric functions. The most basic of these is the Hopf superalgebra of odd symmetric functions. This algebra is neither (super-)commutative nor (super-)cocommutative, yet its combinatorics still exhibit many of the striking integrality and positivity properties of the usual symmetric functions. In particular, we give odd analogues of Schur functions, Kostka numbers, and Littlewood-Richardson coefficients. Using an odd analogue of the nilHecke algebra, we give a categorification of the integral divided powers form of U_q^+(sl_2) inequivalent to the one due to Khovanov-Lauda. Along the way, we develop a graphical calculus for indecomposable modules for the odd nilHecke algebra.Mathematicsape2104MathematicsDissertationsLocalization and Heegaard Floer Homology
https://academiccommons.columbia.edu/catalog/ac:160806
Hendricks, Kristenhttp://hdl.handle.net/10022/AC:P:20103Wed, 01 May 2013 13:43:29 +0000In this thesis we use Seidel-Smith localization for Lagrangian Floer cohomology to study invariants of cyclic branched covers of three-manifolds and symmetry groups of knots by constructing localization spectral sequences in Heegaard Floer homology.Mathematicskeh2141MathematicsDissertationsp-adic Heights of Heegner points on Shimura curves
https://academiccommons.columbia.edu/catalog/ac:160525
Disegni, Danielhttp://hdl.handle.net/10022/AC:P:20097Wed, 01 May 2013 13:11:24 +0000Let f be a primitive Hilbert modular form of weight 2 and level N for the totally real field F, and let p be an odd rational prime such that f is ordinary at all primes dividing p. When E is a CM extension of F of relative discriminant prime to Np, we give an explicit construction of the p-adic Rankin-Selberg L-function L_p(f_E,-) and prove that when the sign of its functional equation is -1, its central derivative is given by the p-adic height of a Heegner point on the abelian variety A associated to f. This p-adic Gross-Zagier formula generalises the result obtained by Perrin-Riou when F=Q and N satisfies the so-called Heegner condition. We deduce applications to both the p-adic and the classical Birch and Swinnerton-Dyer conjectures for A.Mathematicsdd2438MathematicsDissertationsOn optimal arbitrage under constraints
https://academiccommons.columbia.edu/catalog/ac:160495
Sadhukhan, Subhankarhttp://hdl.handle.net/10022/AC:P:20076Wed, 01 May 2013 11:07:50 +0000In this thesis, we investigate the existence of relative arbitrage opportunities in a Markovian model of a financial market, which consists of a bond and stocks, whose prices evolve like Itô processes. We consider markets where investors are constrained to choose from among a restricted set of investment strategies. We show that the upper hedging price of (i.e. the minimum amount of wealth needed to superreplicate) a given contingent claim in a constrained market can be expressed as the supremum of the fair price of the given contingent claim under certain unconstrained auxiliary Markovian markets. Under suitable assumptions, we further characterize the upper hedging price as viscosity solution to certain variational inequalities. We, then, use this viscosity solution characterization to study how the imposition of stricter constraints on the market affect the upper hedging price. In particular, if relative arbitrage opportunities exist with respect to a given strategy, we study how stricter constraints can make such arbitrage opportunities disappear.Applied mathematics, Financess3240Statistics, MathematicsDissertationsSato-Tate Problem for GL(3)
https://academiccommons.columbia.edu/catalog/ac:160461
Zhou, Fanhttp://hdl.handle.net/10022/AC:P:20048Tue, 30 Apr 2013 14:18:13 +0000Based upon the work of Goldfeld and Kontorovich on the Kuznetsov trace formula of Maass forms for SL(3,Z), we prove a weighted vertical equidistribution theorem (with respect to the generalized Sato-Tate measure) for the p-th Hecke eigenvalue of Maass forms, with the rate of convergence. With a conjectured orthogonality relation between the Fourier coefficients of Maass forms for SL(N,Z) for N≥4, we generalize the above equidistribution theorem to N≥4.Mathematicsfz2133MathematicsDissertationsExploring the String Landscape: The Dynamics, Statistics, and Cosmology of Parallel Worlds
https://academiccommons.columbia.edu/catalog/ac:155499
