Academic Commons Search Results
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Academic Commons Search Resultsen-usGeneralized Volatility-Stabilized Processes
http://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
http://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
http://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
http://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.Mathematicszam2104MathematicsDissertationsPurity of the stratification by Newton polygons and Frobenius-periodic vector bundles
http://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
http://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
http://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
http://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
http://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
http://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)
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://academiccommons.columbia.edu/catalog/ac:132323
Gilmore, Allison Leighhttp://hdl.handle.net/10022/AC:P:10411Wed, 18 May 2011 14:07:07 +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
http://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.Mathematicsal2495MathematicsDissertationsBordered Sutured Floer Homology
http://academiccommons.columbia.edu/catalog/ac:132302
Zarev, Rumenhttp://hdl.handle.net/10022/AC:P:10404Tue, 17 May 2011 17:51:10 +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;)×[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
http://academiccommons.columbia.edu/catalog/ac:132266
Bloom, Jonathan Michaelhttp://hdl.handle.net/10022/AC:P:10392Tue, 17 May 2011 15:17:35 +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.Mathematicsjmb2177MathematicsDissertationsSoergel Diagrammatics for Dihedral Groups
http://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.Mathematicsbse2103MathematicsDissertationsOn Using Graphical Calculi: Centers, Zeroth Hochschild Homology and Possible Compositions of Induction and Restriction Functors in Various Diagrammatical Algebras
http://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
http://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
http://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
http://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
http://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
http://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
http://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