Change search
Refine search result
1234 1 - 50 of 162
CiteExportLink to result list
Permanent link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Rows per page
  • 5
  • 10
  • 20
  • 50
  • 100
  • 250
Sort
  • Standard (Relevance)
  • Author A-Ö
  • Author Ö-A
  • Title A-Ö
  • Title Ö-A
  • Publication type A-Ö
  • Publication type Ö-A
  • Issued (Oldest first)
  • Issued (Newest first)
  • Created (Oldest first)
  • Created (Newest first)
  • Last updated (Oldest first)
  • Last updated (Newest first)
  • Disputation date (earliest first)
  • Disputation date (latest first)
  • Standard (Relevance)
  • Author A-Ö
  • Author Ö-A
  • Title A-Ö
  • Title Ö-A
  • Publication type A-Ö
  • Publication type Ö-A
  • Issued (Oldest first)
  • Issued (Newest first)
  • Created (Oldest first)
  • Created (Newest first)
  • Last updated (Oldest first)
  • Last updated (Newest first)
  • Disputation date (earliest first)
  • Disputation date (latest first)
Select
The maximal number of hits you can export is 250. When you want to export more records please use the Create feeds function.
  • 1.
    Abebaw, Tilahun
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics. Addis Ababa University, Ethiopia.
    Bøgvad, Rikard
    Stockholm University, Faculty of Science, Department of Mathematics.
    Decomposition factors of D-modules on hyperplane configurations in general position2012In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 140, no 8, p. 2699-2711Article in journal (Refereed)
    Abstract [en]

    Let alpha(1), ... , alpha(m) be linear functions on C-n and X = C-n \ V(alpha), where alpha = Pi(m)(i=1) alpha(i) and V(alpha) = {p is an element of C-n : alpha(p) = 0}. The coordinate ring O-X = C[x](alpha) of X is a holonomic A(n)-module, where A(n) is the n-th Weyl algebra, and since holonomic A(n)-modules have finite length, O-X has finite length. We consider a twisted variant of this A(n)-module which is also holonomic. Define M-alpha(beta) to be the free rank 1 C[x](alpha)-module on the generator alpha(beta) (thought of as a multivalued function), where alpha(beta) = alpha(beta 1)(1) ... alpha(beta m)(m) and the multi-index beta = (beta(1), ... , beta(m)) is an element of C-m. It is straightforward to describe the decomposition factors of M-alpha(beta), when the linear functions alpha(1), ... , alpha(m) define a normal crossing hyperplane configuration, and we use this to give a sufficient criterion on beta for the irreducibility of M-alpha(beta), in terms of numerical data for a resolution of the singularities of V(alpha).

  • 2. Aceto, Luca
    et al.
    Della Monica, Dario
    Goranko, Valentin
    Stockholm University, Faculty of Humanities, Department of Philosophy. University of Johannesburg, South Africa.
    Ingólfsdóttir, Anna
    Montanari, Angelo
    Sciavicco, Guido
    A complete classification of the expressiveness of interval logics of Allen’s relations: the general and the dense cases2016In: Acta Informatica, ISSN 0001-5903, E-ISSN 1432-0525, Vol. 53, no 3, p. 207-246Article in journal (Refereed)
    Abstract [en]

    Interval temporal logics take time intervals, instead of time points, as their primitive temporal entities. One of the most studied interval temporal logics is Halpern and Shoham’s modal logic of time intervals HS, which associates a modal operator with each binary relation between intervals over a linear order (the so-called Allen’s interval relations). In this paper, we compare and classify the expressiveness of all fragments of HS on the class of all linear orders and on the subclass of all dense linear orders. For each of these classes, we identify a complete set of definabilities between HS modalities, valid in that class, thus obtaining a complete classification of the family of all 4096 fragments of HS with respect to their expressiveness. We show that on the class of all linear orders there are exactly 1347 expressively different fragments of HS, while on the class of dense linear orders there are exactly 966 such expressively different fragments.

  • 3. Afshari, Bahareh
    et al.
    Enqvist, Sebastian
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    Leigh, Graham E.
    Cyclic proofs for the first-order µ-calculus2024In: Logic journal of the IGPL (Print), ISSN 1367-0751, E-ISSN 1368-9894, Vol. 32, no 1, p. 1-34Article in journal (Refereed)
    Abstract [en]

    We introduce a path-based cyclic proof system for first-order μ-calculus, the extension of first-order logic by second-order quantifiers for least and greatest fixed points of definable monotone functions. We prove soundness of the system and demonstrate it to be as expressive as the known trace-based cyclic systems of Dam and Sprenger. Furthermore, we establish cut-free completeness of our system for the fragment corresponding to the modal μ-calculus.

  • 4. Ahmed, Chwas
    et al.
    Fröberg, Ralf
    Stockholm University, Faculty of Science, Department of Mathematics.
    Rafiq Namiq, Mohammed
    The graded Betti numbers of truncation of ideals in polynomial rings2023In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 57, no 4, p. 1303-1312Article in journal (Refereed)
    Abstract [en]

    Let R=K[x1,…,xn], a graded algebra S=R/I satisfies Nk,p if I is generated in degree k, and the graded minimal resolution is linear the first p steps, and the k-index of S is the largest p such that S satisfies Nk,p. Eisenbud and Goto have shown that for any graded ring R/I, then R/I≥k, where I≥k=I∩Mk and M=(x1,…,xn), has a k-linear resolution (satisfies Nk,p for all p) if k≫0. For a squarefree monomial ideal I, we are here interested in the ideal Ik which is the squarefree part of I≥k. The ideal I is, via Stanley–Reisner correspondence, associated to a simplicial complex ΔI. In this case, all Betti numbers of R/Ik for k>min{deg(u)∣u∈I}, which of course are a much finer invariant than the index, can be determined from the Betti diagram of R/I and the f-vector of ΔI. We compare our results with the corresponding statements for I≥k. (Here I is an arbitrary graded ideal.) In this case, we show that the Betti numbers of R/I≥k can be determined from the Betti numbers of R/I and the Hilbert series of R/I≥k.