Ahlqvist, Stein Pontushttp://hdl.handle.net/10022/AC:P:15784Tue, 15 Jan 2013 10:32:23 +0000This dissertation explores various facets of the low-energy solutions in string theory known as the string landscape. Three separate questions are addressed - the tunneling dynamics between these vacua, the statistics of their location in moduli space, and the potential realization of slow-roll inflation in the flux potentials generated in string theory. We find that the tunneling transitions that occur between a certain class of supersymmetric vacua related to each other via monodromies around the conifold point are sensitive to the details of warping in the near-conifold regime. We also study the impact of warping on the distribution of vacua near the conifold and determine that while previous work has concluded that the conifold point acts as an accumulation point for vacua, warping highly dilutes the distribution in precisely this regime. Finally we investigate a novel form of inflation dubbed spiral inflation to see if it can be realized near the conifold point. We conclude that for our particular models, spiral inflation seems to rely on a de Sitter-like vacuum energy. As a result, whenever spiral inflation is realized, the inflation is actually driven by a vacuum energy.Physicsspa2111Physics, MathematicsDissertationsOn Fourier-Mukai type functors
https://academiccommons.columbia.edu/catalog/ac:146747
Rizzardo, Alicehttp://hdl.handle.net/10022/AC:P:13173Mon, 07 May 2012 17:05:29 +0000In this thesis we study functors between bounded derived categories of sheaves and how they can be expressed in a geometric way, namely whether they are isomorphic to a Fourier-Mukai transform. Specifically, we describe the behavior of a functor between derived categories of smooth projective varieties when restricted to the derived category of the generic point of the second variety, when this last variety is a curve, a point or a rational surface. We also compute in general some sheaves that play the role of the cohomology sheaves of the kernel of a Fourier-Mukai transform and are then able to exhibit a class of functors that are neither faithful nor full, that are isomorphic to a Fourier-Mukai transform.Mathematicsar2532MathematicsDissertationsBordered Heegaard Floer Homology, Satellites, and Decategorification
https://academiccommons.columbia.edu/catalog/ac:146701
Petkova, Tsvetelina Vanevahttp://hdl.handle.net/10022/AC:P:13158Mon, 07 May 2012 12:39:26 +0000We use the methods of bordered Floer homology to provide a formula for both τ and HFK of certain satellite knots. In many cases, this formula determines the 4-ball genus of the satellite knot. In parallel, we explore the structural aspects of the bordered theory, developing the notion of an Euler characteristic for the modules associated to a bordered manifold. The Euler characteristic is an invariant of the underlying space, and shares many properties with the analogous invariants for closed 3-manifolds. We study the TQFT properties of this invariant corresponding to gluing, as well as its connections to sutured Floer homology. As one application, we show that the pairing theorem for bordered Floer homology categorifies the classical Alexander polynomial formula for satellites.Mathematicstvp2103MathematicsDissertationsEigenvarieties and twisted eigenvarieties
https://academiccommons.columbia.edu/catalog/ac:146594
Xiang, Zhengyuhttp://hdl.handle.net/10022/AC:P:13124Thu, 03 May 2012 14:44:03 +0000For an arbitrary reductive group G, we construct the full eigenvariety E, which parameterizes all p-adic overconvergent cohomological eigenforms of G in the sense of Ash-Stevens and Urban. Further, given an algebraic automorphism a of G, we construct the twisted eigenvariety E^a, a rigid subspace of E, which parameterizes all eigenforms that are invariant under a. In particular, in the case G = GLn, we prove that every self-dual automorphic representation can be deformed into a family of self-dual cuspidal forms containing a Zariski dense subset of classical points. This is the inverse of Ash-Pollack-Stevens conjecture. We also give some hint to this conjecture.Mathematicszx2108MathematicsDissertationsForbidden Substructures in Graphs and Trigraphs, and Related Coloring Problems
https://academiccommons.columbia.edu/catalog/ac:146465
