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  • 1. Aczel, Peter
    et al.
    Crosilla, Laura
    Ishihara, Hajime
    Palmgren, Erik
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematisk logik.
    Schuster, Peter
    Binary refinement implies discrete exponentiation2006Ingår i: Studia Logica, Vol. 84, s. 361-368Artikel i tidskrift (Refereegranskat)
  • 2.
    Ahlman, Ove
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    A zero-one law for l-colourable structures with a vectorspace pregeometry2012Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)
  • 3.
    Ahlman, Ove
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Homogenizable structures and model completeness2016Ingår i: Archive for mathematical logic, ISSN 0933-5846, E-ISSN 1432-0665, Vol. 55, nr 7-8, s. 977-995Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A homogenizable structure M is a structure where we may add a finite amount of new relational symbols to represent some 0-definable relations in order to make the structure homogeneous. In this article we will divide the homogenizable structures into different classes which categorize many known examples and show what makes each class important. We will show that model completeness is vital for the relation between a structure and the amalgamation bases of its age and give a necessary and sufficient condition for an countably categorical model-complete structure to be homogenizable.

  • 4.
    Ahlman, Ove
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    >k-homogeneous infinite graphs2018Ingår i: Journal of combinatorial theory. Series B (Print), ISSN 0095-8956, E-ISSN 1096-0902, Vol. 128, s. 160-174Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this article we give an explicit classification for the countably infinite graphs G which are, for some k, ≥k-homogeneous. It turns out that a ≥k  -homogeneous graph M is non-homogeneous if and only if it is either not 1-homogeneous or not 2-homogeneous, both cases which may be classified using ramsey theory.

  • 5.
    Ahlman, Ove
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Limit Laws, Homogenizable Structures and Their Connections2018Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    This thesis is in the field of mathematical logic and especially model theory. The thesis contain six papers where the common theme is the Rado graph R. Some of the interesting abstract properties of R are that it is simple, homogeneous (and thus countably categorical), has SU-rank 1 and trivial dependence. The Rado graph is possible to generate in a probabilistic way. If we let K be the set of all finite graphs then we obtain R as the structure which satisfy all properties which hold with assymptotic probability 1 in K. On the other hand, since the Rado graph is homogeneous, it is also possible to generate it as a Fraïssé-limit of its age.

    Paper I studies the binary structures which are simple, countably categorical, with SU-rank 1 and trivial algebraic closure. The main theorem shows that these structures are all possible to generate using a similar probabilistic method which is used to generate the Rado graph. Paper II looks at the simple homogeneous structures in general and give certain technical results on the subsets of SU-rank 1.

    Paper III considers the set K consisting of all colourable structures with a definable pregeometry and shows that there is a 0-1 law and almost surely a unique definable colouring. When generating the Rado graph we almost surely have only rigid structures in K. Paper IV studies what happens if the structures in K are only the non-rigid finite structures. We deduce that the limit structures essentially try to stay as rigid as possible, given the restriction, and that we in general get a limit law but not a 0-1 law.

    Paper V looks at the Rado graph's close cousin the random t-partite graph and notices that this structure is not homogeneous but almost homogeneous. Rather we may just add a definable binary predicate, which hold for any two elemenets which are in the same part, in order to make it homogeneous. This property is called being homogenizable and in Paper V we do a general study of homogenizable structures. Paper VI conducts a special case study of the homogenizable graphs which are the closest to being homogeneous, providing an explicit classification of these graphs.

    Delarbeten
    1. Simple structures axiomatized by almost sure theories
    Öppna denna publikation i ny flik eller fönster >>Simple structures axiomatized by almost sure theories
    2016 (Engelska)Ingår i: Annals of Pure and Applied Logic, ISSN 0168-0072, E-ISSN 1873-2461, Vol. 167, nr 5, s. 435-456Artikel i tidskrift (Refereegranskat) Published
    Abstract [en]

    In this article we give a classification of the binary, simple, ω-categorical structures with SU-rank 1 and trivial algebraic closure. This is done both by showing that they satisfy certain extension properties, but also by noting that they may be approximated by the almost sure theory of some sets of finite structures equipped with a probability measure. This study give results about general almost sure theories, but also considers certain attributes which, if they are almost surely true, generate almost sure theories with very specific properties such as ω-stability or strong minimality.