  • 5. Ahrens, Benedikt
    et al.
    Huber, Simon
    Mörtberg, Anders
    Stockholm University, Faculty of Science, Department of Mathematics.
    Preface to the MSCS Issue 31.1 (2021) Homotopy Type Theory and Univalent Foundations2021In: Mathematical Structures in Computer Science, ISSN 0960-1295, E-ISSN 1469-8072, Vol. 31, no 1, p. 1-2Article in journal (Other academic)
  • 6. Ahrens, Benedikt
    et al.
    Lumsdaine, Peter Lefanu
    Stockholm University, Faculty of Science, Department of Mathematics.
    Displayed Categories2019In: Logical Methods in Computer Science, E-ISSN 1860-5974, Vol. 15, no 1, article id 20Article in journal (Refereed)
    Abstract [en]

    We introduce and develop the notion of displayed categories. A displayed category over a category C is equivalent to 'a category D and functor F : D -> C', but instead of having a single collection of 'objects of D' with a map to the objects of C, the objects are given as a family indexed by objects of C, and similarly for the morphisms. This encapsulates a common way of building categories in practice, by starting with an existing category and adding extra data/properties to the objects and morphisms. The interest of this seemingly trivial reformulation is that various properties of functors are more naturally defined as properties of the corresponding displayed categories. Grothendieck fibrations, for example, when defined as certain functors, use equality on objects in their definition. When defined instead as certain displayed categories, no reference to equality on objects is required. Moreover, almost all examples of fibrations in nature are, in fact, categories whose standard construction can be seen as going via displayed categories. We therefore propose displayed categories as a basis for the development of fibrations in the type-theoretic setting, and similarly for various other notions whose classical definitions involve equality on objects. Besides giving a conceptual clarification of such issues, displayed categories also provide a powerful tool in computer formalisation, unifying and abstracting common constructions and proof techniques of category theory, and enabling modular reasoning about categories of multi-component structures. As such, most of the material of this article has been formalised in Coq over the UniMath library, with the aim of providing a practical library for use in further developments.

  • 7.
    Alexandersson, Per
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hopkins, Sam
    Zaimi, Gjergji
    Restricted Birkhoff Polytopes and Ehrhart Period Collapse2023In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444Article in journal (Refereed)
    Abstract [en]

    We show that the polytopes obtained from the Birkhoff polytope by imposing additional inequalities restricting the “longest increasing subsequence” have Ehrhart quasi-polynomials which are honest polynomials, even though they are just rational polytopes in general. We do this by defining a continuous, piecewise-linear bijection to a certain Gelfand–Tsetlin polytope. This bijection is not an integral equivalence but it respects lattice points in the appropriate way to imply that the two polytopes have the same Ehrhart (quasi-)polynomials. In fact, the bijection is essentially the Robinson–Schensted–Knuth correspondence.

  • 8.
    Arone, Gregory
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Barnea, Ilan
    Schlank, Tomer M.
    Suspension spectra of matrix algebras, the rank filtration, and rational noncommutative CW-spectra2023In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 386, no 1-2, p. 35-84Article in journal (Refereed)
    Abstract [en]

    In a companion paper (Arone et al. in Noncommutative CW-spectra as enriched presheaves on matrix algebras, arXiv:2101.09775, 2021) we introduced the stable ∞-category of noncommutative CW-spectra, which we denoted NSp. Let M denote the full spectrally enriched subcategory of NSp whose objects are the non-commutative suspension spectra of matrix algebras. In Arone et al. (2021) we proved that NSp is equivalent to the ∞-category of spectral presheaves on M. In this paper we investigate the structure of M, and derive some consequences regarding the structure of NSp. To begin with, we introduce a rank filtration of M. We show that the mapping spectra of M map naturally to the connective K-theory spectrum ku, and that the rank filtration of M is a lift of the classical rank filtration of ku. We describe the subquotients of the rank filtration in terms of spaces of direct-sum decompositions which also arose in the study of K-theory and of Weiss’s orthogonal calculus. We prove that the rank filtration stabilizes rationally after the first stage. Using this we give an explicit model of the rationalization of NSp as presheaves of rational spectra on the category of finite-dimensional Hilbert spaces and unitary transformations up to scaling. Our results also have consequences for the p-localization and the chromatic localization of M.

  • 9.
    Awodey, Steve
    Stockholm University, Faculty of Science, Department of Mathematics. Carnegie Mellon University, USA.
    A cubical model of homotopy type theory2016Report (Other academic)
    Download full text (pdf)
    Awodey-Stockholm-Lectures-2016
  • 10. Baba, Srinath
    et al.
    Granath, Håkan
    Stockholm University, Faculty of Science, Department of Mathematics.
    Picard-Fuchs equations for Shimura curves over Q2021In: Bulletin of the London Mathematical Society, ISSN 0024-6093, E-ISSN 1469-2120, Vol. 53, no 2, p. 408-415Article in journal (Refereed)
    Abstract [en]

    We show that Picard-Fuchs equations of periods of certain families of abelian surfaces with quaternionic multiplication have fractional powers of algebraic modular forms as solutions. We give several applications of this modularity property, including a computation of exceptional sets of certain transcendental functions.

  • 11.
    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, p. 3158-3180Article 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.

    Download full text (pdf)
    fulltext
  • 12.
    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.

    Download full text (pdf)
    fulltext
    Download (jpg)
    Omslagsframsida
  • 13.
    Balletti, Gabriele
    Stockholm University, Faculty of Science, Department of Mathematics.
    Connectivity through bounds for the Castelnuovo–Mumford regularity2017In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 147, p. 46-54Article in journal (Refereed)
    Abstract [en]

    In this note we generalize and unify two results on connectivity of graphs: one by Balinsky and Barnette, one by Athanasiadis. This is done through a simple proof using commutative algebra tools. In particular we use bounds for the Castelnuovo-Mumford regularity of their Stanley-Reisner rings. As a result, if Delta is a simplicial d-pseudomanifold and s is the largest integer such that A has a missing face of size s, then the 1-skeleton of Delta is inverted right perpendicular(s)/((s+1)d)inverted left perpendicular-connected. We also show that this value is tight.

