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  • 1.
    Abathun, Addisalem
    Stockholm University, Faculty of Science, Department of Mathematics.
    Asymptotic distribution of zeros of a certain class of hypergeometric polynomials2014Licentiate thesis, monograph (Other academic)
    Abstract [en]

    The thesis consists of two papers, both treating hypergeometric polynomials, and a short introduction. The main results are as follows.In the first paper,we study the asymptotic zero distribution of a family of hypergeometric polynomials in one complex variable as their degree goes to infinity,using the associated differential equations that hypergeometric polynomials satisfy.   We describe in particular the curve complex on which the zeros cluster, as level curves associated to integrals on an algebraic curve derived from the equation.   The new result is first of all that we are able to formulate results on the location of zeros of generalized hypergeometric polynomials in greater generality than before (earlier results are mainly concerned with the Gauss hypergeometric case.) Secondly, we are able to formulate a precise conjucture giving the asymptotic behaviour of zeros in the generalized case of our polynomials, which covers previous results.In the second paper we partly prove one of the  conjectures in the first paper by using Euler integral representation of the Gauss hypergeometric functions together with the Saddle point method.

  • 2.
    Abouzaid, Mohammed
    et al.
    Columbia Univ, Dept Math, New York, NY 10027 USA.
    Kragh, Thomas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    On the immersion classes of nearby Lagrangians2016In: Journal of Topology, ISSN 1753-8416, E-ISSN 1753-8424, Vol. 9, no 1, 232-244 p.Article in journal (Refereed)
    Abstract [en]

    We show that the transfer map on Floer homotopy types associated to an exact Lagrangian embedding is an equivalence. This provides an obstruction to representing isotopy classes of Lagrangian immersions by Lagrangian embeddings, which, unlike previous obstructions, is sensitive to information that cannot be detected by Floer cochains. We show this by providing a concrete computation in the case of spheres.

  • 3. Abramov, V.
    et al.
    Paal, E.Tallinn University of Technology.Silvestrov, Sergei D.Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.Stolin, A.Chalmers University of Techology.
    Proceedings of the 3rd Baltic-Nordic Workshop “Algebra, Geometry, and Mathematical Physics”2008Conference proceedings (editor) (Refereed)
  • 4. 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, 1-8 p.Article 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.

  • 5.
    Alm, Johan
    Stockholm University, Faculty of Science, Department of Mathematics.
    Universal algebraic structures on polyvector fields2014Doctoral thesis, monograph (Other academic)
    Abstract [en]

    The theory of operads is a conceptual framework that has become a kind of universal language, relating branches of topology and algebra. This thesis uses the operadic framework to study the derived algebraic properties of polyvector fields on manifolds.The thesis is divided into eight chapters. The first is an introduction to the thesis and the research field to which it belongs, while the second chapter surveys the basic mathematical results of the field.The third chapter is devoted to a novel construction of differential graded operads, generalizing an earlier construction due to Thomas Willwacher. The construction highlights and explains several categorical properties of differential graded algebras (of some kind) that come equipped with an action by a differential graded Lie algebra. In particular, the construction clarifies the deformation theory of such algebras and explains how such algebras can be twisted by Maurer-Cartan elements.The fourth chapter constructs an explicit strong homotopy deformation of polynomial polyvector fields on affine space, regarded as a two-colored noncommutative Gerstenhaber algebra. It also constructs an explicit strong homotopy quasi-isomorphism from this deformation to the canonical two-colored noncommmutative Gerstenhaber algebra of polydifferential operators on the affine space. This explicit construction generalizes Maxim Kontsevich's formality morphism.The main result of the fifth chapter is that the deformation of polyvector fields constructed in the fourth chapter is (generically) nontrivial and, in a sense, the unique such deformation. The proof is based on some cohomology computations involving Kontsevich's graph complex and related complexes. The chapter ends with an application of the results to properties of a derived version of the Duflo isomorphism.The sixth chapter develops a general mathematical framework for how and when an algebraic structure on the germs at the origin of a sheaf on Cartesian space can be "globalized" to a corresponding algebraic structure on the global sections over an arbitrary smooth manifold. The results are applied to the construction of the fourth chapter, and it is shown that the construction globalizes to polyvector fields and polydifferential operators on an arbitrary smooth manifold.The seventh chapter combines the relations to graph complexes, explained in chapter five, and the globalization theory of chapter six, to uncover a representation of the Grothendieck-Teichmüller group in terms of A-infinity morphisms between Poisson cohomology cochain complexes on a manifold.Chapter eight gives a simplified version of a construction of a family of Drinfel'd associators due to Carlo Rossi and Thomas Willwacher. Our simplified construction makes the connections to multiple zeta values more transparent--in particular, one obtains a fairly explicit family of evaluations on the algebra of formal multiple zeta values, and the chapter proves certain basic properties of this family of evaluations.

