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  • 1.
    Aas, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Linusson, Svante
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Continuous multi-line queues and TASEP2018In: ANNALES DE L INSTITUT HENRI POINCARE D, ISSN 2308-5827, Vol. 5, no 1, p. 127-152Article in journal (Refereed)
    Abstract [en]

    In this paper, we study a distribution Xi of labeled particles on a continuous ring. It arises in three different ways, all related to the multi-type TASEP on a ring. We prove formulas for the probability density function for some permutations and give conjectures for a larger class. We give a complete conjecture for the probability of two particles i, j being next to each other on the cycle, for which we prove some cases. We also find that two natural events associated to the process have exactly the same probability expressed as a Vandermonde determinant. It is unclear whether this is just a coincidence or a consequence of a deeper connection.

  • 2. Aghajani, A.
    et al.
    Razani, Abdolrahman
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Department of Mathematics, Faculty of Science, Imam Khomeini International University, Iran .
    Some completeness theorems in the Menger probabilistic metric space2008In: Applied Sciences: APPS, ISSN 1454-5101, E-ISSN 1454-5101, Vol. 10, p. 1-8Article in journal (Refereed)
    Abstract [en]

    In this article, some new completeness theorems in probabilistic normed space are proved. Moreover, the existence of a constrictive Monger probabilistic normed space is shown.

  • 3. Ammann, Bernd
    et al.
    Dahl, Mattias
    Humbert, Emmanuel
    Smooth yamabe invariant and surgery2013In: Journal of differential geometry, ISSN 0022-040X, E-ISSN 1945-743X, Vol. 94, no 1, p. 1-58Article in journal (Refereed)
    Abstract [en]

    We prove a surgery formula for the smooth Yamabe invariant sigma(M) of a compact manifold M. Assume that N is obtained from M by surgery of codimension at least 3. We prove the existence of a positive constant Lambda(n), depending only on the dimension n of M, such that sigma(N) >= min{sigma(M), Lambda(n)}.

  • 4.
    Andersson, Lars
    et al.
    KTH, Superseded Departments, Mathematics.
    Dahl, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Howard, Ralph
    Boundary and lens rigidity of Lorentzian surfaces1996In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 348, p. 2307-2329Article in journal (Refereed)
    Abstract [en]

    Let g be a Lorentzian metric on the plane ℝ2 that agrees with the standard metric g0 = -dx2 + dy2 outside a compact set and so that there are no conjugate points along any time-like geodesic of (ℝ2, g). Then (ℝ2, g) and (ℝ2, g0) are isometric. Further, if (M*, g*) and (M*, p*) are two dimensional compact time oriented Lorentzian manifolds with space-like boundaries and so that all time-like geodesies of (M, g) maximize the distances between their points and (M, g) and (M*, g*) are "boundary isometric", then there is a conformal diffeomorphism between (M, g) and (M*, g*) and they have the same areas. Similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds.

  • 5.
    Arnlind, Joakim
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Hoppe, Jens
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Huisken, Gerhard
    Max Planck Institute for Gravitational Physics.
    Multi linear formulation of differential geometry and matrix regularizations2012In: Journal of differential geometry, ISSN 0022-040X, E-ISSN 1945-743X, Vol. 91, no 1, p. 1-39Article in journal (Refereed)
    Abstract [en]

    We prove that many aspects of the differential geometry of em-bedded Riemannian manifolds can be formulated in terms of multilinear algebraic structures on the space of smooth functions. Inparticular, we find algebraic expressions for Weingarten’s formula,the Ricci curvature and the Codazzi-Mainardi equations.For matrix analogues of embedded surfaces we define discretecurvatures and Euler characteristics, and a non-commutative Gauss–Bonnet theorem is shown to follow. We derive simple expressionsfor the discrete Gauss curvature in terms of matrices representingthe embedding coordinates, and explicit examples are provided.Furthermore, we illustrate the fact that techniques from differen-tial geometry can carry over to matrix analogues by proving thata bound on the discrete Gauss curvature implies a bound on theeigenvalues of the discrete Laplace operator.

  • 6.
    Beltran Jimenez, Jose
    et al.
    Univ Autonoma Madrid, Inst Fis Teor, UAM CSIC, E-28049 Madrid, Spain.;Univ Salamanca, Dept Fis Fundamental, E-37008 Salamanca, Spain..
    Heisenberg, Lavinia
    Swiss Fed Inst Technol, Inst Theoret Studies, Clausiusstr 47, CH-8092 Zurich, Switzerland..
    Koivisto, Tomi S.
    KTH, Centres, Nordic Institute for Theoretical Physics NORDITA.
    Coincident general relativity2018In: Physical Review D: covering particles, fields, gravitation, and cosmology, ISSN 2470-0010, E-ISSN 2470-0029, Vol. 98, no 4, article id 044048Article in journal (Refereed)
    Abstract [en]

    The metric-affine variational principle is applied to generate teleparallel and symmetric teleparallel theories of gravity. From the latter we discover an exceptional class which is consistent with a vanishing affine connection. Based on this remarkable property, this work proposes a simpler geometrical formulation of general relativity that is oblivious to the affine spacetime structure, thus fundamentally depriving gravity of any inertial character. The resulting theory is described by the Hilbert action purged from the boundary term and is more robustly underpinned by the spin-2 field theory, where an extra symmetry is now manifest, possibly related to the double-copy structure of the gravity amplitudes. This construction also provides a novel starting point for modified gravity theories, and the paper presents new and simple generalizations where analytical self-accelerating cosmological solutions arise naturally in the early-and late-time Universe.