    Download full text (pdf)
    fulltext
  • 14. Barucci, Valentina
    et al.
    Fröberg, Ralf
    Stockholm University, Faculty of Science, Department of Mathematics.
    Associated graded rings of one-dimensional analytically irreducible rings2006In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 304, no 1, p. 349-358Article in journal (Refereed)
  • 15. Bauer, Andrej
    et al.
    Gross, Jason
    LeFanu Lumsdaine, Peter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Shulman, Michael
    Sozeau, Matthieu
    Spitters, Bas
    The HoTT Library: A formalization of homotopy type theory in Coq2017In: Proceedings of the 6th ACM SIGPLAN Conference on Certified Programs and Proofs / [ed] Yves Bertot, Viktor Vafeiadis, New York, NY, USA: Association for Computing Machinery (ACM), 2017, p. 164-172Conference paper (Refereed)
    Abstract [en]

    We report on the development of the HoTT library, a formalization of homotopy type theory in the Coq proof assistant. It formalizes most of basic homotopy type theory, including univalence, higher inductive types, and significant amounts of synthetic homotopy theory, as well as category theory and modalities. The library has been used as a basis for several independent developments. We discuss the decisions that led to the design of the library, and we comment on the interaction of homotopy type theory with recently introduced features of Coq, such as universe polymorphism and private inductive types.

  • 16.
    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, p. 2511-2548Article 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

  • 17.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Börjeson, Kaj
    Stockholm University, Faculty of Science, Department of Mathematics.
    Koszul A-algebras and free loop space homologyManuscript (preprint) (Other academic)
    Abstract [en]

    We introduce a notion of Koszul A-algebra that generalizes Priddy’s notion of a Koszul algebra and we use it to construct small A- algebra models for Hochschild cochains. As an application, this yields new techniques for computing free loop space homology algebras of manifolds that are either formal or coformal (over a field or over the integers). We illustrate these techniques in two examples. 

  • 18.
    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.

  • 19.
    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.

  • 20.
    Bergström, Jonas
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Dummigan, Neil
    Farmer, David
    Koutsoliotas, Sally
    GL2 x GSp2 L-values and Hecke eigenvalue congruences2019In: Journal de Théorie des Nombres de Bordeaux, ISSN 1246-7405, E-ISSN 2118-8572, Vol. 31, no 3, p. 751-775Article in journal (Refereed)
    Abstract [en]

    We find experimental examples of congruences of Hecke eigenvalues between automorphic representations of groups such as GSp2(A), SO(4,3)(A) and SO(5,4)(A), where the prime modulus should, for various reasons, appear in the algebraic part of a critical “tensor-product” L-value associated to cuspidal automorphic representations of GL2(A) and GSp2(A). Using special techniques for evaluating L-functions with few known coefficients, we compute sufficiently good approximations to detect the anticipated prime divisors.

  • 21.
    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.

    Download full text (pdf)
    fulltext
  • 22. Boij, Mats
    et al.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    A classification of the weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms2023In: Algebra & Number Theory, ISSN 1937-0652, E-ISSN 1944-7833, Vol. 17, no 1, p. 111-126Article in journal (Refereed)
    Abstract [en]

    We use Macaulay’s inverse system to study the Hilbert series for almost complete intersections generated by uniform powers of general linear forms. This allows us to give a classification of the weak Lefschetz property for these algebras, settling a conjecture by Migliore, Miró-Roig, and Nagel.

  • 23.
    Booij, Auke
    et al.
    University of Birmingham.
    Escardó, Martín
    University of Birmingham.
    Lumsdaine, Peter LeFanu
    Stockholm University, Faculty of Science, Department of Mathematics.
    Shulman, Michael
    University of San Diego.
    Parametricity, automorphisms of the universe, and excluded middleManuscript (preprint) (Other academic)
    Abstract [en]

    It is known that one can construct non-parametric functions by assuming classical axioms. Our work is a converse to that: we prove classical axioms in dependent type theory assuming specific instances of non-parametricity. We also address the interaction between classical axioms and the existence of automorphisms of a type universe. We work over intensional Martin-Löf dependent type theory, and in some results assume further principles including function extensionality, propositional extensionality, propositional truncation, and the univalence axiom.

  • 24. Bouarroudj, Sofiane
    et al.
    Grozman, Pavel
    Lebedev, Alexei
    Leites, Dimitry
    Stockholm University, Faculty of Science, Department of Mathematics. New York University Abu Dhabi, United Arab Emirates.
    Derivations and Central Extensions of Symmetric Modular Lie Algebras and Superalgebras (with an Appendix by Andrey Krutov)2023In: Symmetry, Integrability and Geometry: Methods and Applications, E-ISSN 1815-0659, Vol. 19, article id 032Article in journal (Refereed)
    Abstract [en]

    Over algebraically closed fields of positive characteristic, for simple Lie (super)algebras, and certain Lie (super)algebras close to simple ones, with symmetric root systems (such that for each root, there is minus it of the same multiplicity) and of ranks less than or equal to 8—most needed in an approach to the classification of simple vectorial Lie superalgebras (i.e., Lie superalgebras realized by means of vector fields on a supermanifold),—we list the outer derivations and nontrivial central extensions. When the conjectural answer is clear for the infinite series, it is given for any rank. We also list the outer derivations and nontrivial central extensions of one series of non-symmetric (except when considered in characteristic 2), namely periplectic, Lie superalgebras—the one that preserves the nondegenerate symmetric odd bilinear form, and of the Lie algebras obtained from them by desuperization. We also list the outer derivations and nontrivial central extensions of an analog of the rank 2 exceptional Lie algebra discovered by Shen Guangyu. Several results indigenous to positive characteristic are of particular interest being unlike known theorems for characteristic 0, some results are, moreover, counterintuitive.