  • 6. 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, 1-58 p.Article 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)}.

  • 7.
    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, 2307-2329 p.Article 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.

  • 8.
    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, 1-39 p.Article 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.

  • 9.
    Arnlind, Joakim
    et al.
    Albert Einstein Institute, Golm, Germany..
    Silvestrov, Sergei
    Centre for Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden .
    Affine transformation crossed product type algebras and noncommutative surfaces2009In: Operator structures and dynamical systems :: July 21-25 2008, Lorentz Center, Leiden, the Netherlands, satellite conference of the fifth European Congress of Mathematics, American Mathematical Society (AMS), 2009, 503, 1-25 p.Chapter in book (Refereed)
    Abstract [en]

    Several classes of *-algebras associated to teh action of an affine transformation are considered, and an investigation of the interplay between the different classes is initiated. Connections are established that relate representations of *-algebras, geometry of algebraic surfaces, dynamics of affine transformations, graphs and algebras coming from a quantization procedure of Poisson structures. In particular, algebras related to surgaced being inverse images of fourth order polynomials (in ) are studied in detail, and a close link between representation theory and geometric properties is established for compact as well as non-compact surfaces.

  • 10.
    Arnlind, Joakim
    et al.
    Albert Einstein Institute, Golm, Germany.
    Silvestrov, Sergei
    Lund University.
    Affine transformation crossed product type algebras and noncommutative surfaces2009In: Operator structures and dynamical systems: July 21-25 2008, Lorentz Center, Leiden, The Netherlands, satellite conference of the fifth European Congress of Mathematics, Amer. Math. Soc. , 2009, Vol. 503, 1-25 p.Chapter in book (Refereed)
  • 11.
    Asekritova, Irina
    et al.
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Cerdá, Joan
    Barcelona University, Spain.
    Kruglyak, Natan
    Linköping University.
    The Riesz-Herz equivalence for capacitary maximal functions2012In: Revista Matemática Complutense, ISSN 1139-1138, Vol. 25, no 1, 43-59 p.Article in journal (Refereed)
    Abstract [en]

    We prove a Riesz-Herz estimate for the maximal function associated toa capacity ConRn,MCf(x)=supQxC(Q)−1Q|f|, which extends the equivalence (Mf )∗(t)f∗∗(t) for the usual Hardy-Littlewood maximal function Mf. The proof is based on an extension of the Wiener-Stein estimates for the distribution function of the maximal function, obtained using a convenient family of dyadiccubes. As a byproduct we obtain a description of the norm of the interpolationspace (L1,L1,C)1/p,p,  where L1,C denotes the Morrey space based on a capacity.

  • 12.
    Asekritova, Irina
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Cobos, Fernando
    Complutense University of Madrid.
    Kruglyak, Natan
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Interpolation of Closed Subspaces and Invertibility of Operators2015In: Journal of Analysis and its Applications, ISSN 0232-2064, Vol. 34, no 2015, 1-15 p.Article in journal (Refereed)
    Abstract [en]

    Let (Y0,Y1) be a Banach couple and let Xj be a closed complemented subspace of Yj, (j = 0,1). We present several results for the general problem of finding necessary and sufficient conditions on the parameters (θ, q) such that the real interpolation space (X0,X1)θ,q is a closed subspace of (Y0,Y1)θ,q. In particular, we establish conditions which are necessary and sufficient for the equality (X0,X1)θ,q = (Y0,Y1)θ,q, with the proof based on a previous result by Asekritova and Kruglyak on invertibility of operators. We also generalize the theorem by Ivanov and Kalton where this problem was solved under several rather restrictive conditions, such as that X1 = Y1 and X0 is a subspace of codimension one in Y0. 

  • 13.
    Asplund, Johan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Contact Homology of Legendrian Knots in Five-Dimensional Circle Bundles2016Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 14. Avdonina, Elena D.
    et al.
    Ibragimov, Nail H.
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Natural Sciences.
    Heat conduction in anisotropic media: Nonlinear self-adjointness and conservation laws2012In: Discontinuity, Nonlinearity and Complexity, ISSN 2164-6376, Vol. 1, no 3, 237-251 p.Article in journal (Refereed)
    Abstract [en]

    Nonlinear self-adjointness of the anisotropic nonlinear heat equation is investigated. Mathematical models of heat conduction in anisotropic media with a source are considered and a class of self-adjoint models is identified. Conservation laws corresponding to the symmetries of the equations in question are computed.