Penev, Irenahttp://hdl.handle.net/10022/AC:P:13082Tue, 01 May 2012 16:34:11 +0000Given a graph G, χ(G) denotes the chromatic number of G, and ω(G) denotes the clique number of G (i.e. the maximum number of pairwise adjacent vertices in G). A graph G is perfect provided that for every induced subgraph H of G, χ(H) = ω(H). This thesis addresses several problems from the theory of perfect graphs and generalizations of perfect graphs. The bull is a five-vertex graph consisting of a triangle and two vertex-disjoint pendant edges; a graph is said to be bull-free provided that no induced subgraph of it is a bull. The first result of this thesis is a structure theorem for bull-free perfect graphs. This is joint work with Chudnovsky, and it first appeared in [12]. The second result of this thesis is a decomposition theorem for bull-free perfect graphs, which we then use to give a polynomial time combinatorial coloring algorithm for bull-free perfect graphs. We remark that de Figueiredo and Maffray [33] previously solved this same problem, however, the algorithm presented in this thesis is faster than the algorithm from [33]. We note that a decomposition theorem that is very similar (but slightly weaker) than the one from this thesis was originally proven in [52], however, the proof in this thesis is significantly different from the one in [52]. The algorithm from this thesis is very similar to the one from [52]. A class G of graphs is said to be χ-bounded provided that there exists a function f such that for all G in G, and all induced subgraphs H of G, we have that χ(H) ≤ f(ω(H)). χ-bounded classes were introduced by Gyarfas [41] as a generalization of the class of perfect graphs (clearly, the class of perfect graphs is χ-bounded by the identity function). Given a graph H, we denote by Forb*(H) the class of all graphs that do not contain any subdivision of H as an induced subgraph. In [57], Scott proved that Forb*(T) is χ-bounded for every tree T, and he conjectured that Forb*(H) is χ-bounded for every graph H. Recently, a group of authors constructed a counterexample to Scott's conjecture [51]. This raises the following question: for which graphs H is Scott's conjecture true? In this thesis, we present the proof of Scott's conjecture for the cases when H is the paw (i.e. a four-vertex graph consisting of a triangle and a pendant edge), the bull, and a necklace (i.e. a graph obtained from a path by choosing a matching such that no edge of the matching is incident with an endpoint of the path, and for each edge of the matching, adding a vertex adjacent to the ends of this edge). This is joint work with Chudnovsky, Scott, and Trotignon, and it originally appeared in [13]. Finally, we consider several operations (namely, "substitution," "gluing along a clique," and "gluing along a bounded number of vertices"), and we show that the closure of a χ-bounded class under any one of them, as well as under certain combinations of these three operations (in particular, the combination of substitution and gluing along a clique, as well as the combination of gluing along a clique and gluing along a bounded number of vertices) is again χ-bounded. This is joint work with Chudnovsky, Scott, and Trotignon, and it originally appeared in [14].Mathematicsip2158Mathematics, Industrial Engineering and Operations ResearchDissertationsThe Asymptotic Cone of Teichmuller Space: Thickness and Divergence
https://academiccommons.columbia.edu/catalog/ac:146378
Sultan, Harold Markhttp://hdl.handle.net/10022/AC:P:13058Mon, 30 Apr 2012 11:56:38 +0000Using the geometric model of the pants complex, we study the Asymptotic Cone of Teichmüller space equipped with the Weil Petersson metric. In particular, we provide a characterization of the canonical finest pieces in the tree-graded structure of the asymptotic cone of Teichmüller space along the same lines as similar characterizations for right angled Artin groups by Behrstock-Charney and for mapping class groups by Behrstock-Kleiner-Minsky-Mosher. As a corollary of the characterization, we complete the thickness classification of Teichmüller spaces for all surfaces of finite type. In particular, we prove that Teichmüller space of the genus two surface with one boundary component (or puncture) can be uniquely characterized in the following two senses: it is thick of order two, and it