  • 25. Bouarroudj, Sofiane
    et al.
    Grozman, Pavel
    Leites, Dimitry
    Stockholm University, Faculty of Science, Department of Mathematics. New York University Abu Dhabi, United Arab Emirates.
    Deformations of Symmetric Simple Modular Lie (Super)Algebras2023In: Symmetry, Integrability and Geometry: Methods and Applications, E-ISSN 1815-0659, Vol. 19, article id 031Article in journal (Refereed)
    Abstract [en]

    We say that a Lie (super)algebra is ''symmetric'' if with every root (with respect to the maximal torus) it has the opposite root of the same multiplicity. Over algebraically closed fields of positive characteristics (up to 7 or 11, enough to formulate a general conjecture), we computed the cohomology corresponding to the infinitesimal deformations of all known simple finite-dimensional symmetric Lie (super)algebras of rank <9, except for superizations of the Lie algebras with ADE root systems, and queerified Lie algebras, considered only partly. The moduli of deformations of any Lie superalgebra constitute a supervariety. Any infinitesimal deformation given by any odd cocycleis integrable. All deformations corresponding to odd cocycles are new. Among new results are classifications of the cocycles describing deforms (results of deformations) of the 29-dimensional Brown algebra in characteristic 3, of Weisfeiler-Kac algebras and orthogonal Lie algebras without Cartan matrix in characteristic 2. Open problems: describe non-isomorphic deforms and equivalence classes of cohomology theories. Appendix: For several modular analogs of complex simple Lie algebras, and simple Lie algebras indigenous to characteristics 3 and 2, we describe the space of cohomology with trivial coefficients. We show that the natural multiplication in this space is very complicated.

  • 26.
    Brage, Jens
    Stockholm University, Faculty of Science, Department of Mathematics.
    A Natural Interpretation of Classical Proofs2006Doctoral thesis, monograph (Other academic)
    Abstract [en]

    In this thesis we use the syntactic-semantic method of constructive type theory to give meaning to classical logic, in particular Gentzen's LK.

    We interpret a derivation of a classical sequent as a derivation of a contradiction from the assumptions that the antecedent formulas are true and that the succedent formulas are false, where the concepts of truth and falsity are taken to conform to the corresponding constructive concepts, using function types to encode falsity. This representation brings LK to a manageable form that allows us to split the succedent rules into parts. In this way, every succedent rule gives rise to a natural deduction style introduction rule. These introduction rules, taken together with the antecedent rules adapted to natural deduction, yield a natural deduction calculus whose subsequent interpretation in constructive type theory gives meaning to classical logic.

    The Gentzen-Prawitz inversion principle holds for the introduction and elimination rules of the natural deduction calculus and allows for a corresponding notion of convertibility. We take the introduction rules to determine the meanings of the logical constants of classical logic and use the induced type-theoretic elimination rules to interpret the elimination rules of the natural deduction calculus. This produces an interpretation injective with respect to convertibility, contrary to an analogous translation into intuitionistic predicate logic.

    From the interpretation in constructive type theory and the interpretation of cut by explicit substitution, we derive a full precision contraction relation for a natural deduction version of LK. We use a term notation to formalize the contraction relation and the corresponding cut-elimination procedure.

    The interpretation can be read as a Brouwer-Heyting-Kolmogorov (BHK) semantics that justifies classical logic. The BHK semantics utilizes a notion of classical proof and a corresponding notion of classical truth akin to Kolmogorov's notion of pseudotruth. We also consider a second BHK semantics, more closely connected with Kolmogorov's double-negation translation.

    The first interpretation reinterprets the consequence relation while keeping the constructive interpretation of truth, whereas the second interpretation reinterprets the notion of truth while keeping the constructive interpretation of the consequence relation. The first and second interpretations act on derivations in much the same way as Plotkin's call-by-value and call-by-name continuation-passing-style translations, respectively.

    We conclude that classical logic can be given a constructive semantics by laying down introduction rules for the classical logical constants. This semantics constitutes a proof interpretation of classical logic.

    Download full text (pdf)
    FULLTEXT01
  • 27. Brunerie, Guillaume
    et al.
    Ljungström, Axel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Mörtberg, Anders
    Stockholm University, Faculty of Science, Department of Mathematics.
    Synthetic Integral Cohomology in Cubical Agda2022In: 30th EACSL Annual Conferenceon Computer Science Logic: CSL 2022, February 14–19, 2022, Göttingen, Germany (Virtual Conference) / [ed] Florin Manea; Alex Simpson, Saarbrücken/Wadern: Dagstuhl Publishing , 2022, p. 11:1-11:19Conference paper (Refereed)
    Abstract [en]

    This paper discusses the formalization of synthetic cohomology theory in a cubical extension of Agda which natively supports univalence and higher inductive types. This enables signifcant simplifcations of many proofs from Homotopy Type Theory and Univalent Foundations as steps that used to require long calculations now hold simply by computation. To this end, we give a new defnition of the group structure for cohomology with Z-coefcients, optimized for efcient computations. We also invent an optimized defnition of the cup product which allows us to give the frst complete formalization of the axioms needed to turn the integral cohomology groups into a graded commutative ring. Using this, we characterize the cohomology groups of the spheres, torus, Klein bottle and real/complex projective planes. As all proofs are constructive we can then use Cubical Agda to distinguish between spaces by computation. 

  • 28. Bulling, Nils
    et al.
    Goranko, Valentin
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    Jamroga, Wojciech
    Logics for reasoning about strategic abilities in multi-player games2015In: Models of strategic reasoning: logics, games and communities / [ed] Johan van Benthem, Sujata Ghosh, Rineke Verbrugge, Berlin: Springer, 2015, p. 93-136Chapter in book (Refereed)
    Abstract [en]

    We introduce and discuss basic concepts, ideas, and logical formalisms used for reasoning about strategic abilities in multi-player games. In particular, we present concurrent game models and the alternating time temporal logic ATL∗ and its fragment ATL. We discuss variations of the language and semantics of ATL∗ that take into account the limitations and complications arising from incomplete information, perfect or imperfect memory of players, reasoning within dynamically changing strategy contexts, or using stronger, constructive concepts of strategy. Finally, we briefly summarize some technical results regarding decision problems for some variants of ATL.