  • 15.
    Backelin, Jörgen
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Oneto, Alessandro
    Stockholm University, Faculty of Science, Department of Mathematics.
    On a class of power ideals2015In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 219, no 8, 3158-3180 p.Article in journal (Refereed)
    Abstract [en]

    In this paper we study the class of power ideals generated by the k(n) forms (x(0) + xi(g1) x(1) + ... + xi(gn) x(n))((k-1)d) where xi is a fixed primitive kth-root of unity and 0 <= g(j) <= k - 1 for all j. For k = 2, by using a Z(k)(n+1)-grading on C[x(0),..., x(n)], we compute the Hilbert series of the associated quotient rings via a simple numerical algorithm. We also conjecture the extension for k > 2. Via Macaulay duality, those power ideals are related to schemes of fat points with support on the k(n) points [1 : xi(g1) : ... : xi(gn)] in P-n. We compute Hilbert series, Betti numbers and Grobner basis for these 0-dimensional schemes. This explicitly determines the Hilbert series of the power ideal for all k: that this agrees with our conjecture for k > 2 is supported by several computer experiments.

  • 16. Backlund, Ulf
    et al.
    Carlsson, Linus
    Fällström, Anders
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Persson, Håkan
    Semi-Bloch Functions in Several Complex Variables2016In: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 26, no 1, 463-473 p.Article in journal (Refereed)
    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.

  • 17.
    Backman, Theo
    Stockholm University, Faculty of Science, Department of Mathematics.
    Configuration spaces, props and wheel-free deformation quantization2016Doctoral thesis, monograph (Other academic)
    Abstract [en]

    The main theme of this thesis is higher algebraic structures that come from operads and props.

    The first chapter is an introduction to the mathematical framework needed for the content of this thesis. The chapter does not contain any new results.

    The second chapter is concerned with the construction of a configuration space model for a particular 2-colored differential graded operad encoding the structure of two A algebras with two A morphisms and a homotopy between the morphisms. The cohomology of this operad is shown to be the well-known 2-colored operad encoding the structure of two associative algebras and of an associative algebra morphism between them.

    The third chapter is concerned with deformation quantization of (potentially) infinite dimensional (quasi-)Poisson manifolds. Our proof employs a variation on the transcendental methods pioneered by M. Kontsevich for the finite dimensional case. The first proof of the infinite dimensional case is due to B. Shoikhet. A key feature of the first proof is the construction of a universal L structure on formal polyvector fields. Our contribution is a simplification of B. Shoikhet proof by considering a more natural configuration space and a simpler choice of propagator. The result is also put into a natural context of the dg Lie algebras coming from graph complexes; the L structure is proved to come from a Maurer-Cartan element in the oriented graph complex.

    The fourth chapter also deals with deformation quantization of (quasi-)Poisson structures in the infinite dimensional setting. Unlike the previous chapter, the methods used here are purely algebraic. Our main theorem is the possibility to deformation quantize quasi-Poisson structures by only using perturbative methods; in contrast to the transcendental methods employed in the previous chapter. We give two proofs of the theorem via the theory of dg operads, dg properads and dg props. We show that there is a dg prop morphism from a prop governing star-products to a dg prop(erad) governing (quasi-)Poisson structures. This morphism gives a theorem about the existence of a deformation quantization of (quasi-)Poisson structure. The proof proceeds by giving an explicit deformation quantization of super-involutive Lie bialgebras and then lifting that to the dg properad governing quasi-Poisson structures. The prop governing star-products was first considered by S.A. Merkulov, but the properad governing quasi-Poisson structures is a new construction.

    The second proof of the theorem employs the Merkulov-Willwacher polydifferential functor to transfer the problem of finding a morphism of dg props to that of finding a morphism of dg operads.We construct an extension of the well known operad of A algebras such that the representations of it in V are equivalent to an A structure on V[[ħ]]. This new operad is also a minimal model of an operad that can be seen as the extension of the operad of associative algebras by a unary operation. We give an explicit map of operads from the extended associative operad to the operad we get when applying the Merkulov-Willwacher polydifferential functor to the properad of super-involutive Lie bialgebras. Lifting this map so as to go between their respective models gives a new proof of the main theorem.