    Nyckelord
    Random structure, Almost sure theory, Pregeometry, Supersimple, Countably categorical
    Nationell ämneskategori
    Algebra och logik
    Forskningsämne
    Matematik
    Identifikatorer
    urn:nbn:se:uu:diva-276995 (URN)10.1016/j.apal.2016.02.001 (DOI)000372680500001 ()
    Tillgänglig från: 2016-03-01 Skapad: 2016-02-17 Senast uppdaterad: 2017-11-30Bibliografiskt granskad
    2. On sets with rank one in simple homogeneous structures
    Öppna denna publikation i ny flik eller fönster >>On sets with rank one in simple homogeneous structures
    2015 (Engelska)Ingår i: Fundamenta Mathematicae, ISSN 0016-2736, E-ISSN 1730-6329, Vol. 228, s. 223-250Artikel i tidskrift (Refereegranskat) Published
    Abstract [en]

    We study definable sets D of SU-rank 1 in Meq, where M is a countable homogeneous and simple structure in a language with finite relational vocabulary. Each such D can be seen as a 'canonically embedded structure', which inherits all relations on D which are definable in Meq, and has no other definable relations. Our results imply that if no relation symbol of the language of M has arity higher than 2, then there is a close relationship between triviality of dependence and D being a reduct of a binary random structure. Somewhat more precisely: (a) if for every n≥2, every n-type p(x1,...,xn) which is realized in D is determined by its sub-2-types q(xi,xj)⊆p, then the algebraic closure restricted to D is trivial; (b) if M has trivial dependence, then D is a reduct of a binary random structure.

    Nyckelord
    model theory, homogeneous structure, simple theory, pregeometry, rank, reduct, random structure
    Nationell ämneskategori
    Algebra och logik
    Forskningsämne
    Matematik
    Identifikatorer
    urn:nbn:se:uu:diva-243006 (URN)10.4064/fm228-3-2 (DOI)000352858400002 ()
    Tillgänglig från: 2015-02-03 Skapad: 2015-02-03 Senast uppdaterad: 2017-12-05Bibliografiskt granskad
    3. Random l-colourable structures with a pregeometry
    Öppna denna publikation i ny flik eller fönster >>Random l-colourable structures with a pregeometry
    2017 (Engelska)Ingår i: Mathematical logic quarterly, ISSN 0942-5616, E-ISSN 1521-3870, Vol. 63, nr 1-2, s. 32-58Artikel i tidskrift (Refereegranskat) Published
    Abstract [en]

    We study finite -colourable structures with an underlying pregeometry. The probability measure that is usedcorresponds to a process of generating such structures by which colours are first randomly assigned to all1-dimensional subspaces and then relationships are assigned in such a way that the colouring conditions aresatisfied but apart from this in a random way. We can then ask what the probability is that the resulting structure,where we now forget the specific colouring of the generating process, has a given property. With this measurewe get the following results: (1) A zero-one law. (2) The set of sentences with asymptotic probability 1 has anexplicit axiomatisation which is presented. (3) There is a formula ξ (x, y) (not directly speaking about colours)such that, with asymptotic probability 1, the relation “there is an -colouring which assigns the same colourto x and y” is defined by ξ (x, y). (4) With asymptotic probability 1, an -colourable structure has a unique-colouring (up to permutation of the colours).

    Ort, förlag, år, upplaga, sidor
    Wiley-VCH Verlagsgesellschaft, 2017
    Nationell ämneskategori
    Algebra och logik
    Forskningsämne
    Matematisk logik
    Identifikatorer
    urn:nbn:se:uu:diva-321515 (URN)10.1002/malq.201500006 (DOI)000400361900003 ()
    Tillgänglig från: 2017-05-06 Skapad: 2017-05-06 Senast uppdaterad: 2017-11-28Bibliografiskt granskad
    4. Limit laws and automorphism groups of random nonrigid structures
    Öppna denna publikation i ny flik eller fönster >>Limit laws and automorphism groups of random nonrigid structures
    2015 (Engelska)Ingår i: Journal of Logic and Analysis, ISSN 1759-9008, E-ISSN 1759-9008, Vol. 7, nr 2, s. 1-53, artikel-id 1Artikel i tidskrift (Refereegranskat) Published
    Abstract [en]

    A systematic study is made, for an arbitrary finite relational language with at least one symbol of arity at least 2, of classes of nonrigid finite structures. The well known results that almost all finite structures are rigid and that the class of finite structures has a zero-one law are, in the present context, the first layer in a hierarchy of classes of finite structures with increasingly more complex automorphism groups. Such a hierarchy can be defined in more than one way. For example, the kth level of the hierarchy can consist of all structures having at least k elements which are moved by some automorphism. Or we can consider, for any finite group G, all finite structures M such that G is a subgroup of the group of automorphisms of M; in this case the "hierarchy" is a partial order. In both cases, as well as variants of them, each "level" satisfies a logical limit law, but not a zero-one law (unless k = 0 or G is trivial). Moreover, the number of (labelled or unlabelled) n-element structures in one place of the hierarchy divided by the number of n-element structures in another place always converges to a rational number or to infinity as n -> infinity. All instances of the respective result are proved by an essentially uniform argument.