  • 29.
    Börjeson, Kaj
    Stockholm University, Faculty of Science, Department of Mathematics.
    A Algebras Derived from Associative Algebras with a Non-Derivation Differential2015In: Journal of Generalized Lie Theory and Applications, ISSN 1736-5279, E-ISSN 1736-4337, Vol. 9, no 1, article id 214Article in journal (Refereed)
    Abstract [en]

    Given an associative graded algebra equipped with a degree +1 differential delta we define an A-structure that measures the failure of delta to be a derivation. This can be seen as a non-commutative analog of generalized BV-algebras. In that spirit we introduce a notion of associative order for the operator delta and prove that it satisfies properties similar to the commutative case. In particular when it has associative order 2 the new product is a strictly associative product of degree +1 and there is compatibility between the products, similar to ordinary BV-algebras. We consider several examples of structures obtained in this way. In particular we obtain an A-structure on the bar complex of an A-algebra that is strictly associative if the original algebra is strictly associative. We also introduce strictly associative degree +1 products for any degree +1 action on a graded algebra. Moreover, an A∞-structure is constructed on the Hochschild cocomplex of an associative algebra with a non-degenerate inner product by using Connes’ B-operator.

  • 30.
    Börjeson, Kaj
    Stockholm University, Faculty of Science, Department of Mathematics.
    Free loop spaces, Koszul duality and A-infinity algebras2017Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of four papers on the topics of free loop spaces, Koszul duality and A-algebras. 

    In Paper I we consider a definition of differential operators for noncommutative algebras. This definition is inspired by the connections between differential operators of commutative algebras, L-algebras and BV-algebras. We show that the definition is reasonable by establishing results that are analoguous to results in the commutative case. As a by-product of this definition we also obtain definitions for noncommutative versions of Gerstenhaber and BV-algebras. 

    In Paper II we calculate the free loop space homology of (n-1)-connected manifolds of dimension of at least 3n-2. The Chas-Sullivan loop product and the loop bracket are calculated. Over a field of characteristic zero the BV-operator is determined as well. Explicit expressions for the Betti numbers are also established, showing that they grow exponentially. 

    In Paper III we restrict our coefficients to a field of characteristic 2. We study the Dyer-Lashof operations that exist on free loop space homology in this case. Explicit calculations are carried out for manifolds that are connected sums of products of spheres. 

    In Paper IV we extend the Koszul duality methods used in Paper II by incorporating A-algebras and A-coalgebras. This extension of Koszul duality enables us to compute free loop space homology of manifolds that are not necessarily formal and coformal. As an example we carry out the computations for a non-formal simply connected 7-manifold. 

    Download full text (pdf)
    Free loop spaces, Koszul duality and A-infinity algebras
    Download (jpg)
    Omslagsframsida
  • 31.
    Börjeson, Kaj
    Stockholm University, Faculty of Science, Department of Mathematics.
    Restricted Gerstenhaber algebra structure on the free loop homology of (Sn×Sn)#mManuscript (preprint) (Other academic)
    Abstract [en]

    We compute the 2-primary restricted Gerstenhaber algebra structure on the free loop homology of (Sn×Sn)#m. To this end we construct a small complex with an explicit retract from the Hochschild cohomology complex of the cohomology algebra. The methods involved are Koszul duality, PBW-bases and the perturbation lemma. 

  • 32. Campos, Ricardo
    et al.
    Ward, Benjamin C.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Gravity formality2018In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 331, p. 439-483Article in journal (Refereed)
    Abstract [en]

    We show that Willwacher's cyclic formality theorem can be extended to preserve natural Gravity operations on cyclic multivector fields and cyclic multidifferential operators. We express this in terms of a homotopy Gravity quasiisomorphism with explicit local formulas. For this, we develop operadic tools related to mixed complexes and cyclic homology and prove that the operad M(O )of natural operations on cyclic operators is formal and hence quasi-isomorphic to the Gravity operad.

  • 33. 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, p. 650-658Article 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.

    Download full text (pdf)
    fulltext
  • 34. Cavallo, Evan
    et al.
    Mörtberg, Anders
    Stockholm University, Faculty of Science, Department of Mathematics. Carnegie Mellon University, USA.
    Swan, Andrew W
    Unifying Cubical Models of Univalent Type Theory2020In: 28th EACSL Annual Conference on Computer Science Logic (CSL 2020) / [ed] Maribel Fernández; Anca Muscholl, Saarbrücken/Wadern: Schloss Dagstuhl – Leibniz-Zentrum für Informatik GmbH , 2020, p. 14:1-14:17Conference paper (Refereed)
    Abstract [en]

    We present a new constructive model of univalent type theory based on cubical sets. Unlike prior work on cubical models, ours depends neither on diagonal cofibrations nor connections. This is made possible by weakening the notion of fibration from the cartesian cubical set model, so that it is not necessary to assume that the diagonal on the interval is a cofibration. We have formally verified in Agda that these fibrations are closed under the type formers of cubical type theory and that the model satisfies the univalence axiom. By applying the construction in the presence of diagonal cofibrations or connections and reversals, we recover the existing cartesian and De Morgan cubical set models as special cases. Generalizing earlier work of Sattler for cubical sets with connections, we also obtain a Quillen model structure.

  • 35. Crispin Quiñonez, Veronica
    et al.
    Lundqvist, Samuel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Nenashev, Gleb
    Stockholm University, Faculty of Science, Department of Mathematics.
    On ideals generated by two generic quadratic forms in the exterior algebraManuscript (preprint) (Other academic)
    Abstract [en]

    Based on the structure theory of pairs of skew-symmetric matrices, we give a conjecture for the Hilbert series of the exterior algebra modulo the ideal generated by two generic quadratic forms. We show that the conjectured series is an upper bound in the coefficient-wise sense, and we determine a majority of the coefficients. We also conjecture that the series is equal to the series of the squarefree polynomial ring modulo the ideal generated by the squares of two generic linear forms.

  • 36.
    de Boer, Menno
    Stockholm University, Faculty of Science, Department of Mathematics.
    A Proof and Formalization of the Initiality Conjecture of Dependent Type Theory2020Licentiate thesis, monograph (Other academic)
    Abstract [en]

    In this licentiate thesis we present a proof of the initiality conjecture for Martin-Löf’s type theory with 0, 1, N, A+B, AB, AB, IdA(u,v), countable hierarchy of universes (Ui)iєN closed under these type constructors and with type of elements (ELi(a))iєN. We employ the categorical semantics of contextual categories. The proof is based on a formalization in the proof assistant Agda done by Guillaume Brunerie and the author. This work was part of a joint project with Peter LeFanu Lumsdaine and Anders Mörtberg, who are developing a separate formalization of this conjecture with respect to categories with attributes and using the proof assistant Coq over the UniMath library instead. Results from this project are planned to be published in the future.