  • 18.
    Bartolini, Gabriel
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    On the Branch Loci of Moduli Spaces of Riemann Surfaces2012Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    The spaces of conformally equivalent Riemann surfaces, Mg where g ≥ 1, are not manifolds. However the spaces of the weaker Teichmüller equivalence, Tg are known to be manifolds. The Teichmüller space Tg is the universal covering of Mg and Mg is the quotient space by the action of the modular group. This gives Mg an orbifold structure with a branch locus Bg. The branch loci Bg can be identified with Riemann surfaces admitting non-trivial automorphisms for surfaces of genus g ≥ 3. In this thesis we consider the topological structure of Bg. We study the connectedness of the branch loci in general by considering families of isolated strata and we we establish that connectedness is a phenomenon for low genera. Further, we give the orbifold structure of the branch locus of surfaces of genus 4 and genus 5 in particular, by studying the equisymmetric stratification of the branch locus.

    Paper 1. In this paper we show that the strata corresponding to actions of order 2 and 3 belong to the same connected component for arbitrary genera. Further we show that the branch locus is connected with the exception of one isolated point for genera 5 and 6, it is connected for genus 7 and it is connected with the exception of two isolated points for genus 8.

    Paper 2. This paper contains a collection of results regarding components of the branch loci, some of them proved in detail in other papers. It is shown that for any integer d if p is a prime such that p > (d + 2)2, there there exist isolated strata of dimension d in the moduli space of Riemann surfaces of genus (d + 1)(p − 1)/2. It is also shown that if we consider Riemann surfaces as Klein surfaces, the branch loci are connected for every genera due to reflections.

    Paper 3. Here we consider surfaces of genus 4 and 5. Here we study the automorphism groups of Riemann surfaces of genus 4 and 5 up to topological equivalence and determine the complete structure of the equisymmetric stratification of the branch locus.

    Paper 4. In this paper we establish that the connectedness of the branch loci is a phenomenon for low genera. More precisely we prove that the only genera g where Bg is connected are g = 3, 4, 13, 17, 19, 59.

  • 19.
    Bartolini, Gabriel
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Costa, Antonio F.
    Departamento de Matematicas Fundamentales, UNED, Madrid, Spain.
    Izquierdo, Milagros
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    On isolated strata of p-gonal Riemann surfaces in the branch locus of moduli spaces2012In: Albanian Journal of Mathematics, ISSN 1930-1235, Vol. 6, 11-19 p.Article in journal (Refereed)
  • 20.
    Bartolini, Gabriel
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Costa, Antonio F.
    Departamento Matematicas Fundamentales, UNED, Madrid, Spain.
    Izquierdo, Milagros
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    On automorphisms groups of cyclic p-gonal Riemann surfaces2013In: Journal of symbolic computation, ISSN 0747-7171, E-ISSN 1095-855X, Vol. 57, 61-69 p.Article in journal (Refereed)
    Abstract [en]

    In this work we obtain the group of conformal and anticonformal automorphisms of real cyclic p-gonal Riemann surfaces, where p⩾3p⩾3 is a prime integer and the genus of the surfaces is at least (p−1)2+1(p−1)2+1. We use Fuchsian and NEC groups, and cohomology of finite groups.

  • 21.
    Bartolini, Gabriel
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Costa, Antonio F.
    Departamento de Matematicas Fundamentales, UNED, Madrid, Spain.
    Izquierdo, Milagros
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    On isolated strata of pentagonal Riemann surfaces in the branch locus of moduli spaces2012In: Contemporary Mathematics, ISSN 0271-4132, Vol. 572, 19-24 p.Article in journal (Refereed)
  • 22.
    Baum, Helga
    et al.
    Humboldt-University Berlin.
    Juhl, Andreas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Conformal Differential Geometry: Q-curvature and Conformal Holonomy2010 (ed. 1)Book (Refereed)
  • 23.
    Bergh, Daniel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Motivic classes of some classifying stacksManuscript (preprint) (Other academic)
    Abstract [en]

    We prove that the class of the classifying stack BPGLn is themultiplicative inverse of the class of the projective linear group PGL_nin the Grothendieck ring of stacks K0(Stack_k) for n = 2 and n = 3 under mild conditions on the base field k. In particular, although it is know that the multiplicativity relation {T} = {S}{PGL_n} does not hold for all PGLn-torsors T -> S, it holds for the universal PGLn-torsors for said n.

  • 24.
    Bergh, Daniel
    Stockholm University, Faculty of Science, Department of Mathematics.
    The Binomial Theorem and motivic classes of universal quasi-split toriManuscript (preprint) (Other academic)
    Abstract [en]

    Taking symmetric powers of varieties can be seen as a functor from the category of varieties to the category of varieties with an action by the symmetric group. We study a corresponding map between the Grothendieck groups of these categories. In particular, we derive a binomial formula and use it to give explicit expressions for the classes of universal quasi-split tori in the equivariant Grothendieck group of varieties.