    Nyckelord
    finite model theory, limit law, zero-one law, random structure, automorphism group
    Nationell ämneskategori
    Algebra och logik
    Forskningsämne
    Matematisk logik
    Identifikatorer
    urn:nbn:se:uu:diva-248078 (URN)10.4115/jla.2015.7.2 (DOI)000359802400001 ()
    Tillgänglig från: 2015-03-26 Skapad: 2015-03-26 Senast uppdaterad: 2017-12-04Bibliografiskt granskad
    5. Homogenizable structures and model completeness
    Öppna denna publikation i ny flik eller fönster >>Homogenizable structures and model completeness
    2016 (Engelska)Ingår i: Archive for mathematical logic, ISSN 0933-5846, E-ISSN 1432-0665, Vol. 55, nr 7-8, s. 977-995Artikel i tidskrift (Refereegranskat) Published
    Abstract [en]

    A homogenizable structure M is a structure where we may add a finite amount of new relational symbols to represent some 0-definable relations in order to make the structure homogeneous. In this article we will divide the homogenizable structures into different classes which categorize many known examples and show what makes each class important. We will show that model completeness is vital for the relation between a structure and the amalgamation bases of its age and give a necessary and sufficient condition for an countably categorical model-complete structure to be homogenizable.

    Nyckelord
    Homogenizable, Model-complete, Amalgamation class, Quantifier-elimination
    Nationell ämneskategori
    Algebra och logik
    Forskningsämne
    Matematik
    Identifikatorer
    urn:nbn:se:uu:diva-303714 (URN)10.1007/s00153-016-0507-6 (DOI)000385155700010 ()
    Tillgänglig från: 2016-09-22 Skapad: 2016-09-22 Senast uppdaterad: 2017-11-28Bibliografiskt granskad
    6. >k-homogeneous infinite graphs
    Öppna denna publikation i ny flik eller fönster >>>k-homogeneous infinite graphs
    2018 (Engelska)Ingår i: Journal of combinatorial theory. Series B (Print), ISSN 0095-8956, E-ISSN 1096-0902, Vol. 128, s. 160-174Artikel i tidskrift (Refereegranskat) Published
    Abstract [en]

    In this article we give an explicit classification for the countably infinite graphs G which are, for some k, ≥k-homogeneous. It turns out that a ≥k  -homogeneous graph M is non-homogeneous if and only if it is either not 1-homogeneous or not 2-homogeneous, both cases which may be classified using ramsey theory.

    Nyckelord
    >k-homomogeneous, countably infinite graph
    Nationell ämneskategori
    Algebra och logik
    Forskningsämne
    Matematisk logik
    Identifikatorer
    urn:nbn:se:uu:diva-329362 (URN)10.1016/j.jctb.2017.08.007 (DOI)000417771100009 ()
    Tillgänglig från: 2017-09-26 Skapad: 2017-09-26 Senast uppdaterad: 2018-04-05Bibliografiskt granskad
  • 6.
    Ahlman, Ove
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Simple structures axiomatized by almost sure theories2016Ingår i: Annals of Pure and Applied Logic, ISSN 0168-0072, E-ISSN 1873-2461, Vol. 167, nr 5, s. 435-456Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this article we give a classification of the binary, simple, ω-categorical structures with SU-rank 1 and trivial algebraic closure. This is done both by showing that they satisfy certain extension properties, but also by noting that they may be approximated by the almost sure theory of some sets of finite structures equipped with a probability measure. This study give results about general almost sure theories, but also considers certain attributes which, if they are almost surely true, generate almost sure theories with very specific properties such as ω-stability or strong minimality.

  • 7.
    Ahlman, Ove
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    To infinity and back: Logical limit laws and almost sure theories2014Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
  • 8.
    Ahlman, Ove
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Koponen, Vera
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Limit laws and automorphism groups of random nonrigid structures2015Ingår i: Journal of Logic and Analysis, ISSN 1759-9008, E-ISSN 1759-9008, Vol. 7, nr 2, s. 1-53, artikel-id 1Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A systematic study is made, for an arbitrary finite relational language with at least one symbol of arity at least 2, of classes of nonrigid finite structures. The well known results that almost all finite structures are rigid and that the class of finite structures has a zero-one law are, in the present context, the first layer in a hierarchy of classes of finite structures with increasingly more complex automorphism groups. Such a hierarchy can be defined in more than one way. For example, the kth level of the hierarchy can consist of all structures having at least k elements which are moved by some automorphism. Or we can consider, for any finite group G, all finite structures M such that G is a subgroup of the group of automorphisms of M; in this case the "hierarchy" is a partial order. In both cases, as well as variants of them, each "level" satisfies a logical limit law, but not a zero-one law (unless k = 0 or G is trivial). Moreover, the number of (labelled or unlabelled) n-element structures in one place of the hierarchy divided by the number of n-element structures in another place always converges to a rational number or to infinity as n -> infinity. All instances of the respective result are proved by an essentially uniform argument.