has superquadratic yet at most cubic divergence. In addition, we characterize strongly contracting quasi-geodesics in Teichmüller space, generalizing results of Brock-Masur-Minsky. As a tool in the thesis, we develop a natural relative of the curve complex called the complex of separating multicurves which may be of independent interest. The final chapter includes various related and independent results including, under mild hypotheses, a proof of the equivalence of wideness and unconstrictedness in the CAT(0) setting, as well as adapted versions of three preprints. Specifically, in the three preprints we characterize hyperbolic type quasi-geodesics in CAT(0) spaces, we prove that the separating curve complex of the genus two surface satisfies a quasidistance formula and is Gromov-hyperbolic, and we study the net of separating pants decompositions in the pants complex.Mathematicshms2121MathematicsDissertationsArithmetic inner product formula for unitary groups
https://academiccommons.columbia.edu/catalog/ac:146317
Liu, Yifenghttp://hdl.handle.net/10022/AC:P:13049Thu, 19 Apr 2012 15:09:40 +0000We study central derivatives of L-functions of cuspidal automorphic representations for unitary groups of even variables defined over a totally real number field, and their relation with the canonical height of special cycles on Shimura varieties attached to unitary groups of the same size. We formulate a precise conjecture about an arithmetic analogue of the classical Rallis' inner product formula, which we call arithmetic inner product formula, and confirm it for unitary groups of two variables. In particular, we calculate the Néron-Tate height of special points on Shimura curves attached to certain unitary groups of two variables. For an irreducible cuspidal automorphic representation of a quasi-split unitary group, we can associate it an ε-factor, which is either 1 or -1, via the dichotomy phenomenon of local theta liftings. If such factor is -1, the central L-value of the representation always vanishes and the Rallis' inner product formula is not interesting. Therefore, we are motivated to consider its central derivative, and propose the arithmetic inner product formula. In the course of such formulation, we prove a modularity theorem of the generating series on the level of Chow groups. We also show the cohomological triviality of the arithmetic theta lifting, which is a necessary step to consider the canonical height. As evidence, we also prove an arithmetic local Siegel-Weil formula at archimedean places for unitary groups of arbitrary sizes, which contributes as a part of the local comparison of the conjectural arithmetic inner product formula.MathematicsMathematicsDissertationsLimiting Properties of Certain Geometric Flows in Complex Geometry
https://academiccommons.columbia.edu/catalog/ac:146314
Jacob, Adam Joshuahttp://hdl.handle.net/10022/AC:P:13047Thu, 19 Apr 2012 14:43:11 +0000In this thesis, we study convergence results of certain non-linear geometric flows on vector bundles over complex manifolds. First we consider the case of a semi-stable vector bundle E over a compact Kahler manifold X of arbitrary dimension. We show that in this case Donaldson's functional is bounded from below. This allows us to construct an approximate Hermitian-Einstein structure on E along the Donaldson heat flow, generalizing a classic result of Kobayashi for projective manifolds to the Kahler case. Next we turn to general unstable bundles. We show that along a solution of the Yang-Mills flow, the trace of the curvature approaches in L2 an endomorphism with constant eigenvalues given by the slopes of the quotients from the Harder-Narasimhan filtration of E. This proves a sharp lower bound for the Hermitian-Yang-Mills functional and thus the Yang-Mills functional, generalizing to arbitrary dimension a formula of Atiyah and Bott first proven on Riemann surfaces. Furthermore, we show any reflexive extension to all of X of the limiting bundle is isomorphic to the double dual of the graded quotients from the Harder-Narasimhan-Seshadri filtration, verifying a conjecture of Bando and Siu. Our work on semi-stable bundles plays an important part of this result. For the final section of this thesis, we show that, in the case where X is an arbitrary Hermitian manifold equipped with a Gauduchon metric, given a stable Higgs bundle the Donaldson heat flow converges along a subsequence of times to a Hermitian-Einstein connection. This allows us to extend to the non-Kahler case the correspondence between stable Higgs bundles and (possibly) non-unitary Hermitian-Einstein connections first proven by Simpson on Kahler manifolds.Mathematicsajj2107MathematicsDissertationsAn Algebraic Circle Method
