This thesis is devoted to quantitative questions about the discrete spectrum of Schrödinger-type operators. In Paper I we show that the Lieb-Thirring inequalities on moments of negative eigen¬values remain true, with possibly different constants, when the critical Hardy weight is subtracted from the Laplace operator. In Paper II we prove that the one-dimensional analog of this inequality holds even for the critical value of the moment parameter. In Paper III we establish Hardy-Lieb-Thirring inequalities for fractional powers of the Laplace operator and, in particular, relativistic Schrödinger operators. We do so by first establishing Hardy-Sobolev inequalities for such operators. We also allow for the inclu¬sion of magnetic fields. As an application, in Paper IV we give a proof of stability of relativistic matter with magnetic fields up to the critical value of the nuclear charge. In Paper V we derive inequalities for moments of the real part and the modulus of the eigen¬values of Schrödinger operators with complex-valued potentials. / QC 20100708

This thesis contains three scientific papers devoted to the study of different spectral theoretical aspects of differential operators in Hilbert spaces.The first paper concerns the magnetic Schrödinger operator (i∇ + A)2 in L2(ℝn). It is proved that given certain conditions on the decay of A, the set [0,∞) is an essential support of the absolutely continuous part of the spectral measure corresponding to the operator.The second paper considers a regular d-dimensional metric tree Γ and defines Schrödinger operators - Δ - V on it.  Here, V is a symmetric, non-negative potential on Γ. It is assumed that V decays like lxl-Γ at infinity, where 1 < γ ≤ d ≤2, γ ≠ 2. A weak coupling constant α is introduced in front of V, and the asymptotics of the bottom of the spectrum as α → 0+ is described.The third, and last, paper revolves around fourth-order differential operators in the space L2(ℝn), where n = 1 or n = 3.  In particular, the operator (-Δ)2 - C|x|-4 - V(x) is studied, where C is the sharp constant in the Hardy-Rellich inequality. A Lieb-Thirring inequality for this operator is proved, and as a consequence a Sobolev-type inequality is obtained. / QC 20100712

QC 20100812

In this thesis we study graded Cohen-Macaulay modules and their possible Hilbert functions and graded Betti numbers. In most cases the Cohen-Macaulay modules we study are level modules. In order to use dualization to study Hilbert functions of artinian level algebras we extend the notion of level sequences and cancellable sequences, introduced by Geramita and Lorenzini, to include Hilbert functions of certain level modules. As in the case of level algebras, a level sequence is cancellable, but now by dualization its reverse is also cancellable which gives a new condition on level sequences. We also give a characterization of the cancellable sequences. We prove that a sequence of positive integers (h0, h1, . . . ,hc) is the Hilbert function of an artinian level module of codimension two if and only if hi−1 − 2hi + hi+1<= 0 for all 0 <= i <= c, where we assume that h−1 = hc+1 = 0. This generalizes a result already known for artinian level algebras. Zanello gives a lower bound for Hilbert functions of generic level quotients of artinian level algebras. We give a new and more straightforward proof of Zanello’s result. Conjectures on the possible graded Betti numbers of Cohen-Macaulay modules up to multiplication by a positive rational number are given. The idea is that the Betti diagrams should be non-negative linear combinations of pure diagrams. The conjectures are verified for modules of codimension two, for Gorenstein algebras of codimension three and for complete intersections. The motivation for the conjectures comes from the Multiplicity conjecture of Herzog, Huneke and Srinivasan. The h-vectors and graded Betti numbers of level modules up to multiplication by a rational number are studied. Assuming the conjecture, mentioned above, on the set of possible graded Betti numbers of Cohen-Macaulay modules we get a description of the possible h-vectors of level modules up to multiplication by a rational number. We determine, again up to multiplication by a rational number, the cancellable h-vectors and the h-vectors of level modules with the weak Lefschetz property. Furthermore, we prove that level modules of codimension three satisfy the upper bound of the Multiplicity conjecture and that the lower bound holds if the module, in addition, has the weak Lefschetz property. / QC 20100819

This thesis treats various aspects of stable reduction of curves, and consists of two separate papers. In Paper I of this thesis, we study stable reduction of curves in the case where a tamely ramified base extension is sufficient. If X is a smooth curve defined over the fraction field of a strictly henselian discrete valuation ring, there is a criterion, due to T. Saito, that describes precisely, in terms of the geometry of the minimal model with strict normal crossings of X, when a tamely ramified extension suffices in order for X to obtain stable reduction. For such curves we construct an explicit extension that realizes the stable reduction, and we furthermore show that this extension is minimal. We also obtain a new proof of Saito's criterion, avoiding the use of adic cohomology and vanishing cycles. In Paper II, we study group actions on regular models of curves. If X is a smooth curve defined over the fraction field K of a complete discrete valuation ring R, every tamely ramified field extension K0=K with Galois group G induces a G-action on the extension XK0 of X to K0. We study the extension of this G-action to certain regular models of XK0 . In particular, we are interested in the induced action on the cohomology groups of the structure sheaf of the special fiber of such a regular model. We obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups. Inspired by this global study, we also consider similar group actions on the cohomology of the structure sheaf of the exceptional locus of a tame cyclic quotient singularity, and obtain an explicit polynomial formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups. We apply these results to study a natural filtration of the special fiber of the Néronmodel of the Jacobian of X by closed, unipotent subgroup schemes. We show that the jumps in this filtration only depend on the fiber type of the special fiber of the minimal regular model with strict normal crossings for X over Spec(R), and in particular are independent of the residue characteristic. Furthermore, we obtain information about where these jumps can occur. We also compute the actual jumps for each of the finitely many possible fiber types for curves of genus 1 and 2. / QC 20100712