    We start by carefully setting up the syntax and rules for the dependent type theory in question followed by an introduction to contextual categories. We then define the partial interpretation of raw syntax into a contextual category and we prove that this interpretation is total on well-formed input. By doing so, we define a functor from the term model, which is built out of the syntax, into any contextual category and we show that any two such functors are equal. This establishes that the term model is initial among contextual categories. At the end we discuss details of the formalization and future directions for research. In particular, we discuss a memory issue that arose in type checking the formalization and how it was resolved.

    Download full text (pdf)
    fulltext
  • 37.
    Ekdahl, Anna-Lena
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Elevers skilda sätt att erfara talmönster - en studie av elever i årskurs 3 och 42012Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [sv]

    Matematiken handlar i mångt och mycket om att lösa problem och se mönster. Talmönster är en viktig del inom algebran och aritmetiken och är det fenomen som jag i denna studie vill undersöka elevers uppfattningar av. Syftet med föreliggande kvalitativa studie är att skapa kunskap om elevers skilda sätt att erfara talmönster, såväl talföljder som visuella talmönster. Därutöver syftar studien till att identifiera kritiska aspekter utifrån de skilda sätt som talmönstren erfars av eleverna.

    Nio elever i årskurs 3 och 4 har intervjuats utifrån ett antal talmönster. Fenomenografin och variationsteorin utgör studiens teoretiska utgångspunkter och har använts för att analysera materialet. I analysen har förutom likheter och skillnader mellan sätten att erfara, innehållet i elevutsagorna analyserats utifrån erfarandets referentiella och strukturella aspekt.

    Resultatet av den fenomenografiska analysen har utmynnat i följande sex beskrivningskategorier:

    Jämn förflyttning, Konstant eller icke-konstant skillnad, Kombination av delar, Relation mellan vissa delar, Olika del- och helhetsstrukturer och Utöver angiven helhet. I analysen har de aspekter som eleverna fokuserat på varit vägledande för att skilja kategorierna åt och identifiera sex kritiska aspekter. En av dessa kritiska aspekter handlar om att urskilja att förhållandet mellan delarna i mönstret kan se olika ut. En annan kritisk aspekt är fråga om att kunna urskilja delarnas inbördes relation, relationernas förhållande till helheten och den icke angivna helheten. En tredje innebär att delarna behöver urskiljas samtidigt som helheten. Inte nödvändigtvis samtliga delar, men tillräckligt många för att se en regelbundenhet.

    Studiens resultat har gett didaktiska implikationer om vad eleverna i en undervisningssituation behöver ges möjlighet att urskilja för att utveckla ett mer innehållsrikt och differentierat sätt att erfara talmönster.

    Resultatet diskuteras utifrån tidigare internationella undersökningar. Det förs även en diskussion om vad studiens resultat kan tillföra och de didaktiska implikationer resultatet ger.

    Download full text (pdf)
    Talmönster årskurs 3och 4
  • 38.
    Emmenegger, Jacopo
    Stockholm University, Faculty of Science, Department of Mathematics.
    A category-theoretic version of the identity type weak factorization systemManuscript (preprint) (Other academic)
    Abstract [en]

    Gambino and Garner proved that the syntactic category of a dependent type theory with identity types can be endowed with a weak factorization system structure, called identity type weak factorization system. In this paper we consider an enrichment of Joyal's notion of tribe which allow us to prove a purely category-theoretic version of the identity type weak factorization system, thus generalizing Gambino and Garner's result. We investigate then how it relates with other well-known weak factorization systems, namely those arising from Quillen model structures on the category of topological spaces and on the category of small groupoids.

    Download full text (pdf)
    fulltext
  • 39.
    Emmenegger, Jacopo
    Stockholm University, Faculty of Science, Department of Mathematics.
    Exact completion and type-theoretic structures2019Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of four papers and is a contribution to the study of representations of extensional properties in intensional type theories using, mainly, the language and tools from category theory. Our main focus is on exact completions of categories with weak finite limits as a category-theoretic description of the setoid construction in Martin-Löf's intensional type theory.

    Paper I, which is joint work with Erik Palmgren, provides sufficient conditions for such an exact completion to produce a model of the system CETCS (Constructive Elementary Theory of the Category of Sets), a finite axiomatisation of the theory of well-pointed locally cartesian closed pretoposes with a natural numbers object and enough projectives. In particular, we use a condition inspired by Aczel's set-theoretic Fullness Axiom to obtain the local cartesian closure of an exact completion. As an application, we obtain a simple  uniform proof that the category of setoids is a model of CETCS.

    Paper II was prompted by the discovery of an overlooked issue in the characterisationof local cartesian closure for exact completions due to Carboni and Rosolini. In this paper we clarify the problem, show that their characterisation is still valid when the base category has finite limits, and provide a complete solution in the general case of a category with weak finite limits.

    In paper III we generalise the approach used in paper I to obtain the local cartesian closure of an exact completion to arbitrary categories with finite limits. We then show how this condition inspired by the Fullness Axiom naturally arises in several homotopy categories and apply this result to obtain the local cartesian closure of the exact completion of the homotopy category of spaces, thus answering a question left open by Marino Gran and Enrico Vitale.

    Finally, in paper IV we abandon the pure category-theoretic approach and instead present a type-theoretic construction, formalised in Coq, of W-types in the category of setoids from dependent W-types in the underlying intensional theory. In particular, contrary to previous approaches, this construction does not require the assumption of Uniqueness of Identity Proofs nor recursion into a type universe.