https://academiccommons.columbia.edu/catalog/ac:134194
Pugin, Thibauthttp://hdl.handle.net/10022/AC:P:10557Mon, 20 Jun 2011 13:12:23 +0000In this thesis we present an adaptation of the Hardy-Littlewood Circle Method to give estimates for the number of curves in a variety over a finite field. The key step in the classical Circle Method is to prove that some cancellation occurs in some exponential sums. Using a theorem of Katz, we reduce this to bounding the dimension of some singular loci. The method is fully carried out to estimate the number of rational curves in a Fermat hypersurface of low degree and some suggestions are given as to how to handle other cases. We draw geometrical consequences from the main estimates, for instance the irreducibility of the space of rational curves on a Fermat hypersurface in a given degree range, and a bound on the dimension of the singular locus of the moduli space.Mathematicstfp2102MathematicsDissertationsKnot Floer Homology and Categorification
https://academiccommons.columbia.edu/catalog/ac:132323
Gilmore, Allison Leighhttp://hdl.handle.net/10022/AC:P:10411Wed, 18 May 2011 10:07:07 +0000With the goal of better understanding the connections between knot homology theories arising from categorification and from Heegaard Floer homology, we present a self-contained construction of knot Floer homology in the language of HOMFLY-PT homology. Using the cube of resolutions for knot Floer homology defined by Ozsváth and Szabó, we first give a purely algebraic proof of invariance that does not depend on Heegaard diagrams, holomorphic disks, or grid diagrams. Then, taking Khovanov's HOMFLY-PT homology as our model, we define a category of twisted Soergel bimodules and construct a braid group action on the homotopy category of complexes of twisted Soergel bimodules. We prove that the category of twisted Soergel bimodules categorifies the Hecke algebra with an extra indeterminate and its inverse adjoined. The braid group action, which is defined via twisted Rouquier complexes, is simultaneously a natural extension of the knot Floer cube of resolutions and a mild modification of the action by Rouquier complexes used by Khovanov in defining HOMFLY-PT homology. Finally, we introduce an operation Qu to play the role that Hochschild homology plays in HOMFLY-PT homology. We conjecture that applying Qu to the twisted Rouquier complex associated to a braid produces the knot Floer cube of resolutions chain complex associated to its braid closure. We prove a partial result in this direction.MathematicsMathematicsDissertationsModuli Spaces of Dynamical Systems on Pn
https://academiccommons.columbia.edu/catalog/ac:132314
Levy, Alonhttp://hdl.handle.net/10022/AC:P:10408Tue, 17 May 2011 19:12:15 +0000This thesis studies the space of morphisms on Pn defined by polynomials of degree d and its quotient by the conjugation action of PGL(n+1), which should be thought of as coordinate change. First, we construct the quotient using geometric invariant theory, proving that it is a geometric quotient and that the stabilizer group in PGL(n+1) of each morphism is finite and bounded in terms of n and d. We then show that when n = 1, the quotient space is rational over a field of any characteristic. We then study semistable reduction in this space. For every complete curve C in the semistable completion of the quotient space, we can find curves upstairs mapping down to it; this leads to an abstract complete curve D with a projective vector bundle parametrizing maps on the curve. The bundle is trivial iff there exists a complete curve D in the semistable space upstairs mapping down to C; we show that for every n and d we can find a C for which no such D exists. Finally, in the case where D does exist, we show that, whenever it lies in the stable space, the map from D to C is ramified only over points with unusually large stabilizer, which for a fixed rational C will bound the degree of the map from D to C.Mathematicsal2495MathematicsDissertationsSoergel Diagrammatics for Dihedral Groups