  • 9.
    Ahlman, Ove
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Koponen, Vera
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    On sets with rank one in simple homogeneous structures2015Ingår i: Fundamenta Mathematicae, ISSN 0016-2736, E-ISSN 1730-6329, Vol. 228, s. 223-250Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study definable sets D of SU-rank 1 in Meq, where M is a countable homogeneous and simple structure in a language with finite relational vocabulary. Each such D can be seen as a 'canonically embedded structure', which inherits all relations on D which are definable in Meq, and has no other definable relations. Our results imply that if no relation symbol of the language of M has arity higher than 2, then there is a close relationship between triviality of dependence and D being a reduct of a binary random structure. Somewhat more precisely: (a) if for every n≥2, every n-type p(x1,...,xn) which is realized in D is determined by its sub-2-types q(xi,xj)⊆p, then the algebraic closure restricted to D is trivial; (b) if M has trivial dependence, then D is a reduct of a binary random structure.

  • 10.
    Ahlman, Ove
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Koponen, Vera
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Random l-colourable structures with a pregeometry2017Ingår i: Mathematical logic quarterly, ISSN 0942-5616, E-ISSN 1521-3870, Vol. 63, nr 1-2, s. 32-58Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study finite -colourable structures with an underlying pregeometry. The probability measure that is usedcorresponds to a process of generating such structures by which colours are first randomly assigned to all1-dimensional subspaces and then relationships are assigned in such a way that the colouring conditions aresatisfied but apart from this in a random way. We can then ask what the probability is that the resulting structure,where we now forget the specific colouring of the generating process, has a given property. With this measurewe get the following results: (1) A zero-one law. (2) The set of sentences with asymptotic probability 1 has anexplicit axiomatisation which is presented. (3) There is a formula ξ (x, y) (not directly speaking about colours)such that, with asymptotic probability 1, the relation “there is an -colouring which assigns the same colourto x and y” is defined by ξ (x, y). (4) With asymptotic probability 1, an -colourable structure has a unique-colouring (up to permutation of the colours).

  • 11.
    Ahlsén, Daniel
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Classifying Categories: The Jordan-Hölder and Krull-Schmidt-Remak Theorems for Abelian Categories2018Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)
  • 12.
    Ahlsén, Daniel
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Limitless Analysis2014Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
  • 13.
    Al Homsi, Rania
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Equation Solving in Indian Mathematics2018Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
  • 14.
    Alsaody, Seidon
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Corestricted Group Actions and Eight-Dimensional Absolute Valued Algebras2012Rapport (Övrigt vetenskapligt)
    Abstract [en]

    A condition for when two eight-dimensional absolute valued algebras are isomorphic was given in [4]. We use this condition to deduce a description (in the sense of Dieterich, [9]) of the category of such algebras, and show how previous descriptions of some full subcategories fit in this description. Led by the structure of these examples, we aim at systematically constructing new subcategories whose classification is manageable. To this end we propose, in greater generality, the definition of sharp stabilizers for group actions, and use these to obtain conditions for when certain subcategories of groupoids are full. This we apply to the category of eight-dimensional absolute valued algebras and obtain a class of subcategories, for which we simplify, and partially solve, the classification problem.

  • 15.
    Alsaody, Seidon
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Morphisms in the Category of Finite Dimensional Absolute Valued Algebras2011Rapport (Övrigt vetenskapligt)
    Abstract [en]

    This is a study of morphisms in the category of nite dimensional absolute valued algebras, whose codomains have dimension four. We begin by citing and transferring a classication of an equivalent category. Thereafter, we give a complete description of morphisms from one-dimensional algebras, partly via solutions of real polynomials, and a complete, explicit description of morphisms from two-dimensional algebras. We then give an account of the reducibility of the morphisms, and for the morphisms from two-dimensional algebras we describe the orbits under the actions of the automorphism groups involved. Parts of these descriptions rely on a suitable choice of a cross-section of four-dimensional absolute valued algebras, and we thus end by providing an explicit means of transferring these results to algebras outside this crosssection.

  • 16.
    Alsaody, Seidon
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    On Finite-Dimensional Absolute Valued Algebras2012Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
  • 17.
    Alsätra, Tova
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Knots, Reidemeister Moves and Knot Invariants2019Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
  • 18.
    Andersson, Anders
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematisk logik.
    On second-order generalized quantifiers and finite structures2002Ingår i: Ann. Pure Appl. Logic, Vol. 115, nr 1-3, s. 1-32Artikel i tidskrift (Refereegranskat)
  • 19.
    Andersson, Evelina
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Random sampling of finite graphs with constraints2014Självständigt arbete på avancerad nivå (magisterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)
  • 20. Armstrong, Alasdair
    et al.
    Struth, Georg
    Weber, Tjark
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Datalogi.
    Programming and automating mathematics in the Tarski-Kleene hierarchy2014Ingår i: Journal of Logical and Algebraic Methods in Programming, ISSN 2352-2208, Vol. 83, nr 2, s. 87-102Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present examples from a reference implementation of variants of Kleene algebras and Tarski's relation algebras in the theorem proving environment Isabelle/HOL. For Kleene algebras we show how models can be programmed, including sets of traces and paths, languages, binary relations, max-plus and min-plus algebras, matrices, formal power series. For relation algebras we discuss primarily proof automation in a comprehensive library and present an advanced formalisation example. 

  • 21.
    Asplund, Johan
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Linking and Morse Theory2014Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
  • 22.
    Asplund, Teo
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Extremal Hypergraphs2013Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
  • 23.
    Awodey, Steve
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematisk logik.
    Eliasson, Jonas
    Ultrasheaves and double negation2002Rapport (Övrigt vetenskapligt)
  • 24. Awodey, Steve
    et al.
    Eliasson, Jonas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Ultrasheaves and double negation2004Ingår i: Notre Dame Journal of Formal Logic, ISSN 0029-4527, E-ISSN 1939-0726, Vol. 45, nr 4, s. 235-245Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Moerdijk has introduced a topos of sheaves on a category of filters. Following his suggestion, we prove that its double negation subtopos is the topos of sheaves on the subcategory of ultrafilters—the ultrasheaves. We then use this result to establish a double negation translation of results between the topos of ultrasheaves and the topos on filters.

  • 25.
    Berg, Sandra
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Om talet Π2016Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
  • 26.
    Berglund, Björn
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Arkimedes metod2017Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
  • 27.
    Bjerkeng van Keppel, Alvar
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    The Unprovability of the Continuum Hypothesis Using the Method of Forcing2016Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
  • 28.
    Björklund, Johan
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Flexible isotopy classification of flexible links2016Ingår i: Journal of knot theory and its ramifications, ISSN 0218-2165, Vol. 25, nr 8, artikel-id 1650049Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper, we define and study flexible links and flexible isotopy in RP3 subset of CP3. Flexible links are meant to capture the topological properties of real algebraic links. We classify all flexible links up to flexible isotopy using Ekholm's interpretation of Viro's encomplexed writhe.

  • 29.
    Blanck, J
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematisk logik.
    Stoltenberg-Hansen, V
    Tucker, J.V.
    Domain representations of partial functions, with applications to spatial objects and contructive volume geometry2001Rapport (Övrigt vetenskapligt)
  • 30.
    Blanck, Jens
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematisk logik.
    Stoltenberg-Hansen, Viggo
    Tucker, John V.
    Domain representations of partial functions, with applications to spatial objects and constructive volume geometry2002Ingår i: Theoret. Comput. Sci., Vol. 284, nr 2, s. 207-240Artikel i tidskrift (Refereegranskat)
  • 31.
    Block, Max
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Undecidability of finite satisfiability and characterization of NP in finite model theory2015Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
  • 32.
    Blomqvist, Christer
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Miniräknarmania: eller hur att få ut så mycket som möjligt från en enkel reklamräknare2014Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
  • 33.
    Bäck, Viktor
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Localization of Multiscale Screened Poisson Equation2012Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
  • 34.
    Carlbaum, Wilhelm
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Primtal och kryptografi2016Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
  • 35.
    Chen, Chih-Whi
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri. Xiamen Univ, Sch Math Sci, Xiamen 361005, Fujian, Peoples R China.
    Lam, Ngau
    Natl Cheng Kung Univ, Dept Math, Tainan 70101, Taiwan.
    Projective modules over classical Lie algebras of infinite rank in the parabolic category2020Ingår i: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 224, nr 1, s. 125-148Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study the truncation functors and show the existence of projective cover with a finite Verma flag of each irreducible module in parabolic BGG category O over infinite rank Lie algebra of types a, b, c, d. Moreover, O is a Koszul category. As a consequence, the corresponding parabolic BGG category (O) over bar over infinite rank Lie superalgebra of types a, b, c, d through the super duality is also a Koszul category. 

  • 36.
    Chen, Chih-Whi
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri. Natl Ctr Theoret Sci, Math Div, Taipei 10617, Taiwan.
    Peng, Yung-Ning
    Natl Cent Univ, Dept Math, Chungli 32054, Taiwan..
    Affine periplectic Brauer algebras2018Ingår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 501, s. 345-372Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We formulate Nazarov-Wenzl type algebras (P) over cap (-)(d) for the representation theory of the periplectic Lie superalgebras p(n). We establish an Arakawa-Suzuki type functor to provide a connection between p(n)-representations and (P) over cap (-)(d)-representations. We also consider various tensor product representations for (P) over cap (-)(d). The periplectic Brauer algebra A(d) developed by Moon is a quotient of (P) over cap (-)(d). In particular, actions induced by Jucys-Murphy elements can also be recovered under the tensor product representation of (P) over cap (-)(d). Moreover, a Poincare-Birkhoff-Witt type basis for (P) over cap (-)(d) is obtained. A diagram realization of PI is also obtained.

  • 37.
    Chen, Gang
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Fysiska sektionen, Institutionen för fysik och astronomi, Teoretisk fysik. Zhejiang Normal Univ, Dept Phys, Jinhua, Zhejiang, Peoples R China;Queen Mary Univ London, Sch Phys & Astron, Ctr Res String Theory, Mile End Rd, London E1 4NS, England.
    Johansson, Henrik
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Fysiska sektionen, Institutionen för fysik och astronomi, Teoretisk fysik. Stockholm Univ, NORDITA, Roslagstullsbacken 23, S-10691 Stockholm, Sweden;KTH Royal Inst Technol, Roslagstullsbacken 23, S-10691 Stockholm, Sweden.
    Teng, Fei
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Fysiska sektionen, Institutionen för fysik och astronomi, Teoretisk fysik.
    Wang, Tianheng
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Fysiska sektionen, Institutionen för fysik och astronomi, Teoretisk fysik. Nanjing Univ, Dept Phys, Nanjing, Jiangsu, Peoples R China;Humboldt Univ, Inst Phys, Zum Grossen Windkanal 6, D-12489 Berlin, Germany;Humboldt Univ, IRIS Adlershof, Zum Grossen Windkanal 6, D-12489 Berlin, Germany.
    On the kinematic algebra for BCJ numerators beyond the MHV sector2019Ingår i: Journal of High Energy Physics (JHEP), ISSN 1126-6708, E-ISSN 1029-8479, nr 11, artikel-id 55Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The duality between color and kinematics present in scattering amplitudes of Yang-Mills theory strongly suggests the existence of a hidden kinematic Lie algebra that controls the gauge theory. While associated BCJ numerators are known on closed forms to any multiplicity at tree level, the kinematic algebra has only been partially explored for the simplest of four-dimensional amplitudes: up to the MHV sector. In this paper we introduce a framework that allows us to characterize the algebra beyond the MHV sector. This allows us to both constrain some of the ambiguities of the kinematic algebra, and better control the generalized gauge freedom that is associated with the BCJ numerators. Specifically, in this paper, we work in dimension-agnostic notation and determine the kinematic algebra valid up to certain ? ((epsilon i .epsilon j )(2)) terms that in four dimensions compute the next-to-MHV sector involving two scalars. The kinematic algebra in this sector is simple, given that we introduce tensor currents that generalize standard Yang-Mills vector currents. These tensor currents control the generalized gauge freedom, allowing us to generate multiple different versions of BCJ numerators from the same kinematic algebra. The framework should generalize to other sectors in Yang-Mills theory.

  • 38. Coquand, Thiery
    et al.
    Palmgren, Erik
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematisk logik.
    Metric boolean algebras and constructive measure theory2002Ingår i: Arch. Math. Logic, Vol. 41, s. 687-704Artikel i tidskrift (Refereegranskat)
  • 39.
    Coulembier, Kevin
    et al.
    Univ Ghent, Dept Math Anal, Krijgslaan 281, B-9000 Ghent, Belgium..
    Mazorchuk, Volodymyr
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Primitive ideals, twisting functors and star actions for classical Lie superalgebras2016Ingår i: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, Vol. 718, s. 207-253Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study three related topics in representation theory of classical Lie superalgebras. The first one is classification of primitive ideals, i.e. annihilator ideals of simple modules, and inclusions between them. The second topic concerns Arkhipov's twisting functors on the BGG category O. The third topic addresses deformed orbits of the Weyl group. These take over the role of the usual Weyl group orbits for Lie algebras, in the study of primitive ideals and twisting functors for Lie superalgebras.

  • 40.
    Coulembier, Kevin
    et al.
    Univ Sydney, Sch Math & Stat, Sydney, NSW 2006, Australia.
    Mazorchuk, Volodymyr
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    The G-Centre and Gradable Derived Equivalences2018Ingår i: Journal of the Australian Mathematical Society, ISSN 1446-7887, E-ISSN 1446-8107, Vol. 105, nr 3, s. 289-315Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We propose a generalisation for the notion of the centre of an algebra in the setup of algebras graded by an arbitrary abelian group G. Our generalisation, which we call the G-centre, is designed to control the endomorphism category of the grading shift functors. We show that the G-centre is preserved by gradable derived equivalences given by tilting modules. We also discuss links with existing notions in superalgebra theory.

  • 41.
    Coulembier, Kevin
    et al.
    Univ Sydney, Sch Math & Stat, Sydney, NSW 2006, Australia.
    Mazorchuk, Volodymyr
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Zhang, Xiaoting
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Indecomposable manipulations with simple modules in category O2019Ingår i: Mathematical Research Letters, ISSN 1073-2780, E-ISSN 1945-001X, Vol. 26, nr 2, s. 447-499Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study the problem of indecomposability of translations of simple modules in the principal block of BGG category O for sl(n), as conjectured in [KiM1]. We describe some general techniques and prove a few general results which may be applied to study various special cases of this problem. We apply our results to verify indecomposability for n <= 6. We also study the problem of indecomposability of shufflings and twistings of simple modules and obtain some partial results.

  • 42.
    Courte, Sylvain
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Legendrian submanifolds with Hamiltonian isotopic symplectizations2016Ingår i: Algebraic and Geometric Topology, ISSN 1472-2747, E-ISSN 1472-2739, Vol. 16, nr 6, s. 3641-3652Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In any closed contact manifold of dimension at least 1 1, we construct examples of closed Legendrian submanifolds which are not diffeomorphic but whose Lagrangian cylinders in the symplectization are Hamiltonian isotopic.

  • 43.
    Crispin Quiñonez, Veronica
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Minimal reductions of monomial ideals in dimension two2016Rapport (Refereegranskat)
  • 44.
    Crispin Quiñonez, Veronica
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Lundqvist, Samuel
    Stockholms universitet.
    Nenashev, Gleb
    Massachusetts Institute of Technology.
    On ideals generated by two generic quadratic forms in the exterior algebra2019Ingår i: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 223, nr 12, s. 5067-5082Artikel i tidskrift (Refereegranskat)
    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.

  • 45.
    Dahlgren, Fredrik
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Effective Distribution Theory2007Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    In this thesis we introduce and study a notion of effectivity (or computability) for test functions and for distributions. This is done using the theory of effective (Scott-Ershov) domains and effective domain representations.

    To be able to construct effective domain representations of the spaces of test functions considered in distribution theory we need to develop the theory of admissible domain representations over countable pseudobases. This is done in the first paper of the thesis. To construct an effective domain representation of the space of distributions, we introduce and develop a notion of partial continuous function on domains. This is done in the second paper of the thesis. In the third paper we apply the results from the first two papers to develop an effective theory of distributions using effective domains. We prove that the vector space operations on each space, as well as the standard embeddings into the space of distributions effectivise. We also prove that the Fourier transform (as well as its inverse) on the space of tempered distributions is effective. Finally, we show how to use convolution to compute primitives on the space of distributions. In the last paper we investigate the effective properties of a structure theorem for the space of distributions with compact support. We show that each of the four characterisations of the class of compactly supported distributions in the structure theorem gives rise to an effective domain representation of the space. We then use effective reductions (and Turing-reductions) to study the reducibility properties of these four representations. We prove that three of the four representations are effectively equivalent, and furthermore, that all four representations are Turing-equivalent. Finally, we consider a similar structure theorem for the space of distributions supported at 0.

    Delarbeten
    1. Admissible Domain Representations of Inductive Limit Spaces
    Öppna denna publikation i ny flik eller fönster >>Admissible Domain Representations of Inductive Limit Spaces
    Ingår i: Mathematical Structures in Computer ScienceArtikel i tidskrift (Refereegranskat) Submitted
    Identifikatorer
    urn:nbn:se:uu:diva-96205 (URN)
    Tillgänglig från: 2007-09-21 Skapad: 2007-09-21Bibliografiskt granskad
    2. Partial Continuous Functions and Admissible Domain Representations
    Öppna denna publikation i ny flik eller fönster >>Partial Continuous Functions and Admissible Domain Representations
    2007 (Engelska)Ingår i: Journal of logic and computation (Print), ISSN 0955-792X, E-ISSN 1465-363X, Vol. 17, nr 6, s. 1063-1081Artikel i tidskrift (Refereegranskat) Published
    Abstract [en]

    It is well known that to be able to represent continuous functions between domain representable spaces it is critical that the domain representations of the spaces we consider are dense. In this article we show how to develop a representation theory over a category of domains with morphisms partial continuous functions. The raison detre for introducing partial continuous functions is that by passing to partial maps, we are free to consider totalities which are not dense. We show that the category of admissibly representable spaces with morphisms functions which are representable by it partial continuous function is Cartesian closed. Finally, we consider the question of effectivity.