    Download full text (pdf)
    Exact completion and type-theoretic structures
    Download (jpg)
    Omslagsframsida
  • 40.
    Emmenegger, Jacopo
    Stockholm University, Faculty of Science, Department of Mathematics.
    On the local cartesian closure of exact completionsManuscript (preprint) (Other academic)
    Abstract [en]

    A characterisation of cartesian closure of exact completions as a property of the projective objects was given by Carboni and Rosolini. We show that the argument used to prove that characterisation is equivalent to the projectives being closed under binary products (equivalently, being internally projective). The property in question is the existence of weak simple products (a slight strengthening of weak exponentials) and the argument used relies on two claims: that weak simple products endow the internal logic with universal quantification, and that an exponential is the quotient of a weak exponential. We show that either these claims hold if and only if the projectives are internally projectives, which entails that Carboni and Rosolini's characterisation only applies to ex/lex completions. We then argue that this limitation depends on the universal property of weak simple products, and derive from this observation an alternative notion, which we call generalised weak simple product. We conclude by showing that existence of generalised weak simple products in the subcategory of projectives is equivalent to the cartesian closure of the exact category, thus obtaining a complete characterisation of (local) cartesian closure for exact completions of categories with weak finite limits.

    Download full text (pdf)
    fulltext
  • 41.
    Emmenegger, Jacopo
    Stockholm University, Faculty of Science, Department of Mathematics.
    The Fullness Axiom and exact completions of homotopy categoriesManuscript (preprint) (Other academic)
    Abstract [en]

    We use a category-theoretic formulation of Aczel's Fullness Axiom from Constructive Set Theory to derive the local cartesian closure of an exact completion. As an application, we prove that such a formulation is valid in the homotopy category of any model category satisfying mild requirements, thus obtaining in particular the local cartesian closure of the exact completion of topological spaces and homotopy classes of maps. Under a type-theoretic reading, these results provide a general motivation for the local cartesian closure of the category of setoids. However, results and proofs are formulated solely in the language of categories, and no knowledge of type theory or constructive set theory is required on the reader's part.

    Download full text (pdf)
    fulltext
  • 42.
    Emmenegger, Jacopo
    Stockholm University, Faculty of Science, Department of Mathematics.
    W-types in setoidsManuscript (preprint) (Other academic)
    Abstract [en]

    W-types and their categorical analogue, initial algebras for polynomial endofunctors, are an important tool in predicative systems to replace transfinite recursion on well-orderings. Current arguments to obtain W-types in quotient completions rely on assumptions, like Uniqueness of Identity Proofs, or on constructions that involve recursion into a universe, that limit their applicability to a specific setting. We present an argument, verified in Coq, that instead uses dependent W-types in the underlying type theory to construct W-types in the setoid model. The immediate advantage is to have a proof more type-theoretic in flavour, which directly uses recursion on the underlying W-type to prove initiality. Furthermore, taking place in intensional type theory and not requiring any recursion into a universe, it may be generalised to various categorical quotient completions, with the aim of finding a uniform construction of extensional W-types.

    Download full text (pdf)
    fulltext
  • 43.
    Emmenegger, Jacopo
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Palmgren, Erik
    Stockholm University, Faculty of Science, Department of Mathematics.
    Exact completion and constructive theories of setsManuscript (preprint) (Other academic)
    Abstract [en]

    In the present paper we use the theory of exact completions to study categorical properties of small setoids in Martin-Loef type theory and, more generally, of models of the Constructive Elementary Theory of the Category of Sets, in terms of properties of their subcategories of choice objects (i.e. objects satisfying the axiom of choice). Because of these intended applications, we deal with categories that lack equalisers and just have weak ones, but whose objects can be regarded as collections of global elements. In this context, we study the internal logic of the categories involved, and employ this analysis to give a sufficient condition for the local cartesian closure of an exact completion. Finally, we apply this result to show when an exact completion produces a model of CETCS.

    Download full text (pdf)
    fulltext
  • 44.
    Enqvist, Sebastian
    et al.
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    Goranko, Valentin
    Stockholm University, Faculty of Humanities, Department of Philosophy. University of Johannesburg, South Africa.
    The Temporal Logic of Coalitional Goal Assignments in Concurrent Multiplayer Games2022In: ACM Transactions on Computational Logic, ISSN 1529-3785, E-ISSN 1557-945X, Vol. 23, no 4, article id 21Article in journal (Refereed)
    Abstract [en]

    We introduce and study a natural extension of the Alternating time temporal logic ATL, called Temporal Logic of Coalitional Goal Assignments (TLCGA). It features one new and quite expressive coalitional strategic operator, called the coalitional goal assignment operator ⦉ γ ⦊, where γ is a mapping assigning to each set of players in the game its coalitional goal, formalised by a path formula of the language of TLCGA, i.e., a formula prefixed with a temporal operator X, U, or G, representing a temporalised objective for the respective coalition, describing the property of the plays on which that objective is satisfied. Then, the formula ⦉ γ ⦊ intuitively says that there is a strategy profile Σ for the grand coalition Agt such that for each coalition C, the restriction Σ |C of Σ to C is a collective strategy of C that enforces the satisfaction of its objective γ (C) in all outcome plays enabled by Σ |C.

    We establish fixpoint characterizations of the temporal goal assignments in a μ-calculus extension of TLCGA, discuss its expressiveness and illustrate it with some examples, prove bisimulation invariance and Hennessy–Milner property for it with respect to a suitably defined notion of bisimulation, construct a sound and complete axiomatic system for TLCGA, and obtain its decidability via finite model property.

  • 45. Enstad, Ulrik
    et al.
    Favre, Gabriel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Raum, Sven
    Free actions of polynomial growth Lie groups and classifiable C*-algebras2023In: Article, review/survey (Refereed)
    Abstract [en]

    We show that any free action of a connected Lie group of polynomial growth on a finite dimensional locally compact space has finite tube dimension. This is shown to imply that theassociated crossed product C*-algebra has finite nuclear dimension. As an application we showthat C*-algebras associated with certain aperiodic point sets in connected Lie groups of polynomial growth are classifiable. Examples include cut-and-project sets constructed from irreduciblelattices in products of connected nilpotent Lie groups. 