https://academiccommons.columbia.edu/catalog/ac:132257
Elias, Benhttp://hdl.handle.net/10022/AC:P:10389Tue, 17 May 2011 14:54:36 +0000We give a diagrammatic presentation for the category of Soergel bimodules for the dihedral group W, finite or infinite. The (two-colored) Temperley-Lieb category is embedded inside this category as the degree 0 morphisms between color-alternating objects. The indecomposable Soergel bimodules are the images of Jones-Wenzl projectors. When W is finite, the Temperley-Lieb category must be taken at an appropriate root of unity, and the negligible Jones-Wenzl projector yields the Soergel bimodule for the longest element of W.Mathematicsbse2103MathematicsDissertationsBordered Sutured Floer Homology
https://academiccommons.columbia.edu/catalog/ac:132302
Zarev, Rumenhttp://hdl.handle.net/10022/AC:P:10404Tue, 17 May 2011 13:51:10 +0000We investigate the relationship between two versions of Heegaard Floer homology for 3-manifolds with boundary--the sutured Floer homology of Juhasz, and the bordered Heegaard Floer homology of Lipshitz, Ozsvath, and Thurston. We define a new invariant called Bordered sutured Floer homology which encompasses these two invariants as special cases. Using the properties of this new invariant we prove a correspondence between the original bordered and sutured homologies. In one direction we prove that for a 3-manifold Y with connected boundary F = dY , and sutures Gamma in dY , we can compute the sutured Floer homology SFH(Y ) from the bordered invariant CFA(Y )A(F ) . The chain complex SFC(Y, Gamma) defining SFH is quasi-isomorphic to the derived tensor product CFA(Y )xCFD(Gamma) where A(F )CFD(Gamma) is a module associated to Gamma. In the other direction we give a description of the bordered invariants in terms of sutured Floer homology. If F is a closed connected surface, then the bordered algebra A(F) is a direct sum of certain sutured Floer complexes. These correspond to the 3-manifold (F \ D2;)Ã—[0,1], where the sutures vary in a finite collection. Similarly, if Y is a connected 3-manifold with boundary dY = F , the module CFA(Y)A(F) is a direct sum of sutured Floer complexes for Y where the sutures on dY vary over a finite collection. The multiplication structure on A(F) and the action of A(F) on CFA(Y) correspond to a natural gluing map on sutured Floer homology. (Further work of the author shows that this map coincides with the one defined by Honda, Kazez, and Matic, using contact topology and open book decompositions).Mathematicsriz2102MathematicsDissertationsMonopole Floer homology, link surgery, and odd Khovanov homology
https://academiccommons.columbia.edu/catalog/ac:132266
Bloom, Jonathan Michaelhttp://hdl.handle.net/10022/AC:P:10392Tue, 17 May 2011 11:17:35 +0000We construct a link surgery spectral sequence for all versions of monopole Floer homology with mod 2 coefficients, generalizing the exact triangle. The spectral sequence begins with the monopole Floer homology of a hypercube of surgeries on a 3-manifold Y, and converges to the monopole Floer homology of Y itself. This allows one to realize the latter group as the homology of a complex over a combinatorial set of generators. Our construction relates the topology of link surgeries to the combinatorics of graph associahedra, leading to new inductive realizations of the latter. As an application, given a link L in the 3-sphere, we prove that the monopole Floer homology of the branched double-cover arises via a filtered perturbation of the differential on the reduced Khovanov complex of a diagram of L. The associated spectral sequence carries a filtration grading, as well as a mod 2 grading which interpolates between the delta grading on Khovanov homology and the mod 2 grading on Floer homology. Furthermore, the bigraded isomorphism class of the higher pages depends only on the Conway-mutation equivalence class of L. We constrain the existence of an integer bigrading by considering versions of the spectral sequence with non-trivial U action, and determine all monopole Floer groups of branched double-covers of links with thin Khovanov homology. Motivated by this perspective, we show that odd Khovanov homology with integer coefficients is mutation invariant. The proof uses only elementary algebraic topology and leads to a new formula for link signature that is well-adapted to Khovanov homology.Mathematicsjmb2177MathematicsDissertationsOn Using Graphical Calculi: Centers, Zeroth Hochschild Homology and Possible Compositions of Induction and Restriction Functors in Various Diagrammatical Algebras