  • 25.
    Bergh, Daniel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Functorial destackification of tame stacks with abelian stabilisersManuscript (preprint) (Other academic)
    Abstract [en]

    In this article, we study the problem of modifying smooth, algebraic stacks with finite, diagonalisable stabilisers such that their coarse spaces become smooth. The only modifications used are root stacks and blow-ups in smooth centres. If the generic stabiliser of the original stack is trivial, the canonical map from the resulting stack to its coarse space is also a root stack. Hence we can think of the process as removing stackiness from, or destackifying, a smooth stack with help of stacky blow-ups. The construction work over a general base and are functorial in the sense that they respect base change andcompositions with gerbes and smooth, stabiliser preserving maps. As applications, we indicate how this can be used for destackifying general Deligne-Mumford stacks with finite inertia in characteristic zero, and to obtain a weak factorisation theorem for such stacks. Over any field, the method can be used for desingularising locally simplicial toric varieties, without assuming the presence of toroidal structures.

  • 26.
    Bergh, Daniel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Computations in the Grothendieck Group of Stacks2012Licentiate thesis, monograph (Other academic)
    Abstract [en]

    Given an algebraic group, one may consider the class of its classifying stackin the Grothendieck group of stacks. This is an invariant studied byEkedahl. For certain connected groups, called the special groups bySerre and Grothendieck, the invariant simply gives the inverse of the class ofthe group itself. It is natural to ask whether the same is true for otherconnected groups. We investigate this for the groups PGL(2) and PGL(3) under mild restrictions on the choice of base field.In the case of PGL(2), the question turns out to have a positive answer. In the case of PGL(3), we reduce the question to the computation of the invariant for thenormaliser of a maximal torus in PGL(3). The reduction involves determiningthe class of a certain gerbe over the moduli stack of elliptic curves.

  • 27.
    Bergh, Daniel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Destackification and Motivic Classes of Stacks2014Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of three articles treating topics in the theory of algebraic stacks. The first two papers deal with motivic invariants. In the first, we show that the class of the classifying stack BPGLn is the inverse of the class of PGLn in the Grothendieck ring of stacks for n ≤ 3. This shows that the multiplicativity relation holds for the universal torsors, although it is known not to hold for torsors ingeneral for the groups PGL2 and PGL3.

    In the second paper, we introduce an exponential function which can be viewed as a generalisation of Kapranov's motivic zeta function. We use this to derive a binomial theorem for a power operation defined on the Grothendieck ring of varieties. As an application, we give an explicit expression for the motivic class of a universal quasi-split torus, which generalises a result by Rökaeus.

    The last paper treats destackification. We give an algorithm for removing stackiness from smooth, tame stacks with abelian stabilisers by repeatedly applying stacky blow-ups. As applications, we indicate how the result can be used for destackifying general Deligne–Mumford stacks in characteristic zero, and to obtain a weak factorisation theorem for such stacks.

  • 28.
    Berglund, Alexander
    Stockholm University, Faculty of Science, Department of Mathematics.
    Homological perturbation theory for algebras over operads2014In: Algebraic and Geometric Topology, ISSN 1472-2747, E-ISSN 1472-2739, Vol. 14, no 5, 2511-2548 p.Article in journal (Refereed)
    Abstract [en]

    We extend homological perturbation theory to encompass algebraic structures governed by operads and cooperads. The main difficulty is to find a suitable notion of algebra homotopy that generalizes to algebras over operads  O . To solve this problem, we introduce thick maps of  O –algebras and special thick maps that we call pseudo-derivations that serve as appropriate generalizations of algebra homotopies for the purposes of homological perturbation theory. 

       As an application, we derive explicit formulas for transferring  Ω(C) –algebra structures along contractions, where C  is any connected cooperad in chain complexes. This specializes to transfer formulas for  O ∞  –algebras for any Koszul operad O , in particular for A ∞  –,  C ∞  –,  L ∞  – and  G ∞  –algebras. A key feature is that our formulas are expressed in terms of the compact description of  Ω(C) –algebras as coderivation differentials on cofree C –coalgebras. Moreover, we get formulas not only for the transferred structure and a structure on the inclusion, but also for structures on the projection and the homotopy

  • 29.
    Berglund, Alexander
    Stockholm University, Faculty of Science, Department of Mathematics.
    Rational homotopy theory of mapping spaces via Lie theory for L-infinity algebras2015In: Homology, Homotopy and Applications, ISSN 1532-0073, E-ISSN 1532-0081, Vol. 17, no 2, 343-369 p.Article in journal (Refereed)
    Abstract [en]

    We calculate the higher homotopy groups of the Deligne–Getzler ∞-groupoid associated to a nilpotent L∞-algebra. As an application, we present a new approach to the rational homotopy theory of mapping spaces.