    Nationell ämneskategori
    Matematik
    Identifikatorer
    urn:nbn:se:uu:diva-96206 (URN)10.1093/logcom/exm034 (DOI)000252665100004 ()
    Tillgänglig från: 2007-09-21 Skapad: 2007-09-21 Senast uppdaterad: 2017-12-14Bibliografiskt granskad
    3. A Domain Theoretic Approach to Effective Distribution Theory
    Öppna denna publikation i ny flik eller fönster >>A Domain Theoretic Approach to Effective Distribution Theory
    Manuskript (Övrigt vetenskapligt)
    Identifikatorer
    urn:nbn:se:uu:diva-96207 (URN)
    Tillgänglig från: 2007-09-21 Skapad: 2007-09-21 Senast uppdaterad: 2010-01-13Bibliografiskt granskad
    4. Effective Structure Theorems for Spaces of Compactly Supported Distributions
    Öppna denna publikation i ny flik eller fönster >>Effective Structure Theorems for Spaces of Compactly Supported Distributions
    Manuskript (Övrigt vetenskapligt)
    Identifikatorer
    urn:nbn:se:uu:diva-96208 (URN)
    Tillgänglig från: 2007-09-21 Skapad: 2007-09-21 Senast uppdaterad: 2010-01-13Bibliografiskt granskad
  • 46.
    Dahlgren, Fredrik
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematisk logik.
    Partial continuous functions and admissible domain representations: Extended abstract2006Ingår i: Logical approaches to computational barriers: Proceedings / [ed] Beckmann A; Berger U; Lowe B; Tucker JV, 2006, Vol. 3988, s. 94-104Konferensbidrag (Refereegranskat)
    Abstract [en]

    It is well known that to be able to represent continuous functions between domain representable spaces it is critical that the domain representations of the spaces we consider are dense. In this article we show how to develop a representation theory over a category of domains with morphisms partial continuous functions. The reason for introducing partial continuous functions is that by passing to partial maps, we are free to consider totalities which are not dense. We show that there is a natural subcategory of the category of representable spaces with morphisms representable maps which is Cartesian closed. Finally, we consider the question of effectivity.

  • 47.
    Darpö, Erik
    et al.
    University of Oxford.
    Dieterich, Ernst
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    The double sign of a real division algebra of finite dimension greater than one2011Rapport (Övrigt vetenskapligt)
    Abstract [en]

    For any real division algebra A of finite dimension greater than one, the signs of the determinants of left multiplication and right multiplication by an element a E A \ {0} are shown to form an invariant of A, called its double sign. For each n E {2, 4, 8}, the double sign causes the category Dn of all n-dimensional real division algebras to decompose into four blocks. The structures of these blocks are closely related, and their relationship is made precise for a sample of full subcategories of Dn.

  • 48. Darpö, Erik
    et al.
    Dieterich, Ernst
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    The double sign of a real division algebra of finite dimension greater than one2012Ingår i: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 285, nr 13, s. 1635-1642Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    For any real division algebra A of finite dimension greater than one, the signs of the determinants of left multiplication and right multiplication by an element a is an element of A\{0} are shown to form an invariant of A, called its double sign. For each n is an element of {2, 4, 8}, the double sign causes the category D-n of all n-dimensional real division algebras to decompose into four blocks. The structures of these blocks are closely related, and their relationship is made precise for a sample of full subcategories of D-n.

  • 49.
    Dieterich, Ernst
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    A general approach to finite dimensional division algebras2011Rapport (Övrigt vetenskapligt)
    Abstract [en]

    We present a short and rather self-contained introduction to the theory of finite dimensional division algebras, setting out from the basic definitions and leading up to recent results and current directions of research. In sections 2-3 we develop the general theory over an arbitrary ground field k, with emphasis on the trichotomy of fields imposed by the dimensions in which a division algebra exists, the groupoid structure of the level subcategories Dn(k), and the role played by the irreducible morphisms. Sections 4-5 deal with the classical case of real division algebras, emphasizing the double sign decomposition of the level subcategories Dn(R) for n E {2, 4, 8} and the problem of describing their blocks, along with an account of known partial solutions to this problem.

  • 50.
    Dieterich, Ernst
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    A general approach to finite dimensional division algebras2012Ingår i: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 126, nr 1, s. 73-86Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present a short and rather self-contained introduction to the theory of finite-dimensional division algebras, setting out from the basic definitions and leading up to recent results and current directions of research. In Sections 2-3 we develop the general theory over an arbitrary ground field k, with emphasis on the trichotomy of fields imposed by the dimensions in which a division algebra exists, the groupoid structure of the level subcategories D-n(k), and the role played by the irreducible morphisms. Sections 4-5 deal with the classical case of real division algebras, emphasizing the double sign decomposition of the level subcategories D-n(R) for n is an element of {2, 4, 8} and the problem of describing their blocks, along with an account of known partial solutions to this problem. 

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