https://academiccommons.columbia.edu/catalog/ac:132037
Brichard, Joellehttp://hdl.handle.net/10022/AC:P:10337Wed, 11 May 2011 17:57:27 +0000This thesis is divided into three chapters, each using certain graphical calculus in a slightly different way. In the first chapter, we compute the dimension of the center of the 0-Hecke algebra Hn and of the Nilcoxeter algebra NCn using a calculus of diagrams on the Moebius band. In the case of the Nilcoxeter algebra, this calculus is shown to produce a basis for Z(NCn) and the table of multiplication in this basis is shown to be trivial. We conjecture that a basis for Z(Hn) can also be obtained in a specic way from this topological calculus. In the second chapter, we also use a calculus of diagrams on the annulus and the Moebius band to determine the zeroth Hochschild Homology of Kuperberg's webs for rank two Lie algebras. We use results from Sikora and Westbury to prove the linear independence of these webs on these surfaces. In the third chapter, we use other diagrams to attempt to find explicitely the possible compositions of the induction and restriction functors in the cyclotomic quotients of the NilHecke algebra. We use a computer program to obtain partial results.Mathematicsjb2543MathematicsDissertationsBounds for the Spectral Mean Value of Central Values of L-functions
https://academiccommons.columbia.edu/catalog/ac:132025
Lu, Qinghttp://hdl.handle.net/10022/AC:P:10333Wed, 11 May 2011 17:10:10 +0000We prove two results about the boundedness of spectral mean value of Rankin-Selberg L-functions at s = 1/2, which is an analogue for Eisenstein series of X. Li's result for Hecke-Maass forms.Mathematicsql2132MathematicsDissertationsOptimal Trading Strategies Under Arbitrage
https://academiccommons.columbia.edu/catalog/ac:131477
Ruf, Johannes Karl Dominikhttp://hdl.handle.net/10022/AC:P:10250Fri, 29 Apr 2011 18:21:03 +0000This thesis analyzes models of financial markets that incorporate the possibility of arbitrage opportunities. The first part demonstrates how explicit formulas for optimal trading strategies in terms of minimal required initial capital can be derived in order to replicate a given terminal wealth in a continuous-time Markovian context. Towards this end, only the existence of a square-integrable market price of risk (rather than the existence of an equivalent local martingale measure) is assumed. A new measure under which the dynamics of the stock price processes simplify is constructed. It is shown that delta hedging does not depend on the "no free lunch with vanishing risk" assumption. However, in the presence of arbitrage opportunities, finding an optimal strategy is directly linked to the non-uniqueness of the partial differential equation corresponding to the Black-Scholes equation. In order to apply these analytic tools, sufficient conditions are derived for the necessary differentiability of expectations indexed over the initial market configuration. The phenomenon of "bubbles," which has been a popular topic in the recent academic literature, appears as a special case of the setting in the first part of this thesis. Several examples at the end of the first part illustrate the techniques contained therein. In the second part, a more general point of view is taken. The stock price processes, which again allow for the possibility of arbitrage, are no longer assumed to be Markovian, but rather only It^o processes. We then prove the Second Fundamental Theorem of Asset Pricing for these markets: A market is complete, meaning that any bounded contingent claim is replicable, if and only if the stochastic discount factor is unique. Conditions under which a contingent claim can be perfectly replicated in an incomplete market are established. Then, precise conditions under which relative arbitrage and strong relative arbitrage with respect to a given trading strategy exist are explicated. In addition, it is shown that if the market is quasi-complete, meaning that any bounded contingent claim measurable with respect to the stock price filtration is replicable, relative arbitrage implies strong relative arbitrage. It is further demonstrated that markets are quasi-complete, subject to the condition that the drift and diffusion coefficients are measurable with respect to the stock price filtration.Mathematics, Financejkr2115Statistics, MathematicsDissertationsF-virtual Abelian Varieties and Rallis Inner Product Formula