  • 30.
    Berglund, Alexander
    Stockholm University, Faculty of Science, Department of Mathematics.
    Shellability and the strong gcd-condition2009In: The Electronic Journal of Combinatorics, ISSN 1077-8926, Vol. 16, no 2Article in journal (Refereed)
    Abstract [en]

    Shellability is a well-known combinatorial criterion on a simplicial complex for verifying that the associated Stanley-Reisner ring k[] is Cohen-Macaulay. Anotion familiar to commutative algebraists, but which has not received as muchattention from combinatorialists as the Cohen-Macaulay property, is the notion ofa Golod ring. Recently, J¨ollenbeck introduced a criterion on simplicial complexesreminiscent of shellability, called the strong gcd-condition, and he together with theauthor proved that it implies Golodness of the associated Stanley-Reisner ring. Thetwo algebraic notions were earlier tied together by Herzog, Reiner and Welker, whoshowed that if k[∨] is sequentially Cohen-Macaulay, where ∨ is the Alexanderdual of , then k[] is Golod. In this paper, we present a combinatorial companionof this result, namely that if ∨ is (non-pure) shellable then satisfies the stronggcd-condition. Moreover, we show that all implications just mentioned are strict ingeneral but that they are equivalences if is a flag complex.

  • 31.
    Berglund, Alexander
    Stockholm University, Faculty of Science, Department of Mathematics.
    Rational homotopy theory of mapping spaces via Lie theory for L-infinity algebrasArticle in journal (Refereed)
  • 32.
    Berglund, Alexander
    Stockholm University, Faculty of Science, Department of Mathematics.
    Koszul spacesIn: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850Article in journal (Refereed)
  • 33.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Börjeson, Kaj
    Stockholm University, Faculty of Science, Department of Mathematics.
    Free loop space homology of highly connected manifolds2017In: Forum mathematicum, ISSN 0933-7741, E-ISSN 1435-5337, Vol. 29, no 1, 201-228 p.Article in journal (Refereed)
    Abstract [en]

    We calculate the homology of the free loop space of (n - 1)-connected closed manifolds of dimension at most 3 n - 2 (n >= 2), with the Chas-Sullivan loop product and loop bracket. Over a field of characteristic zero, we obtain an expression for the BV-operator. We also give explicit formulas for the Betti numbers, showing they grow exponentially. Our main tool is the connection between formality, coformality and Koszul algebras that was elucidated by the first author [6].

  • 34.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hess, Kathryn
    Homotopical Morita theory for coringsManuscript (preprint) (Other academic)
    Abstract [en]

    A coring (A,C) consists of an algebra A and a coalgebra C in the monoidal category of A-bimodules. Corings and their comodules arise naturally in the study of Hopf-Galois extensions and descent theory, as well as in the study of Hopf algebroids. In this paper, we address the question of when two corings in a symmetric monoidal model category V are homotopically Morita equivalent, i.e., when their respective categories of comodules are Quillen equivalent. The category of comodules over the trivial coring (A,A) is isomorphic to the category of A-modules, so the question above englobes that of when two algebras are homotopically Morita equivalent. We discuss this special case in the first part of the paper, extending previously known results. To approach the general question, we introduce the notion of a 'braided bimodule' and show that adjunctions between A-Mod and B-Mod that lift to adjunctions between (A,C)-Comod and (B,D)-Comod correspond precisely to braided bimodules between (A,C) and (B,D). We then give criteria, in terms of homotopic descent, for when a braided bimodule induces a Quillen equivalence. In particular, we obtain criteria for when a morphism of corings induces a Quillen equivalence, providing a homotopic generalization of results by Hovey and Strickland on Morita equivalences of Hopf algebroids. To illustrate the general theory, we examine homotopical Morita theory for corings in the category of chain complexes over a commutative ring.

  • 35.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hess, Kathryn
    Homotopic Hopf-Galois extensions revisitedManuscript (preprint) (Other academic)
    Abstract [en]

    In this article we revisit the theory of homotopic Hopf-Galois extensions introduced in arXiv:0902.3393v2 [math.AT], in light of the homotopical Morita theory of comodules established in arXiv:1411.6517 [math.AT]. We generalize the theory to a relative framework, which we believe is new even in the classical context and which is essential for treating the Hopf-Galois correspondence in forthcoming work of the second author and Karpova. We study in detail homotopic Hopf-Galois extensions of differential graded algebras over a commutative ring, for which we establish a descent-type characterization analogous to the one Rognes provided in the context of ring spectra. An interesting feature in the differential graded setting is the close relationship between homotopic Hopf-Galois theory and Koszul duality theory. We show that nice enough principal fibrations of simplicial sets give rise to homotopic Hopf-Galois extensions in the differential graded setting, for which this Koszul duality has a familiar form.

  • 36.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Madsen, Ib
    Rational homotopy theory of automorphisms of highly connected manifoldsManuscript (preprint) (Other academic)
    Abstract [en]

    We study the rational homotopy types of classifying spaces of automorphism groups of 2d-dimensional (d-1)-connected manifolds (d > 2). We prove that the rational homology groups of the homotopy automorphisms and the block diffeomorphisms of the manifold #^g S^d x S^d relative to a disk stabilize as g increases. Via a theorem of Kontsevich, we obtain the striking result that the stable rational cohomology of the homotopy automorphisms comprises all unstable rational homology groups of all outer automorphism groups of free groups.

  • 37.
    Berglund, Alexander
    et al.
    Köpenhamns universitet, Danmark.
    Madsen, Ib
    University of Copenhagen.
    Homological stability of diffeomorphism groups2013In: Pure and Applied Mathematics Quarterly, ISSN 1558-8599, Vol. 9, no 1, 1-48 p.Article in journal (Refereed)
  • 38.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Madsen, Ib
    University of Copenhagen.
    Rational homotopy theory of automorphisms of highly connected manifoldsArticle in journal (Refereed)
  • 39.
    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, 119-126 p.Chapter 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.

  • 40.
    Bergström, Jonas
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Faber, Carel
    KTH.
    van der Geer, Gerard
    UvA.
    Siegel modular forms of degree three and the cohomology of local systems2014In: Selecta Mathematica, New Series, ISSN 1022-1824, E-ISSN 1420-9020, Vol. 20, no 1, 83-124 p.Article in journal (Refereed)
    Abstract [en]

    We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space A 3   of principally polarized abelian threefolds. The main term of the formula is a conjectural motive of Siegel modular forms of a certain type; the remaining terms admit a surprisingly simple description in terms of the motivic Euler characteristics for lower genera. The conjecture is based on extensive counts of curves of genus three and abelian threefolds over finite fields. It provides a lot of new information about vector-valued Siegel modular forms of degree three, such as dimension formulas and traces of Hecke operators. We also use it to predict several lifts from genus 1 to genus 3, as well as lifts from G 2   and new congruences of Harder type.

  • 41.
    Bergström, Jonas
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Minabe, Satoshi
    Tokyo Denki University.
    On the cohomology of moduli spaces of (weighted) stable rational curves2013In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 275, no 3-4, 1095-1108 p.Article in journal (Refereed)
    Abstract [en]

    We give a recursive algorithm for computing the character of the cohomology of the moduli space M ¯ ¯ ¯ ¯   0,n   of stable n  -pointed genus zero curves as a representation of the symmetric group S n   on n  letters. Using the algorithm we can show a formula for the maximum length of this character. Our main tool is connected to the moduli spaces of weighted stable curves introduced by Hassett

  • 42.
    Bergström, Jonas
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Minabe, Satoshi
    On the cohomology of the Losev-Manin moduli space2014In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 144, no 1, 241-252 p.Article in journal (Refereed)
    Abstract [en]

    We determine the cohomology of the Losev--Manin moduli space $\overline{M}_{0, 2 | n}$ of pointed genus zero curves as a representation of the product of symmetric groups $\Sg_2 \times \Sg_n$.

  • 43.
    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, Vol. 1, no 3, 651-662 p.Article 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.

  • 44.
    Bergvall, Olof
    Stockholm University, Faculty of Science, Department of Mathematics.
    Cohomology of the moduli space of curves of genus three with level two structure2014Licentiate thesis, monograph (Other academic)
    Abstract [en]

    In this thesis we investigate the moduli space M3[2] of curves of genus 3 equipped with a symplectic level 2 structure. In particular, we are interested in the cohomology of this space. We obtain cohomological information by decomposing M3[2] into a disjoint union of two natural subspaces, Q[2] and H3[2], and then making S7- resp. S8-equivariantpoint counts of each of these spaces separately.

  • 45.
    Blaszak, Maciej
    et al.
    Adam Mickiewicz University, Poland.
    Marciniak, Krzysztof
    Linköping University, Department of Science and Technology, Communications and Transport Systems. Linköping University, Faculty of Science & Engineering.
    Domanski, Ziemowit
    Polish Academic Science, Poland.
    Separable quantizations of Stackel systems2016In: Annals of Physics, ISSN 0003-4916, E-ISSN 1096-035X, Vol. 371, 460-477 p.Article in journal (Refereed)
    Abstract [en]

    In this article we prove that many Hamiltonian systems that cannot be separably quantized in the classical approach of Robertson and Eisenhart can be separably quantized if we extend the class of admissible quantizations through a suitable choice of Riemann space adapted to the Poisson geometry of the system. Actually, in this article we prove that for every quadratic in momenta Stackel system (defined on 2n dimensional Poisson manifold) for which Stackel matrix consists of monomials in position coordinates there exist infinitely many quantizations - parametrized by n arbitrary functions - that turn this system into a quantum separable Stackel system. (C) 2016 Elsevier Inc. All rights reserved.

    The full text will be freely available from 2018-06-14 10:46
  • 46.
    Boman, Anna
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Ändliga kroppar2016Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
  • 47.
    Bourgeois, Frederic
    et al.
    ULB.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Eliashberg, Yakov
    Stanford University.
    Ganatra, Sheel
    MIT.
    Maydanskiy, Maksim
    Stanford University.
    Effect of Legendrian Surgery2012In: Geometry and Topology, ISSN 1465-3060, E-ISSN 1364-0380, Vol. 16, no 1, 301-389 p.Article in journal (Refereed)
    Abstract [en]

    The paper is a summary of the results of the authors concerning computations of symplectic invariants of Weinstein manifolds and contains some examples and applications. Proofs are sketched. The detailed proofs will appear in a forthcoming paper.

    In the Appendix written by S Ganatra and M Maydanskiy it is shown that the results of this paper imply P Seidel’s conjecture from [Proc. Sympos. Pure Math. 80, Amer. Math. Soc. (2009) 415–434].

  • 48.
    Bruhn, Linda
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics Education.
    Lindberg, Monika
    Linnaeus University, Faculty of Technology, Department of Mathematics Education.
    Geometri i förskolan: En studie av Reggio Emilia, Montessori och I Ur och Skur2014Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Syftet med studien är att undersöka olika pedagogiska inriktningars arbete med momentet geometri, för att få en inblick i vilket arbetsätt de har med former, rumsuppfattning och mätning som här representerar begreppet geometri.

    Studien beskriver de olika inriktningarnas kunskapssyn och arbetssätt som framkommit genom intervjuer av pedagoger inom Reggio Emilia, Montessori och I Ur och Skur.

    Alla de pedagogiska inriktningarna arbetade med geometri på skilda sätt utifrån sin inriktning, alla fick med de delar som här representerar geometri men kunskaperna hos pedagogerna kring geometri på djupet var skiftande och det handlar ofta mer om hur insatt pedagogen är i såväl ämnet som arbetssättet.

  • 49.
    Bujalance, Emilio
    et al.
    Departamento de Matematicas Fundamentales, UNED.
    Costa, Antonio F.
    Departamento de Matematicas Fundamentales, UNED.
    Izquierdo, Milagros
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    On Riemann surfaces with 4g automorphisms2017In: Topology and its Applications, ISSN 0166-8641, E-ISSN 1879-3207, Vol. 218, no 1, 18Article in journal (Refereed)
    Abstract [en]

    We determine, for all genus g≥2g≥2 the Riemann surfaces of genus g with exactly 4g automorphisms. For g  ≠ 3,6,12,153,6,12,15 or 30, these surfaces form a real Riemann surface FgFg in the moduli space MgMg: the Riemann sphere with three punctures. We obtain the automorphism groups and extended automorphism groups of the surfaces in the family. Furthermore we determine the topological types of the real forms of real Riemann surfaces in FgFg. The set of real Riemann surfaces in FgFg consists of three intervals its closure in the Deligne–Mumford compactification of MgMg is a closed Jordan curve. We describe the nodal surfaces that are limits of real Riemann surfaces in Fg

    The full text will be freely available from 2018-12-23 11:26
  • 50. Carlini, Enrico
    et al.
    Oneto, Alessandro
    Stockholm University, Faculty of Science, Department of Mathematics. Polytechnic University of Turin, Italy.
    Monomials as Sums of k-th Powers of Forms2015In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 43, no 2, 650-658 p.Article in journal (Refereed)
    Abstract [en]

    Motivated by recent results on the Waring problem for polynomial rings [4] and representation of monomial as sum of powers of linear forms [3], we consider the problem of presenting monomials of degree kd as sums of kth-powers of forms of degree d. We produce a general bound on the number of summands for any number of variables which we refine in the two variables case. We completely solve the k = 3 case for monomials in two and three variables.

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