This thesis treats a family of categories, the chain categories of an A-module M, and functors indexed by them. Among the chain categories are two classical constructions; the category of finitely generated projective Amodules, and the category of finitely generated free A-modules, here denoted by P0(0) and Sing(0) respectively. The focus of this thesis is on how to construct homotopy colimits of functors indexed by chain categories, and taking values in non-negative chain complexes of A-modules. One consequence of Lazard’s theorem is that if M is flat, then all functors over Sing(M) are flat; that is, the homotopy colimits of these functors are weakly equivalent to the ordinairy colimits. A motivating question has been to understand when functors over Sing(M) are flat for non-flat M. In particular, when the forgetful functor UM is flat. One of the results obtain is that if A is Noetherian, then UM is flat over many chain categories, and this property is independent of M. In contrast, if A is commutative, then the pointwise tensor product UM UM is defined, and this is not a flat functor in general, even if UM is flat. The key notion used to study these questions is that of a cofinal functor. Among the main results are the cofinality of various inclusion functors among the chain categories themselves, and the existence, construction and classification of cofinal simplicial objects in P0(M) and Sing(M). Also, a method to construct flat resolutions of functors indexed by P0 and taking values in A-modules is developed (but applicability of this construction depends on severe restrictions on M). These methods are used to compute the homotopy colimits of several functors defined over various chain categories. / QC 20100831

This thesis is a study of approximation and calibration of stochastic processes with applications in finance. It consists of an introduction and four research papers. The introduction is as an overview of the role of mathematics incertain areas of finance. It contains a brief introduction to the mathematicaltheory of option pricing, as well as a description of a mathematicalmodel of a financial exchange. The introduction also includessummaries of the four research papers. In Paper I, Markov decision theory is applied to design algorithmic trading strategies in an order driven market. A high dimensional Markov chain is used to model the state and evolution of the limit order book. Trading strategies are formulated as optimal decision problems. Conditions that guarantee existence of optimal strategies are provided, as well as a value-iterative algorithm that enables numerical construction of optimal strategies. The results are illustrated with numerical experiments on high frequency data from a foreign exchange market. Paper II focuses on asset pricing with Lévy processes. The expected value E[g(XT )] is estimated using a Monte Carlo method, when Xt is a d-dimensional Lévy process having infinite jump activity and a smooth density. Approximating jumps smaller then a parameter ε > 0 by diffusion results in a weak approximation, Xt, of Xt. The main result of the paper is an estimate of the resulting model error E[g(XT )] − E[g(T )], with a computable leading order term. Option prices in exponential Lévy models solve certain partia lintegro-differential equations (PIDEs). A finite difference scheme suitable for solving such PIDEs is studied in Paper III. The main resultsare estimates of the time and space discretization errors, with leading order terms in computable form. If the underlying Lévy process has infinite jump activity, the jumps smaller than some ε > 0 are replacedby diffusion. The size of this diffusion approximation is estimated, as well as its effect on the space and time discretization errors. Combined, the results of the paper are enough to determine how to jointly choose the grid size and the parameter ε. In Paper IV it is demonstrated how optimal control can be used to calibrate a jump-diffusion process to quoted option prices. The calibration problem is formulated as an optimal control problem with the model parameter as a control variable. The corresponding regularized Hamiltonian system is solved with a symplectic Euler method. / QC 20101008

Matematik är ett av de viktigaste ämnena i dagens skola, men det är också ett av de ämnena som eleverna ofta tycker är tråkigast. På senare år har undersökningar visat på brister i elevernas matematikkunskaper och skolans matematik har blivit beskylld för att inte vara tillräckligt tydligt kopplad till verkligheten. Genom att ta del av tidigare forskning kom jag fram till sex riktlinjer för att utforma ett bättre läromedel i matematik. Jag bestämde mig således för att designa ett läromedel: med öppna lösningar där eleverna själva fick formulera problemen, ett läromedel som appellerade till flera sinnen, som var konkret utan abstrakta termer, som gjorde det möjligt för eleverna att använda sin kreativitet, som inbjöd till diskussioner och kommunikation och som fångade elevernas engagemang. Jag bestämde mig för att utforma mitt läromedel kring Felicia, hennes detektivbyrå och fallet med stölderna på hotell Citronella.

Model theory and combinatorial pregeometries are closely related throughthe so called algebraic closure operator on strongly minimal sets. Thestudy of projective and ane pregeometries are especially interestingsince they have a close relation to vectorspaces. In this thesis we willsee how the relationship occur and how model theory can concludea very strong classi cation theorem which divides pregeometries withcertain properties into projective, ane and degenerate (trivial) cases. / Modellteori är ett ämne som är starkt relaterat till studien av kombinatoriska pregeometrier, detta genom den algebraiska tillslutningsoperatorn som agerar på starkt minimala mängder. Studien av projektivaoch affina pregeometrier är speciellt intressant genom dessas relation till vektorrum. I den här uppsatsen kommer vi att se hur denna relation uppstår och hur modellteori kan förklara en väldigt stark klassifikationssats, som delar upp pregeometrier med speciella egenskaper i projektiva, affina och degenererade (triviala) fall.

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