https://academiccommons.columbia.edu/catalog/ac:131441
Wu, Chenyanhttp://hdl.handle.net/10022/AC:P:10237Fri, 29 Apr 2011 16:22:06 +0000This thesis consists of two topics. First we study F-virtual Abelian varieties of GL2-type where F is a number field. We show the relation between these Abelian varieties and those defined over F. We compare their l-adic representations and study the modularity of F-virtual Abelian varieties of GL2-type. Then we construct their moduli spaces and in the case where the moduli space is a surface we give criteria when it is of general type. We also give two examples of surfaces that are rational and one that is neither rational nor of general type. Second we prove a crucial case of Siegel-Weil formula for orthogonal groups and metaplectic groups. With this we can compute the pairing of theta functions and show in this case that it is related to the central value of Langlands L-function. This new case of Rallis inner product formula enables us to relate nonvanishing of L-value to the nonvanishing of theta lifting.Mathematicscw2314MathematicsDissertationsTwo Approaches to Non-Zero-Sum Stochastic Differential Games of Control and Stopping
https://academiccommons.columbia.edu/catalog/ac:131462
Li, Qinghuahttp://hdl.handle.net/10022/AC:P:10245Fri, 29 Apr 2011 13:48:56 +0000This dissertation takes two approaches - martingale and backward stochastic differential equation (BSDE) - to solve non-zero-sum stochastic differential games in which all players can control and stop the reward streams of the games. Existence of equilibrium stopping rules is proved under some assumptions. The martingale part provides an equivalent martingale characterization of Nash equilibrium strategies of the games. When using equilibrium stopping rules, Isaacs' condition is necessary and sufficient for the existence of an equilibrium control set. The BSDE part shows that solutions to BSDEs provide value processes of the games. A multidimensional BSDE with reflecting barrier is studied in two cases for its solution: existence and uniqueness with Lipschitz growth, and existence in a Markovian system with linear growth rate.Mathematicsql2133Statistics, MathematicsDissertationsLattice Subdivisions and Tropical Oriented Matroids, Featuring Products of Simplices
https://academiccommons.columbia.edu/catalog/ac:131450
Piechnik, Lindsay C.http://hdl.handle.net/10022/AC:P:10240Fri, 29 Apr 2011 12:47:09 +0000Subdivisions of products of simplices, and their applications, appear across mathematics. In this thesis, they are the tie between two branches of my research: polytopal lattice subdivisions and tropical oriented matroid theory. The first chapter describes desirable combinatorial properties of subdivisions of lattice polytopes, and how they can be used to address algebraic questions. Chapter two discusses tropical hyperplane arrangements and the tropical oriented matroid theory they inspire, paying particular attention to the previously uninvestigated distinction between the generic and non-generic cases. The focus of chapter three is products of simplices, and their connections and applications to ideas covered in the first two chapters.Mathematicslp2149Mathematics, Mathematics (Barnard College)DissertationsApproximate converse theorem
https://academiccommons.columbia.edu/catalog/ac:130096
Lee, Minhttp://hdl.handle.net/10022/AC:P:9949Fri, 11 Mar 2011 20:53:18 +0000The theme of this thesis is an "approximate converse theorem" for globally unramified cuspidal representations of PGL(n, A), n ≥ 1. For a given set of Langlands parameters for some places of Q, we can compute ε > 0 such that there exists a genuine globally unramified cuspidal representation, whose Langlands parameters are within ε of the given ones for finitely many places.Mathematicsml2660MathematicsDissertations