    Download full text (pdf)
    fulltext
  • 46.
    Eriksson, Anders
    Stockholm University.
    Differential operators on some classes of rings2000Doctoral thesis, comprehensive summary (Other academic)
  • 47.
    Eriksson, Helena
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Att utveckla algebraiskt tänkande genom lärandeverksamhet: En undervisningsutvecklande studie i flerspråkiga klasser i grundskolans tidigaste årskurser2021Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    The aim of this thesis is to develop and explore teaching possible to promote algebraic thinking together with young, multilingual students six to twelve years old. One underlying assumption for the aim is that algebraic thinking can be developed by students participating in learning activities that are characterized by collective mathematical reasoning on relations between quantities of positive whole and rational numbers. Two overall research questions support this work: (1) What in students work indicate algebraic thinking identified in learning activities and as experiences of algebraic thinking? (2) How can learning models manifest in learning activity, in what ways do learning models change and enhance, and which characteristics of learning actions are enabled?   

    Data was produced by interviews and from research lessons with students in lower grades in a multilingual Swedish school. The research lessons were focused on learning activity as suggested by Davydov (1990, 2008/1986), aimed at developing theoretical thinking – here algebraic thinking. They were staged in two research projects conducted as networks of learning studies. In these learning studies, the group of teachers iteratively designed and revised learning activities whereby the students could identify mathematical knowledge and collectively solve mathematical problems. 

    The findings in the articles signal that learning models were developed as rudimentary, preliminary, prototypical and finally symbolic. Rudimentary models were grounded in algebraic thinking when the students analysed problem situations and identified the problem. Preliminary and prototypical models were developed by initiating and formalising actions understood as algebraic thinking. Different tools were initiated by the students and the teachers. These tools were formalised by the students. The students used algebraic symbols and line-segments to think together when comparing different quantities (Article 2). They carried out operations using unknown quantities when reflecting on additive and multiplicative relationships (Article 3). The students also used algebraic symbols to reflect on subtraction as non-commutative (Article 3). The different tools they used interacted on different levels of generalisation (Article 1). Algebraic thinking grounded the students reflections but interacted with, for example, fractional thinking in their arguments during the development of their learning models (Article 4). The different ways of thinking interacted in arguments when developing the rudimentary, the preliminary and the prototypical models. However, in the conclusion of their collective reasoning and in the development of the symbolic learning models, these different ways of thinking were intertwined in the same arguments (Article 4).

    As a conclusion, the four articles signal that learning models including algebraic symbols developed in a learning activity can be used by newly-arrived immigrant students to reflect on structures of numbers.

    Download full text (pdf)
    Att utveckla algebraiskt tänkande genom lärandeverksamhet
    Download (jpg)
    Omslagsframsida
  • 48.
    Espic, Hadrien
    Stockholm University, Faculty of Science, Department of Mathematics.
    Koszul duality for categories and a relative Sullivan-Wilkerson theorem2022Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This PhD thesis consists in a collection of three papers on Koszul duality of categories and on an analogue of the Sullivan-Wilkerson theorem for relative CW-complexes.

    In Paper I, we define a general notion of Koszul dual in the context of a monoidal biclosed model category. We apply it to a category of enriched graphs to define the Koszul dual of an augmented enriched category C. We mostly study the case of categories enriched over a stable model category. We establish the expected adjunctions between categories of modules over C and modules over its Koszul dual K(C), and investigate the question of when the map from C to its double dual K(K(C)) is an equivalence. We also, importantly, show that Koszul duality of operads can be understood as a special case of Koszul duality of categories.

    In Paper II, we investigate further this notion of Koszul duality in the case of categories enriched over a category of chain complexes. In this setting, there is a natural cocategory structure on the bar construction on a category C. We show that the dual of this bar cocategory is equivalent to our definition of the Koszul dual of C.

    In Paper III, coauthored with Bashar Saleh, we prove more general versions of two important consequences of the Sullivan-Wilkerson theorem. Namely, we show that the group of homotopy classes of relative homotopy automorphisms of a simply connected finite CW-complex is finitely presented, and that its associated rationalization map has finite kernel.

    Download full text (pdf)
    Koszul duality for categories and a relative Sullivan-Wilkerson theorem
    Download (jpg)
    Omslagsframsida
  • 49.
    Espic, Hadrien
    Stockholm University, Faculty of Science, Department of Mathematics.
    Koszul duality for categories with a fixed object setManuscript (preprint) (Other academic)
    Abstract [en]

    We define a notion of Koszul dual of a monoid object in a monoidal biclosed model category. Our construction generalizes the classic Yoneda algebra ExtA(k,k). We apply this general construction to define the Koszul dual of a category enriched over spectra or chain complexes. This example relies on the classical observation that enriched categories are monoid objects in a category of enriched graphs. We observe that the category of enriched graphs is biclosed, meaning that it comes with both left and right internal hom objects. Given a category R (which plays the role of the ground field k in classical algebra), and an augmented R-algebra C, we define the Koszul dual of C, denoted K(C), as the R-algebra of derived endomorphisms of R in the category of right C-modules.We establish the expected adjunctions between the categories of modules over C and modules over K(C). We investigate the question of when the map from C to its double dual K(K(C)) is an equivalence. We also show that Koszul duality of operads can be understood as a special case of Koszul duality of categories. In this way we incorporate Koszul duality of operads in a wider context, and possibly clarify some aspects of it.

  • 50.
    Espic, Hadrien
    Stockholm University, Faculty of Science, Department of Mathematics.
    The Koszul dual of an augmented dg-category and the dual of its bar constructionManuscript (preprint) (Other academic)
    Abstract [en]

    We investigate a generalization of the Yoneda algebra (which gives a general notion of Koszul dual of an augmented small category) in the case where the enrichment category is a category of chain complexes. This notion is a category of derived endomorphisms with the idea of generalizing the Yoneda algebra. We recall how the bar construction on a dg-category has a natural cocategory structure. Then, given R a category enriched in chain complexes and C an augmented R-algebra, we construct a weak equivalence from the dual of the bar construction homR(BC,R) to the object of derived endomorphisms RhomC(R,R). We also obtain a result linking this to Koszul duality of algebraic operads.

1234 1 - 50 of 162
CiteExportLink to result list
Permanent link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf