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  • 1.
    Andreasson, Rolf
    et al.
    Department of Mathematical Sciences, Chalmers University of Technology, Gothenburg, Sweden.
    Hultgren, Jakob
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Solvability of Monge-Ampère equations and tropical affine structures on reflexive polytopesManuskript (preprint) (Annet vitenskapelig)
    Abstract [en]

    Given a reflexive polytope with a height function, we prove a necessary and sufficient condition for solvability of the associated Monge-Ampère equation. When the polytope is Delzant, solvability of this equation implies the metric SYZ conjecture for the corresponding family of Calabi-Yau hypersurfaces. We show how the location of the singularities in the tropical affine structure is determined by the PDE in the spirit of a free boundary problem and give positive and negative examples, demonstrating subtle issues with both solvability and properties of the singular set. We also improve on existing results regarding the SYZ conjecture for the Fermat family by showing regularity of the limiting potential.

  • 2. Backlund, Ulf
    et al.
    Carlsson, Linus
    Fällström, Anders
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Persson, Håkan
    Semi-Bloch Functions in Several Complex Variables2016Inngår i: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 26, nr 1, s. 463-473Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Let M be an n-dimensional complex manifold. A holomorphic function f : M -> C is said to be semi-Bloch if for every lambda is an element of C the function g(lambda) = exp(lambda f(z)) is normal on M. We characterize semi-Bloch functions on infinitesimally Kobayashi non-degenerate M in geometric as well as analytic terms. Moreover, we show that on such manifolds, semi-Bloch functions are normal.

  • 3. Cenanovic, Mirza
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Minimal surface computation using a finite element method on an embedded surface2015Inngår i: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 104, nr 7, s. 502-512Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We suggest a finite element method for finding minimal surfaces based on computing a discrete Laplace-Beltrami operator operating on the coordinates of the surface. The surface is a discrete representation of the zero level set of a distance function using linear tetrahedral finite elements, and the finite element discretization is carried out on the piecewise planar isosurface using the shape functions from the background three-dimensional mesh used to represent the distance function. A recently suggested stabilized scheme for finite element approximation of the mean curvature vector is a crucial component of the method.

  • 4.
    Collin, Jan-Ola
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    The Existence of Riemannian Metrics on Real Vector Bundles2018Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
    Abstract [en]

    In this thesis we present a self-contained proof of the existence of Riemannian metrics on real vector bundles.

    Fulltekst (pdf)
    fulltext
  • 5.
    Gerken, Jan
    et al.
    Department of Mathematical Sciences, Chalmers University of Technology, Gothenburg, Sweden; Machine Learning Group at Berlin Institute of Technology, Berlin, Germany; Berlin Institute for the Foundations of Learning and Data (BIFOLD), Berlin, Germany.
    Carlsson, Oscar
    Department of Mathematical Sciences, Chalmers University of Technology, Gothenburg, Sweden.
    Linander, Hampus
    Department of Physics, University of Gothenburg, Gothenburg, Sweden.
    Ohlsson, Fredrik
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Petersson, Christoffer
    Zenseact, Gothenburg, Sweden; Department of Mathematical Sciences, Chalmers University of Technology, Gothenburg, Sweden.
    Persson, Daniel
    Department of Mathematical Sciences, Chalmers University of Technology, Gothenburg, Sweden.
    Equivariance versus augmentation for spherical images2022Inngår i: Proceedings of Machine Learning Resaerch: International Conference on Machine Learning, 17-23 July 2022, Baltimore, Maryland, USA, 2022, Vol. 162, s. 7404-7421Konferansepaper (Fagfellevurdert)
    Abstract [en]

    We analyze the role of rotational equivariance in convolutional neural networks (CNNs) applied to spherical images. We compare the performance of the group equivariant networks known as S2CNNs and standard non-equivariant CNNs trained with an increasing amount of data augmentation. The chosen architectures can be considered baseline references for the respective design paradigms. Our models are trained and evaluated on single or multiple items from the MNIST- or FashionMNIST dataset projected onto the sphere. For the task of image classification, which is inherently rotationally invariant, we find that by considerably increasing the amount of data augmentation and the size of the networks, it is possible for the standard CNNs to reach at least the same performance as the equivariant network. In contrast, for the inherently equivariant task of semantic segmentation, the non-equivariant networks are consistently outperformed by the equivariant networks with significantly fewer parameters. We also analyze and compare the inference latency and training times of the different networks, enabling detailed tradeoff considerations between equivariant architectures and data augmentation for practical problems.

  • 6.
    Hermansson, Robert
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Laborativ geometriundervisning i olika åldersgrupper: En intervjustudie med lärare från förskola till gymnasium2013Independent thesis Basic level (university diploma), 10 poäng / 15 hpOppgave
    Abstract [sv]

    Utvärderingar visar att svenska elevers kunskaper i geometri har försämrats under de senaste decennierna och att dagens elever upplever geometri som ett av de svårare områdena i matematikkurserna. Samtidigt finns det forskningsresultat som säger att laborativa arbetsformer kan öka elevernas intresse, motivation, inställning och självförtroende gällande matematikämnet. Syftet med detta arbete är att undersöka vilka laborativa arbetssätt i matematik, med fokus på geometri, som av lärare upplevs vara mest positiva eller ge bäst resultat i olika åldersgrupper. Med detta avses både vad som väcker störst intresse hos eleverna och vad de verkar lära sig mest av. I denna studie undersöks vilka laborativa arbetssätt som lärare upplever fungera bäst i olika åldersgrupper, vad i det laborativa arbetssättet som de anser vara positivt samt om nämnda arbetsformer tycks ge elever en ökad förståelse av geometri eller förhöjd prestation i matematikämnet. För att få svar på dessa frågor genomfördes djupintervjuer med sju verksamma lärare. Dessa undervisar i olika åldersgrupper och har tillsammans erfarenhet av laborativ matematikundervisning från förskola till gymnasium. Resultaten visar att moderna forskningsrön och verksamma lärares erfarenheter i mångt och mycket överensstämmer med varandra. I alla åldersgrupper finns ett behov av mer laborativt arbete i matematikundervisningen. Naturen och uterummet som inlärningsmiljö framhålls huvudsakligen av lärare för yngre elever. Dessa talar också mer detaljerat om vilken specifik laborativ materiel de använder, till exempel twistband, geobräde eller geometrirep, medan lärare för äldre elever mer talar om arbetssätt, till exempel muntlig kommunikation, färdighetsträning, att återknyta till egengjort arbetsmaterial eller arbeta med verklighetsnära uppgifter. Det laborativa arbetet kan underlätta yngre barns begreppsutveckling och förmåga att tänka i bilder, vara verklighetsnära, aktivera olika sinnen, vara ett stöd för minnet, erbjuda uttråkade (huvudsakligen äldre) elever omväxling, underlätta förståelse, ge en positiv känsla till matematik och skapa ett meningsfullt lärande. Samtliga lärare som deltog i studien anser att laborativa inslag i matematikundervisningen ger eleverna en ökad förståelse, det vill säga detta verkar gälla oavsett ålder.

    Fulltekst (pdf)
    Laborativ geometriundervisning i olika åldersgrupper: En intervjustudie med lärare från förskola till gymnasium
  • 7.
    Hultgren, Jakob
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Duality of Hessian manifolds and optimal transportManuskript (preprint) (Annet vitenskapelig)
    Abstract [en]

    This is an expository paper describing how duality theory for Hessian manifolds provides a natural setting for optimal transport. We explain how this can be used to solve Monge-Ampère equations and survey recent results along these lines with applications to the SYZ-conjecture in mirror symmetry.

  • 8.
    Hultgren, Jakob
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Dept of Mathematics, University of Maryland, MD, College Park, United States.
    Jonsson, Mattias
    Dept of Mathematics, University of Michigan, MI, Ann Arbor, United States.
    Mazzon, Enrica
    Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany.
    McCleerey, Nicholas
    Dept of Mathematics, Purdue University, IN, West Lafayette, United States.
    Tropical and non-Archimedean Monge–Ampère equations for a class of Calabi–Yau hypersurfaces2024Inngår i: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 439, artikkel-id 109494Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    For a large class of maximally degenerate families of Calabi–Yau hypersurfaces of complex projective space, we study non-Archimedean and tropical Monge–Ampère equations, taking place on the associated Berkovich space, and the essential skeleton therein, respectively. For a symmetric measure on the skeleton, we prove that the tropical equation admits a unique solution, up to an additive constant. Moreover, the solution to the non-Archimedean equation can be derived from the tropical solution, and is the restriction of a continuous semipositive toric metric on projective space. Together with the work of Yang Li, this implies the weak metric SYZ conjecture on the existence of special Lagrangian fibrations in our setting.

  • 9.
    Hultgren, Jakob
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Mattias, Jonsson
    Department of Mathematics, University of Michigan, Michigan, USA.
    McCleerey, Nicholas
    Department of Mathematics, University of Michigan, Michigan, USA.
    Mazzon, Enrica
    Department of Mathematics, University of Michigan, Michigan, USA.
    Tropical and non-Archimedean Monge-Ampère equations for a class of Calabi-Yau hypersurfacesManuskript (preprint) (Annet vitenskapelig)
    Abstract [en]

    For a class of maximally degenerate families of Calabi-Yau hypersurfaces of complex projective space, we study associated non-Archimedean and tropical Monge--Ampère equations, taking place on the associated Berkovich space, and the essential skeleton therein, respectively. For a symmetric measure on the skeleton, we prove that the tropical equation admits a unique solution, up to an additive constant. Moreover, the solution to the non-Archimedean equation can be derived from the tropical solution, and is the restriction of a continuous semipositive toric metric on projective space. Together with the work of Yang Li, this implies the weak metric SYZ conjecture on the existence of special Lagrangian fibrations in our setting.

  • 10.
    Johansson, Ingvar
    Umeå universitet, Humanistiska fakulteten, Institutionen för idé- och samhällsstudier.
    Shape is a non-quantifiable physical dimension2011Inngår i: SHAPES 1.0: Proceedings of the First Interdisciplinary Workshop on SHAPES, 2011Konferansepaper (Fagfellevurdert)
    Abstract [en]

    In the natural-scientific community it is often taken for granted that, sooner or later, all basic physical property dimensions can be quantified and turned into a kind-of-quantity; meaning that all their possible determinate properties can be put in a one-to-one correspondence with the real numbers. By using some transfinite mathematics, the paper shows this tacit assumption to be wrong. Shape is a very basic property dimension; but, since it can be proved that there are more possible kinds of determinate shapes than real numbers, shape cannot be quantified. There will never be a shape scale the way we have length and temperature scales. This is the most important conclusion, but more is implied by the proof. Since every n-dimensional manifold has the same cardinality as the real number line, all shapes cannot even be represented in a three-dimensional manifold the way perceivable colors are represented in socalled color solids.

  • 11.
    Jörgenfelt, Erik
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Stokes' Theorem on Smooth Manifolds2016Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
    Abstract [en]

    A proof of Stokes' theorem on smooth manifolds is given, complete with prerequisite results in tensor algebra and differential geometry. The essay assumes familiarity with multi-variable calculus and linear algebra, as well as a basic understanding of point-set topology. Stokes' theorem is then applied to the conservation of energy-momentum in general relativity under the existence of so called Killing vectors.

    Fulltekst (pdf)
    fulltext
  • 12.
    Karlsson, Jesper
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Symplectic Automorphisms of C2n2018Independent thesis Advanced level (degree of Master (Two Years)), 20 poäng / 30 hpOppgave
    Abstract [en]

    This essay is a detailed survey of an article from 1996 published by Franc Forstneric, where he studies symplectic automorphisms of C2n. The vision is to introduce the density property for holomorphic symplectic manifolds. Our idea is that of Dror Varolin when he in 2001 introduced the concept of density property for Stein manifolds. The main result here is the introduction of symplectic shears on C2n equipped with a holomorphic symplectic form and to show that the group generated by finite compositions of symplectic shears is dense in the group of symplectic automorphisms of C2n in the compact-open topology. We give a complete background of the tools from the theory of ordinary differential equations, smooth manifolds, and complex and symplectic geometry that is needed in order to prove this result.

    Fulltekst (pdf)
    fulltext
  • 13.
    Leemans, Dimitri
    et al.
    Département de Mathématique, Université libre de Bruxelles, Algèbre et Combinatoire, Brussels, Belgium.
    Stokes, Klara
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Incidence geometries with trialities coming from maps with Wilson trialities2023Inngår i: Innovations in Incidence Geometry, ISSN 2640-7337, Vol. 20, nr 2-3, s. 325-340Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Triality is a classical notion in geometry that arose in the context of the Lie groups of type D4. Another notion of triality, Wilson triality, appears in the context of reflexible maps. We build a bridge between these two notions, showing how to construct an incidence geometry with a triality from a map that admits a Wilson triality. We also extend a result by Jones and Poulton, showing that for every prime power q, the group L2 (q3) has maps that admit Wilson trialities but no dualities.

  • 14.
    Leijon, Rasmus
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    On the geometry of calibrated manifolds: with applications to electrodynamics2013Independent thesis Advanced level (degree of Master (Two Years)), 20 poäng / 30 hpOppgave
    Abstract [en]

    In this master thesis we study calibrated geometries, a family of Riemannian or Hermitian manifolds with an associated differential form, φ. We show that it isuseful to introduce the concept of proper calibrated manifolds, which are in asense calibrated manifolds where the geometry is derived from the calibration. In particular, the φ-Grassmannian is considered in the case of proper calibratedmanifolds. The impact of proper calibrated manifolds as a model is studied, aswell as the usefulness of pluripotential theory as tools for the model. The specialLagrangian calibration is an example of an important calibration introduced byHarvey and Lawson, which leads to the definition of the special Lagrangian differentialequation. This partial differential equation can be formulated in threeand four dimensions as det(H(u)) = Δu, where H(u) is the Hessian matrix of some potential u. We prove the existence of solutions and some other propertiesof this nonlinear differential equation and present the resulting 6- and 8-dimensional manifolds defined by the graph {x + iu(x)}. We also considerthe physical applications of calibrated geometry, which have so far largely beenrestricted to string theory. However, we consider the manifold (M,g,F), whichis calibrated by the scaled Maxwell 2-form. Some geometrical properties of relativisticand classical electrodynamics are translated into calibrated geometry.

    Fulltekst (pdf)
    Kappa.pdf
  • 15.
    Lundqvist, Signe
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Stokes, Klara
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Öhman, Lars-Daniel
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Applying the pebble game algorithm to rod configurations2023Inngår i: EuroCG 2023: Book of abstracts, 2023, artikkel-id 41Konferansepaper (Fagfellevurdert)
    Abstract [en]

    We present results on rigidity of structures of rigid rods connected in joints: rod configurations. The underlying combinatorial structure of a rod configuration is an incidence structure. Our aim is to find simple ways of determining which rod configurations admit non-trivial motions, using the underlying incidence structure.