  • 7.
    Beltran Jimenez, Jose
    et al.
    Univ Autonoma Madrid, CSIC, Inst Fis Teor, E-28049 Madrid, Spain.;Univ Salamanca, Dept Fis Fundamental, Plaza Merced, E-37008 Salamanca, Spain..
    Heisenberg, Lavinia
    Swiss Fed Inst Technol, Inst Theoret Studies, Clausiusstr 47, CH-8092 Zurich, Switzerland..
    Koivisto, Tomi S.
    KTH, Centres, Nordic Institute for Theoretical Physics NORDITA.
    Teleparallel Palatini theories2018In: Journal of Cosmology and Astroparticle Physics, ISSN 1475-7516, E-ISSN 1475-7516, no 8, article id 039Article in journal (Refereed)
    Abstract [en]

    The Palatini formalism, which assumes the metric and the affine connection as independent variables, is developed for gravitational theories in flat geometries. We focus on two particularly interesting scenarios. First, we fix the connection to be metric compatible, as done in the usual teleparallel theories, but we follow a completely covariant approach by imposing the constraints with suitable Lagrange multipliers. For a general quadratic theory we show how torsion naturally propagates and we reproduce the Teleparallel Equivalent of General Relativity as a particular quadratic action that features an additional Lorentz symmetry. We then study the much less explored theories formulated in a geometry with neither curvature nor torsion, so that all the geometrical information is encoded in the non-metricity. We discuss how this geometrical framework leads to a purely inertial connection that can thus be completely removed by a coordinate gauge choice, the coincident gauge. From the quadratic theory we recover a simpler formulation of General Relativity in the form of the Einstein action, which enjoys an enhanced symmetry that reduces to a second linearised diffeomorphism at linear order. More general theories in both geometries can be formulated consistently by taking into account the inertial connection and the associated additional degrees of freedom. As immediate applications, the new cosmological equations and their Newtonian limit are considered, where the role of the lapse in the consistency of the equations is clarified, and the Schwarzschild black hole entropy is computed by evaluating the corresponding Euclidean action. We discuss how the boundary terms in the usual formulation of General Relativity are related to different choices of coordinates in its coincident version and show that in isotropic coordinates the Euclidean action is finite without the need to introduce boundary or normalisation terms. Finally, we discuss the double-copy structure of the gravity amplitudes and the bootstrapping of gravity within the framework of coincident General Relativity.

  • 8.
    Bergström, Jonas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Brown, Francis
    CNRS.
    Inversion of series and the cohomology of the moduli spaces $\scr M\sb 0,n\sp δ$2010In: Motives, quantum field theory, and pseudodifferential operators, American Mathematical Society (AMS), 2010, Vol. 12, p. 119-126Chapter in book (Other academic)
    Abstract [en]

    For $n\geq 3$, let $\mathcal{M}_{0,n}$ denote the moduli space of genus 0 curves with $n$ marked points, and $\overline{\mathcal{M}}_{0,n}$ its smooth compactification. A theorem due to Ginzburg, Kapranov and Getzler states that the inverse of the exponential generating series for the Poincar\'e polynomial of $H^{\bullet}(\mathcal{M}_{0,n})$ is given by the corresponding series for $H^{\bullet}(\overline{\mathcal{M}}_{0,n})$. In this paper, we prove that the inverse of the ordinary generating series for the Poincar\'e polynomial of $H^{\bullet}(\mathcal{M}_{0,n})$ is given by the corresponding series for $H^{\bullet}(\mathcal{M}^{\delta}_{0,n})$, where $\mathcal{M}_{0,n}\subset \mathcal{M}^{\delta}_{0,n} \subset \overline{\mathcal{M}}_{0,n}$ is a certain smooth affine scheme.

  • 9.
    Bergström, Jonas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    van der Geer, Gerard
    UvA.
    The Euler characteristic of local systems on the moduli of curves and abelian varieties of genus three2008In: Journal of Topology, ISSN 1753-8416, E-ISSN 1753-8424, Vol. 1, no 3, p. 651-662Article in journal (Refereed)
    Abstract [en]

    We show how to calculate the Euler characteristic of a local system V(lambda) associated to an irreducible representation V(lambda) of the symplectic group of genus 3 on the moduli space M(3) of curves of genus 3 and the moduli space A(3) of principally polarised abelian varieties of dimension 3.

  • 10.
    Carvalho, J. Frederico
    et al.
    KTH. KTH, CAS, RPL, Royal Inst Technol, Stocholm, Sweden..
    Vejdemo-Johansson, Mikael
    CUNY Coll Staten Isl, Math Dept, Staten Isl, NY 10314 USA.;CUNY, Grad Ctr, Comp Sci, New York, NY USA..
    Kragic, Danica
    KTH, School of Electrical Engineering and Computer Science (EECS), Robotics, perception and learning, RPL. KTH, CAS, RPL, Royal Inst Technol, Stocholm, Sweden..
    Pokorny, Florian T.
    KTH, School of Electrical Engineering and Computer Science (EECS), Robotics, perception and learning, RPL. KTH, CAS, RPL, Royal Inst Technol, Stocholm, Sweden..
    Path Clustering with Homology Area2018In: 2018 IEEE INTERNATIONAL CONFERENCE ON ROBOTICS AND AUTOMATION (ICRA), IEEE Computer Society, 2018, p. 7346-7353Conference paper (Refereed)
    Abstract [en]

    Path clustering has found many applications in recent years. Common approaches to this problem use aggregates of the distances between points to provide a measure of dissimilarity between paths which do not satisfy the triangle inequality. Furthermore, they do not take into account the topology of the space where the paths are embedded. To tackle this, we extend previous work in path clustering with relative homology, by employing minimum homology area as a measure of distance between homologous paths in a triangulated mesh. Further, we show that the resulting distance satisfies the triangle inequality, and how we can exploit the properties of homology to reduce the amount of pairwise distance calculations necessary to cluster a set of paths. We further compare the output of our algorithm with that of DTW on a toy dataset of paths, as well as on a dataset of real-world paths.

  • 11.
    Dahl, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the space of metrics with invertible Dirac operator2008In: Commentarii Mathematici Helvetici, ISSN 0010-2571, E-ISSN 1420-8946, Vol. 83, p. 451-469Article in journal (Refereed)
    Abstract [en]

    On a compact spin manifold we study the space of Riemannian metrics for which the Dirac operator is invertible. The first main result is a surgery theorem stating that such a metric can be extended over the trace of a surgery of codimension at least three. We then prove that, if non-empty, the space of metrics with invertible Dirac operators is disconnected in dimensions n equivalent to 0, 1, 3, 7 mod 8, n >= 5. As corollaries follow results on the existence of metrics with harmonic spinors by Hitchin and Bar. Finally we use computations of the eta invariant by Botvinnik and Gilkey to find metrics with harmonic spinors on simply connected manifolds with a cyclic group action. In particular this applies to spheres of all dimensions n >= 5.

  • 12.
    Damjanović, Danijela
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Abelian actions with globally hypoelliptic leafwise Laplacian and rigidity2016In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 129, no 1, p. 139-163Article in journal (Refereed)
    Abstract [en]

    In this paper, we prove several results concerning smooth Rk actions on a smooth compact manifold with the property that their leafwise Laplacian is globally hypoelliptic. Such actions are necessarily uniquely ergodic and minimal, and their cohomology is often finite dimensional, even trivial. Further, we consider a class of examples of R2 actions on two-step nilmanifolds that have globally hypoelliptic leafwise Laplacian, and we show transversal local rigidity under certain Diophantine conditions.

  • 13.
    Di Rocco, Sandra
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Eklund, David
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). KTH, Dept Math, S-10044 Stockholm, Sweden..
    Peterson, Chris
    Colorado State Univ, Dept Math, Ft Collins, CO 80523 USA..
    Numerical polar calculus and cohomology of line bundles2018In: Advances in Applied Mathematics, ISSN 0196-8858, E-ISSN 1090-2074, Vol. 100, p. 148-162Article in journal (Refereed)
    Abstract [en]

    Let L-1,..., L-s be line bundles on a smooth complex variety X subset of P-r and let D-1,..., D-s be divisors on X such that D-i represents L-i. We give a probabilistic algorithm for computing the degree of intersections of polar classes which are in turn used for computing the Euler characteristic of linear combinations of L-1,..., L-s. The input consists of generators for the homogeneous ideals I-X, I-Di subset of C[x(0),..., x(r).] defining X and D-i. 

  • 14. Foertsch, T.
    et al.
    Karlsson, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Hilbert metrics and Minkowski norms2005In: Journal of Geometry, ISSN 0047-2468, E-ISSN 1420-8997, Vol. 83, no 1-2, p. 22-31Article in journal (Refereed)
    Abstract [en]

    It is shown that the Hilbert geometry (D,h D ) associated to a bounded convex domain D ⊂ double struck E signn is isometric to a normed vector space (V,∥ ̇ ∥) if and only if D is an open n-simplex. One further result on the asymptotic geometry of Hilbert's metric is obtained with corollaries for the behavior of geodesics. Finally we prove that every geodesic ray in a Hilbert geometry converges to a point of the boundary.

  • 15.
    Greco, Ornella
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Bounds on Hilbert Functions and Betti Numbers of Veronese Modules2014Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    The thesis is a collection of four papers dealing with Hilbert functions and Betti numbers.In the first paper, we study the h-vectors of reduced zero-dimensional schemes in  . In particular we deal with the problem of findingthe possible h-vectors for the union of two sets of points of given h-vectors. To answer to this problem, we give two kinds of bounds on theh-vectors and we provide an algorithm that calculates many possibleh-vectors.In the second paper, we prove a generalization of Green’s Hyper-plane Restriction Theorem to the case of finitely generated modulesover the polynomial ring, providing an upper bound for the Hilbertfunction of the general linear restriction of a module M in a degree dby the corresponding Hilbert function of a lexicographic module.In the third paper, we study the minimal free resolution of theVeronese modules, , where  by giving a formula for the Betti numbers in terms of the reduced homology of the squarefree divisor complex. We prove that is Cohen-Macaulay if and only if k < d, and that its minimal resolutionis linear when k > d(n − 1) − n. We prove combinatorially that the resolution of  is pure. We provide a formula for the Hilbert seriesof the Veronese modules. As an application, we calculate the completeBetti diagrams of the Veronese rings  .In the fourth paper, we apply the same combinatorial techniques inthe study of the properties of pinched Veronese rings, in particular weshow when this ring is Cohen-Macaulay. We also study the canonicalmodule of the Veronese modules.

  • 16.
    Greco, Ornella
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Martino, Ivan
    Syzygies of Veronese modulesManuscript (preprint) (Other academic)
  • 17.
    Heinrich, Katharina
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The Cohen–Macaulay space of twisted cubicsManuscript (preprint) (Other academic)
    Abstract [en]

    In this work, we describe the Cohen-Macaulay space CM of twisted cubics parameterizing curves  together with a finite map  that is generically a closed immersion and such that  has Hilbert polynomial p(t)=3t+1 with respect to . We show that CM is irreducible, smooth and birational to one component of the Hilbert scheme of twisted cubics.

  • 18.
    Heinrich, Katharina
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The space of Cohen–Macaulay curvesManuscript (preprint) (Other academic)
    Abstract [en]

    One can consider the Hilbert scheme as a natural compactification of the space of smooth projective curves with fixed Hilbert polynomial. Here we consider a different modular compactification, namely the functor CM parameterizing curves together with a finite map to  that is generically a closed immersion. We prove that CM is an algebraic space by contructing a scheme W and a representable, surjective and smooth map W -> CM. Moreover, we show that CM satisfies the valuative criterion for properness.

  • 19.
    Heinrich, Katharina
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The space of Cohen-Macaulay curves2012Licentiate thesis, monograph (Other academic)
    Abstract [en]

    In this thesis we discuss a moduli space of projective curves with a map to a given projective space. The functor CM parametrizes curves, that is, Cohen-Macaulay schemes of pure dimension 1, together with a finite map to the projective space that is an isomorphism onto its image away from a finite set of closed points.

    We proof that CM is an algebraic space by contructing a scheme W and a representable, surjective and smooth map from W to CM. 

  • 20.
    Heinrich, Katharina
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The space of Cohen–Macaulay curves and related topics2014Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    The space of Cohen–Macaulay curves is a compactification of the space of curves that are embedded in a given projective space Pn. The idea is similar to that of the Hilbert scheme but instead of adding degenerated curves, one considers only curves without embedded or isolated points. However, the curves need not be embedded into the projective space. Instead, they come with a finite morphism to Pn that is generically a closed immersion. More precisely, the space CM of Cohen–Macaulay curves parameterizes flat families of pairs  where  is a curve without embedded or isolated points and  is a finite morphism that is an isomorphism onto its image away from finitely many closed points and such that  has Hilbert polynomial p(t) with respect to the map .

    In Paper A we show that the moduli functor CM is represented by a proper algebraic space. This is done by constructing a smooth, surjective cover  and by verifying the valuative criterion for properness.

    Paper B studies the moduli space CM in the particular case n = 3 and p(t) = 3t + 1, that is, the Cohen–Macaulay space of twisted cubics. We de- scribe the points of CM and show that they are in bijection with the points on the 12-dimensional component H0 of the Hilbert scheme of twisted cu- bics. Knowing the points of CM, we can then show that the moduli space is irreducible, smooth and has dimension 12.

    Paper C concerns the notion of biequidimensionality of topological spaces and Noetherian schemes. In EGA it is claimed that a topological space X is equidimensional, equicodimensional and catenary if and only if all maximal chains of irreducible closed subsets in X have the same length. We construct examples of topological spaces and Noetherian schemes showing that the sec- ond property is strictly stronger. This gives rise to two different notions of biequidimensionality, and we show how they relate to the dimension formula and the existence of a codimension function.

  • 21.
    Hultman, Axel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Jonsson, Jakob
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The topology of the space of matrices of Barvinok rank two2010In: Beiträge zur Algebra und Geometrie, ISSN 0138-4821, Vol. 51, no 2, p. 373-390Article in journal (Refereed)
    Abstract [en]

    The Barvinok rank of a d x n matrix is the minimum number of  points in Rd such that the tropical convex hull of the points contains all columns of the matrix. The concept originated in work by Barvinok and others on the travelling salesman problem. Our object of study is the space of real d x n matrices of Barvinok rank two. Let Bd,n denote this space modulo rescaling and translation. We show that Bd,n is a manifold, thereby settling a  conjecture due to Develin. In fact, Bd,n is homeomorphic to the quotient of the product of spheres Sd-2 x Sn-2 under the involution which sends each point to its antipode simultaneously in both  components.  In addition, using discrete Morse theory, we compute the integral homology of Bd,n. Assuming d \ge n, for odd d the homology turns out to be   isomorphic to that of Sd-2 x RPn-2. This  is true also for even d up to degree d-3, but the two cases differ from degree d-2 and up. The homology computation straightforwardly extends to more general  complexes of the form (Sd-2 x X)//Z2, where X is a finite cell  complex of dimension at most d-2 admitting a free  Z2-action.

  • 22.
    Langmann, Edwin
    KTH, School of Engineering Sciences (SCI), Physics, Mathematical Physics. KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory.
    Descent equations of Yang-Mills anomalies in noncommutative geometry1997In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, p. 259-279Article in journal (Refereed)
  • 23.
    Langmann, Edwin
    KTH, School of Engineering Sciences (SCI), Physics, Mathematical Physics. KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory.
    Non-commutative Integration Calculus1995In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 36, p. 3822-3835Article in journal (Refereed)
  • 24.
    Larsson, Eric
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Broken causal lens rigidity and sky shadow rigidity of Lorentzian manifoldsManuscript (preprint) (Other academic)
  • 25.
    Larsson, Eric
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Regularity of the boundary of the trapped region in asymptotically Euclidean Riemannian manifolds of arbitrarily large dimensionsManuscript (preprint) (Other academic)
  • 26.
    Larsson, Eric
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Smoothness of Compact Horizons2015In: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 16, no 9, p. 2163-2214Article in journal (Refereed)
    Abstract [en]

    We prove that compact Cauchy horizons in a smooth spacetime satisfying the null energy condition are smooth. As an application, we consider the problem of determining when a cobordism admits Lorentzian metrics with certain properties. In particular, we prove a result originally due to Tipler without the smoothness hypothesis necessary in the original proof.

  • 27.
    Larsson, Eric
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Dahl, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Outermost apparent horizons diffeomorphic to unit normal bundlesManuscript (preprint) (Other academic)
  • 28.
    Lee, Jae Min
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Geometric approach on the global conservative solutions of the Camassa–Holm equation2019In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 142, p. 137-150Article in journal (Refereed)
    Abstract [en]

    In this paper we construct global weak conservative solutions of the Camassa–Holm (CH)equation on the periodic domain. We first express the equation in Lagrangian flow variable η and then transform it using the change of variables ρ=η x . The new variable removes the singularity of the CH equation, and we obtain both global weak conservative solutions and global spatial smoothness of the Lagrangian trajectories of the CH equation. This work is motivated by J. Lenells who proved similar results for the Hunter–Saxton equation using the geometric interpretation.

  • 29.
    Lundman, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Computing Seshardi constants on smooth toric surfacesManuscript (preprint) (Other academic)
    Abstract [en]

    In this paper we compute the Seshadri constants at the general point on many smooth polarized toric surfaces. We consider the case when the degree of jet separation is small or the core of the associated polygon is a line segment. Our main result is that in this case the Seshadri constant at the general point can often be determined in terms of easily computable invariants of the surfaces at hand. Lastly we consider the case that the core of the associated polygon is a point for a smooth polarized toric surface (X, L ). We show that in this case X can be constructed via consecutive equivariant blow-ups of either P^2 or P^1 x P^1. 

  • 30.
    Lundman, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Di Rocco, Sandra
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Bauer, Thomas
    Fachbereich Mathematik und Informatik, Philipps-Universität Marburg.
    Harbourne, Brian
    Department of Mathematics, University of Nebraska-Lincoln.
    Huizenga, Jack
    Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago.
    Pokora, Piotr
    Instytut Matematyki, UP.
    Szemberg, Thomasz
    Instytut Matematyki UP.
    Bounded Negativity and Arrangements of Lines2015In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, Vol. 2015, no 19, p. 9456-9471Article in journal (Refereed)
    Abstract [en]

    The Bounded Negativity Conjecture predicts that for any smooth complex surface X there exists a lower bound for the selfintersection of reduced divisors on X. This conjecture is open. It is also not known if the existence of such a lower bound is invariant in the birational equivalence class of X. In the present note, we introduce certain constants H(X) which measure in effect the variance of the lower bounds in the birational equivalence class of X. We focus on rational surfaces and relate the value of H(ℙ^2) to certain line arrangements. Our main result is Theorem 3.3 and the main open challenge is Problem 3.10.

  • 31.
    Lundman, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Di Rocco, Sandra
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Jabbusch, Kelly
    Department of mathematics & statistics, University of Michigan-Dearborn.
    A note on higher order Gauss MapsManuscript (preprint) (Other academic)
    Abstract [en]

    We study Gauss maps of order k, associated to a projective variety X embedded in projective space via a line bundle L. We show that if X is a smooth, complete complex variety and L is a k-jet spanned line bundle on X, with k > 1, then the Gauss map of order k has finite fibers, unless X = P^n is embedded by the Veronese embedding of order k. In the case where X is a toric variety, we give a combinatorial description of the Gauss maps of order k, its image and the general fibers. 

  • 32.
    Lundman, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Sædén Ståhl, Gustav
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    LatticePolytopes: A package for computations with lattice polytopes in Macaulay2Manuscript (preprint) (Other academic)
    Abstract [en]

    We introduce the package LatticePolytopes for Macaulay2. The package provides methods for computations related to Cayley structures, local positivity and smoothness for lattice polytopes.

  • 33.
    Petersen, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Minimal models, GT-action and formality of the little disk operad2014In: Selecta Mathematica, New Series, ISSN 1022-1824, E-ISSN 1420-9020, Vol. 20, no 3, p. 817-822Article in journal (Refereed)
    Abstract [en]

    We give a new proof of formality of the operad of little disks. The proof makes use of an operadic version of a simple formality criterion for commutative differential graded algebras due to Sullivan. We see that formality is a direct consequence of the fact that the Grothendieck-Teichmuller group operates on the chain operad of little disks.

  • 34.
    Petersen, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The structure of the tautological ring in genus one2014In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 163, no 4, p. 777-793Article in journal (Refereed)
    Abstract [en]

    We prove Getzler's claims about the cohomology of the moduli space of stable curves of genus one, that is, that the even cohomology ring is spanned by the strata classes and that all relations between these classes follow from the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) relation and Getzler's relation. In particular, the even cohomology ring is isomorphic to the tautological ring.

  • 35.
    Petersen, Dan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Topology of moduli spaces and operads2013Doctoral thesis, comprehensive summary (Other academic)
  • 36.
    Petersen, Dan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Tommasi, Orsola
    Leibniz Universität Hannover.
    The Gorenstein conjecture fails for the tautological ring of M̅ 2,n2014In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 196, no 1, p. 139-161Article in journal (Refereed)
    Abstract [en]

    We prove that for N equal to at least one of the integers 8, 12, 16, 20 the tautological ring is not Gorenstein. In fact, our N equals the smallest integer such that there is a non-tautological cohomology class of even degree on . By work of Graber and Pandharipande, such a class exists on , and we present some evidence indicating that N is in fact 20.

  • 37. Pokorny, F. T.
    et al.
    Kragic, Danica
    KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP. KTH, School of Computer Science and Communication (CSC), Centres, Centre for Autonomous Systems, CAS.
    Kavraki, L. E.
    Goldberg, K.
    High-dimensional Winding-Augmented Motion Planning with 2D topological task projections and persistent homology2016In: Proceedings - IEEE International Conference on Robotics and Automation, IEEE conference proceedings, 2016, p. 24-31Conference paper (Refereed)
    Abstract [en]

    Recent progress in motion planning has made it possible to determine homotopy inequivalent trajectories between an initial and terminal configuration in a robot configuration space. Current approaches have however either assumed the knowledge of differential one-forms related to a skeletonization of the collision space, or have relied on a simplicial representation of the free space. Both of these approaches are currently however not yet practical for higher dimensional configuration spaces. We propose 2D topological task projections (TTPs): mappings from the configuration space to 2-dimensional spaces where simplicial complex filtrations and persistent homology can identify topological properties of the high-dimensional free configuration space. Our approach only requires the availability of collision free samples to identify winding centers that can be used to determine homotopy inequivalent trajectories. We propose the Winding Augmented RRT and RRT∗ (WA-RRT/RRT∗) algorithms using which homotopy inequivalent trajectories can be found. We evaluate our approach in experiments with configuration spaces of planar linkages with 2-10 degrees of freedom. Results indicate that our approach can reliably identify suitable topological task projections and our proposed WA-RRT and WA-RRT∗ algorithms were able to identify a collection of homotopy inequivalent trajectories in each considered configuration space dimension.

  • 38.
    Pokorny, Florian T.
    School of Mathematics, The University of Edinburgh.
    The Bergman Kernel on Toric Kähler Manifolds2011Doctoral thesis, monograph (Other academic)
    Abstract [en]

    Let $(L,h)\to (X, \omega)$ be a compact toric polarized Kähler manifold of complex dimension $n$. For each $k\in N$, the fibre-wise Hermitian metric $h^k$ on $L^k$ induces a natural inner product on the vector space $C^{\infty}(X, L^k)$ of smooth global sections of $L^k$ by integration with respect to the volume form $\frac{\omega^n}{n!}$. The orthogonal projection $P_k:C^{\infty}(X, L^k)\to H^0(X, L^k)$ onto the space $H^0(X, L^k)$ of global holomorphic sections of $L^k$ is represented by an integral kernel $B_k$ which is called the Bergman kernel (with parameter $k\in N$). The restriction $\rho_k:X\to R$ of the norm of $B_k$ to the diagonal in $X\times X$ is called the density function of $B_k$.

    On a dense subset of $X$, we describe a method for computing the coefficients of the asymptotic expansion of $\rho_k$ as $k\to \infty$ in this toric setting. We also provide a direct proof of a result which illuminates the off-diagonal decay behaviour of toric Bergman kernels.

    We fix a parameter $l\in N$ and consider the projection $P_{l,k}$ from $C^{\infty}(X, L^k)$ onto those global holomorphic sections of $L^k$ that vanish to order at least $lk$ along some toric submanifold of $X$. There exists an associated toric partial Bergman kernel $B_{l, k}$ giving rise to a toric partial density function $\rho_{l, k}:X\to R$. For such toric partial density functions, we determine new asymptotic expansions over certain subsets of $X$ as $k\to \infty$. Euler-Maclaurin sums and Laplace's method are utilized as important tools for this. We discuss the case of a polarization of $CP^n$ in detail and also investigate the non-compact Bargmann-Fock model with imposed vanishing at the origin.

    We then discuss the relationship between the slope inequality and the asymptotics of Bergman kernels with vanishing and study how a version of Song and Zelditch's toric localization of sums result generalizes to arbitrary polarized Kähler manifolds.

    Finally, we construct families of induced metrics on blow-ups of polarized Kähler manifolds. We relate those metrics to partial density functions and study their properties for a specific blow-up of $C^n$ and $CP^n$ in more detail.

  • 39.
    Pokorny, Florian T.
    et al.
    KTH, School of Computer Science and Communication (CSC), Computer Vision and Active Perception, CVAP. KTH, School of Computer Science and Communication (CSC), Centres, Centre for Autonomous Systems, CAS.
    Singer, Michael
    UCL.
    Toric partial density functions and stability of toric varieties2014In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 358, no 3-4, p. 879-923Article in journal (Refereed)
    Abstract [en]

    Let (L, h) -> (X, omega) denote a polarized toric Kahler manifold. Fix a toric submanifold Y and denote by (rho) over cap (tk) : X -> R the partial density function corresponding to the partial Bergman kernel projecting smooth sections of L-k onto holomorphic sections of L-k that vanish to order at least tk along Y, for fixed t > 0 such that tk is an element of N. We prove the existence of a distributional expansion of (rho) over cap (tk) as k -> infinity, including the identification of the coefficient of k(n-1) as a distribution on X. This expansion is used to give a direct proof that if omega has constant scalar curvature, then (X, L) must be slope semi-stable with respect to Y (cf. Ross and Thomas in J Differ Geom 72(3): 429-466, 2006). Similar results are also obtained for more general partial density functions. These results have analogous applications to the study of toric K-stability of toric varieties.

  • 40.
    Radermacher, Katharina Maria
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the Cosmic No-Hair Conjecture in T2-symmetric non-linear scalar field spacetimesManuscript (preprint) (Other academic)
    Abstract [en]

    We consider spacetimes solving the Einstein non-linear scalar field equations with T2-symmetry and show that they admit an areal time foliation in the expanding direction. In particular, we prove global existence and uniqueness of solutions to the corresponding system of evolution equations for all future times. The only assumption we have to make is that the potential is a non-negative smooth function.

    In the special case of a constant potential, a setting which is equivalent to a linear scalar field on a background with a positive cosmological constant, we achieve detailed asymptotic estimates for the different components of the spacetime metric. This result holds for all T3-Gowdy symmetric metrics and extends to certain T2-symmetric ones satisfying an a priori decay property. Building upon these asymptotic estimates, we show future causal geodesic completeness and prove the Cosmic No-Hair conjecture.

  • 41.
    Radermacher, Katharina Maria
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Orthogonal Bianchi B stiff fluids close to the initial singularityManuscript (preprint) (Other academic)
    Abstract [en]

    In our previous article [Rad16], we investigated the asymptotic behaviour of orthogonal Bianchi class B perfect fluids close to the initial singularity and proved the Strong Cosmic Censorship conjecture in this setting. In several of the statements, the case of a stiff fluid had to be excluded. The present paper fills this gap.

    We work in expansion-normalised variables introduced by Hewitt-Wainwright and find that solutions converge, but show a convergence behaviour very different from the non-stiff case: All solutions tend to limit points in the two-dimensional Jacobs set. A set of full measure, which is also a countable intersection of open and dense sets in the state space, yields convergence to a specific subset of the Jacobs set.

  • 42.
    Radermacher, Katharina Maria
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Strong Cosmic Censorship and Cosmic No-Hair in spacetimes with symmetries2017Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of three articles investigating the asymptotic behaviour of cosmological spacetimes with symmetries arising in Mathematical General Relativity.

    In Paper A and B, we consider spacetimes with Bianchi symmetry and where the matter model is that of a perfect fluid. We investigate the behaviour of such spacetimes close to the initial singularity ('Big Bang'). In Paper A, we prove that the Strong Cosmic Censorship conjecture holds in non-exceptional Bianchi class B spacetimes. Using expansion-normalised variables, we further show detailed asymptotic estimates. In Paper B, we prove similar estimates in the case of stiff fluids.

    In Paper C, we consider T2-symmetric spacetimes satisfying the Einstein equations for a non-linear scalar field. To given initial data, we show global existence and uniqueness of solutions to the corresponding differential equations for all future times. In the special case of a constant potential, a setting which is equivalent to a linear scalar field on a background with a positive cosmological constant, we investigate in detail the asymptotic behaviour towards the future. We prove that the Cosmic No-Hair conjecture holds for solutions satisfying an additional a priori estimate, an estimate which we show to hold in T3-Gowdy symmetry.

  • 43.
    Radermacher, Katharina Maria
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Strong Cosmic Censorship in orthogonal Bianchi class B perfect fluid and vacuum modelsManuscript (preprint) (Other academic)
    Abstract [en]

    The Strong Cosmic Censorship conjecture states that for generic initial data to Einstein's field equations, the maximal globally hyperbolic development is inextendible. We prove this conjecture in the class of orthogonal Bianchi class B perfect fluids and vacuum spacetimes, by showing that unboundedness of certain curvature invariants such as the Kretschmann scalar is a generic property. The only spacetimes where this scalar remains bounded exhibit local rotational symmetry or are of plane wave equilibrium type.

    We further investigate the qualitative behaviour of solutions towards the initial singularity. To this end, we work in the expansion-normalised variables introduced by Hewitt-Wainwright and show that a set of full measure, which is also a countable intersection of open and dense sets in the state space, yields convergence to a specific subarc of the Kasner parabola. We further give an explicit construction enabling the translation between these variables and geometric initial data to Einstein's equations.

  • 44.
    Rydh, David
    Department of Mathematics, University of California, Berkeley.
    Submersions and effective descent of étale morphisms2010In: Bulletin de la Société Mathématique de France, ISSN 0037-9484, E-ISSN 2102-622X, Vol. 138, no 2, p. 181-230Article in journal (Refereed)
    Abstract [en]

    Using the flatification by blow-up result of Raynaud and Gruson, we obtain new results for submersive and subtrusive morphisms. We show that universally subtrusive morphisms, and in particular universally open morphisms, are morphisms of effective descent for the fibered category of étale morphisms. Our results extend and supplement previous treatments on submersive morphisms by Grothendieck, Picavet and Voevodsky. Applications include the universality of geometric quotients and the elimination of noetherian hypotheses in many instances.

  • 45.
    Rydh, David
    Department of Mathematics, University of California, Berkeley.
    The canonical embedding of an unramified morphism in an étale morphism2011In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 268, no 3-4, p. 707-723Article in journal (Refereed)
    Abstract [en]

    We show that every unramified morphism XY has a canonical and universal factorization XEY where the first morphism is a closed embedding and the second is étale (but not separated).

  • 46.
    Staffas, Eric
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Nonspherical black holes and spacetime reconstructions2018Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of three papers in mathematical general relativity. The first paper concerns inverse problems and reconstruction of spacetimes from boundary data. We consider boundary data in the form of broken geodesics with different causal types, which have the physical interpretation of relativistic fireworks. Using this data, we show that it is possible to completely reconstruct the topology, smooth structure, and metric of the spacetime.

    The second paper is about the topology of black holes. When black holes are shown in illustrations, they are typically shown as spherical, and a natural question is whether this spherical topology is necessary. Would it not be possible for black holes to have other topologies? It is previously known that an apparent horizon of a black hole must admit a metric of positive scalar curvature, and this imposes restrictions on the possible topologies. However, there is a large class of topologies which cannot be excluded by this result, but where there are no examples or constructions showing that they are actually possible. This means that it is possible that there are further restrictions which have not yet been found. In the second paper of the thesis, we describe a construction which gives rise to a large class of new examples of topologies for apparent horizons. This decreases the gap between the topologies which are known to be possible and those which are known to be impossible.

    The third paper in the thesis also concerns apparent horizons of black holes, and the main result is more technical in character than those of the other papers. It is a special case of a generalization of a regularity theorem for apparent horizons, which is previously known in low dimensions. It is not immediately obvious from the definition of an apparent horizon that it should have any particularly good properties, but it is previously known that it is a smooth hypersurface if the dimension is sufficiently small. We show in the third paper for time-symmetric initial data in any dimension that an outermost apparent horizon is a smooth hypersurface, apart from a singular set of large codimension.

  • 47.
    Svedberg, Christopher
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Future stability of the Einstein-Maxwell-Scalar field system and non-linear wave equations coupled to generalized massive-massless Vlasov equations2012Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of two articles related to mathematical relativity theory.

    In the first article we prove future stability of certain spatially homogeneous solutionsto Einstein’s field equations. The matter model is assumed to consist of an electromagnetic field and a scalar field with a potential creating an accelerated expansion. Beside this, more general properties concerning Einstein’s field equation coupled to a scalar field and an electromagnetic field are settled. The most important of these questions are the existence of a maximal globally hyperbolic development and the Cauchy stability of solutions to the initial value problem.

    In the second article we consider Einstein’s field equations where the matter model consists of two momentum distribution functions. The first momentum distribution function represents massive matter, for instance galactic dust, and the second represents massless matter, for instance radiation. Furthermore, we require that each of the momentum distribution functions shall satisfy the Vlasov equation. This means that the momentum distribution functions represent collisionless matter. If Einstein’s field equations with such a matter model is expressed in coordinates and if certain gauges are fixed we get a system of integro-partial differential equations we shall call non-linear wave equations coupled to generalized massive-massless Vlasov equations. In the second article we prove that the initial value problem associated to this kind of equations has a unique local solution.

  • 48.
    Svedberg, Christopher
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Non-linear wave equations coupled to generalized massive-massless Vlasov equationsManuscript (preprint) (Other academic)
    Abstract [en]

    Consider Einstein’s field equations where the matter model consists oftwo momentum distribution functions. Let the first momentum distribution function represent massive matter, for instance galactic dust, and let the second represent massless matter, for instance radiation. Furthermore,let us require that each of the momentum distribution functions shall satisfy the Vlasov equation. This means that the momentum distribution functions represent collisionless matter. If Einstein’s field equations withsuch a matter model is expressed in coordinates and if certain gauges arefixed we get a system of integro-partial differential equations we shall call non-linear wave equations coupled to generalized massive-massless Vlasovequations. We prove that the initial value problem associated to this kindof equations has a unique local solution. Moreover, we prove a continuation criterion for the solution.

  • 49.
    Sædén Ståhl, Gustav
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Good Hilbert functorsManuscript (preprint) (Other academic)
    Abstract [en]

    We introduce the good Hilbert functor and prove that it is algebraic. This functor generalizes various versions of the Hilbert moduli problem, such as the multigraded Hilbert scheme and the invariant Hilbert scheme. Moreover, we generalize a result concerning formal GAGA for good moduli spaces.

  • 50.
    Sædén Ståhl, Gustav
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Gotzmann's persistence theorem for finite modulesManuscript (preprint) (Other academic)
    Abstract [en]

    We prove a generalization of Gotzmann's persistence theorem in the case of modules with constant Hilbert polynomial. As a consequence, we show that the defining equations that give the embedding of a Quot scheme of points into a Grassmannian are given by a single Fitting ideal.

12 1 - 50 of 56
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