    Rigidity of graphs in the plane is well understood. Indeed, there is a polynomial time algorithm for deciding whether most realisations of a graph are rigid. One of the results presented here equates rigidity of sufficiently generic rod configurations to rigidity of a related graph. As a consequence, itis possible to determine the rigidity of rod configurations using the previously mentioned polynomial time algorithm. We use this to show that all v3-configurations on up to 15 points and all triangle-free v3-configurations on up to 20 points are rigid in regular position, if such a realisation exists. We also conjecture that the smallest v3-configuration that is flexible in regular position is a previously known 283-configuration. 

  • 16.
    Persson, Aron
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för fysik.
    On the Existence of Electrodynamics on Manifold-like Polyfolds2019Independent thesis Advanced level (professional degree), 20 poäng / 30 hpOppgave
    Abstract [en]

    This essay examines the question whether the classical theory of electrodynamics can be extended to a spacetime which locally changes dimension and if such an endeavour is mathematically possible. Recent research has developed a new generalisation of smooth manifolds, the so called M-polyfolds, which constitutes a sufficient foundation to make this endeavour a physical plausibility. These M-polyfolds then facilitate the capability to define the velocity of a curve going through a dimensionally shifting spacetime. Moreover, necessary extensions to the theory of M-polyfolds is developed in order to tailor the theory to a more physically focused framework. Concluding the essay, Maxwell’s equations on M-polyfolds are defined.

    Fulltekst (pdf)
    fulltext
  • 17.
    Persson, Nicklas
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Shortest paths and geodesics in metric spaces2013Independent thesis Advanced level (degree of Master (Two Years)), 20 poäng / 30 hpOppgave
    Abstract [en]

    This thesis is divided into three part, the first part concerns metric spaces and specically length spaces where the existence of shortest path between points is the main focus. In the second part, an example of a length space, the Riemannian geometry will be given. Here both a classical approach to Riemannian geometry will be given together with specic results when considered as a metric space. In the third part, the Finsler geometry will be examined both with a classical approach and trying to deal with it as a metric space.

    Fulltekst (pdf)
    FULLTEXT.pdf
  • 18.
    Stokes, Klara
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Geometric decoding of subspace codes with explicit Schubert calculus applied to spread codes2023Inngår i: Advances in mathematics of communicationsArtikkel i tidsskrift (Fagfellevurdert)
  • 19.
    Viktor, Vigren Näslund
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Hassler Whitney's Hidden Treasure: Every Differentiable Manifold Can Be Made Smooth2023Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
    Abstract [en]

    In this essay, we give background to differential topology and utilize approximation techniques to prove Hassler Whitney's classic result that a manifold with a $C^r$ differential structure, $r \geq 1$, admits a compatible $C^s$ differential structure, $r < s \leq \infty$. That is, every differentiable manifold is a smooth manifold. 

    Fulltekst (pdf)
    fulltext
  • 20.
    Wahlberg, Mats Karl Anders
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    The Banach-Tarski Paradox: How I Learned to Stop Worrying and Love the Axiom of Choice2022Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
    Abstract [en]

    This thesis presents the strong and weak forms of the Banach-Tarski paradox based on the Hausdorff paradox. It provides modernized proofs of the paradoxes and necessary properties of equidecomposable and paradoxical sets. The historical significance of the paradox for measure theory is covered, along with its incorrect attribution to Banach and Tarski. Finally, the necessity of the axiom of choice is discussed and contrasted with other axiomatic and topological assumptions that enable the paradoxes.

    Fulltekst (pdf)
    fulltext
  • 21.
    Åhag, Per
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Czyż, Rafał
    Institute of Mathematics Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland.
    On a family of quasimetric spaces in generalized potential theory2022Inngår i: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 32, nr 4, artikkel-id 117Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We construct a family of quasimetric spaces in generalized potential theory containing m-subharmonic functions with finite (pm)-energy. These quasimetric spaces will be viewed both in Cn and in compact Kähler manifolds, and their convergence will be used to improve known stability results for the complex Hessian equations.

    Fulltekst (pdf)
    fulltext
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