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  • 1. Agerholm, Troels
    et al.
    Mazorchuk, Volodymyr
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    On selfadjoint functors satisfying polynomial relations2011In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 330, no 1, p. 448-467Article in journal (Refereed)
    Abstract [en]

    We study selfadjoint functors acting on categories of finite dimensional modules over finite dimensional algebras with an emphasis on functors satisfying some polynomial relations. Selfadjoint functors satisfying several easy relations, in particular, idempotents and square roots of a sum of identity functors. are classified. We also describe various natural constructions for new actions using external direct sums, external tensor products. Serre subcategories, quotients and centralizer subalgebras.

  • 2. Batra, Punita
    et al.
    Mazorchuk, Volodymyr
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Blocks and modules for Whittaker pairs2011In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 215, no 7, p. 1552-1568Article in journal (Refereed)
    Abstract [en]

    Inspired by recent activities on Whittaker modules over various (Lie) algebras, we describe a general framework for the study of Lie algebra modules locally finite over a subalgebra. As a special case, we obtain a very general set-up for the study of Whittaker modules, which includes, in particular, Lie algebras with triangular decomposition and simple Lie algebras of Cartan type. We describe some basic properties of Whittaker modules, including a block decomposition of the category of Whittaker modules and certain properties of simple Whittaker modules under some rather mild assumptions. We establish a connection between our general set-up and the general set-up of Harish-Chandra subalgebras in the sense of Drozd, Futorny and Ovsienko. For Lie algebras with triangular decomposition, we construct a family of simple Whittaker modules (roughly depending on the choice of a pair of weights in the dual of the Cartan subalgebra), describe their annihilators, and formulate several classification conjectures. In particular, we construct some new simple Whittaker modules for the Virasoro algebra. Finally, we construct a series of simple Whittaker modules for the Lie algebra of derivations of the polynomial algebra, and consider several finite-dimensional examples, where we study the category of Whittaker modules over solvable Lie algebras and their relation to Koszul algebras.

  • 3.
    Björklund, Johan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Real Algebraic Knots of Low Degree2011In: Journal of knot theory and its ramifications, ISSN 0218-2165, Vol. 20, no 9, p. 1285-1309Article in journal (Refereed)
    Abstract [en]

    In this paper we study rational real algebraic knots in RP3. We show that two real algebraic knots of degree d<6 are rigidly isotopic if and only if their degrees and encomplexed writhes are equal. We also show that any irreducible smooth knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree d<7. Furthermore an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented.

  • 4.
    Björklund, Johan
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Holmgren, Cecilia
    Counterexamples to a monotonicity conjecture for the threshold pebbling number2012In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 312, no 15, p. 2401-2405Article in journal (Refereed)
    Abstract [en]

    Graph pebbling considers the problem of transforming configurations of discrete pebbles to certain target configurations on the vertices of a graph, using the so-called pebbling move, in which two pebbles are removed from a vertex and one is placed on a neighbouring vertex. Given a graph G, the pebbling number pi (G) is the least t such that every initial distribution of t pebbles at the vertices of G is solvable, that is for every target vertex nu, there is some list of pebbling moves that ends with nu having a pebble. Given a graph sequence (G(n)), the pebbling threshold tau (G(n)) is a sequence (a(n)) such that t = a(n) is the smallest number of pebbles such that a random configuration of t pebbles on the vertices of G(n) is solvable with probability at least 1/2, in the probabilistic model where each configuration oft pebbles on the vertices of G(n) is selected uniformly at random. This paper provides counterexamples to the following monotonicity conjecture stated by Hurlbert et al.: If (G(n)) and (H-n) are graph sequences such that pi(G(n)) <= pi(H-n), then it holds that tau(G(n)) is an element of O(tau(H-n)). 

  • 5. Blanck, Jens
    et al.
    Stoltenberg-Hansen, Viggo
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Tucker, John V.
    Stability of representations of effective partial algebras2011In: Mathematical logic quarterly, ISSN 0942-5616, E-ISSN 1521-3870, Vol. 57, no 2, p. 217-231Article in journal (Refereed)
    Abstract [en]

    An algebra is effective if its operations are computable under some numbering. When are two numberings of an effective partial algebra equivalent? For example, the computable real numbers form an effective field and two effective numberings of the field of computable reals are equivalent if the limit operator is assumed to be computable in the numberings (theorems of Moschovakis and Hertling). To answer the question for effective algebras in general, we give a general method based on an algebraic analysis of approximations by elements of a finitely generated subalgebra. Commonly, the computable elements of a topological partial algebra are derived from such a finitely generated algebra and form a countable effective partial algebra. We apply the general results about partial algebras to the recursive reals, ultrametric algebras constructed by inverse limits, and to metric algebras in general.

  • 6. Bourgeois, Frederic
    et al.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Eliashberg, Yakov
    Symplectic homology product via Legendrian surgery2011In: Proceedings of the National Academy of Sciences of the United States of America, ISSN 0027-8424, E-ISSN 1091-6490, Vol. 108, no 20, p. 8114-8121Article in journal (Refereed)
    Abstract [en]

    This research announcement continues the study of the symplectic homology of Weinstein manifolds undertaken by the authors [Bourgeois F, Ekholm T, Eliashberg Y (2009) arXiv:0911.0026] where the symplectic homology, as a vector space, was expressed in terms of the Legendrian homology algebra of the attaching spheres of critical handles. Here, we express the product and Batalin-Vilkovisky operator of symplectic homology in that context.

  • 7.
    Cieliebak, Kai
    et al.
    LMU, Munich.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Latschev, Janko
    Univ Hamburg.
    Compactness for holomorphic curves with switching Lagrangian boundary conditions2010In: The Journal of Symplectic Geometry, ISSN 1527-5256, E-ISSN 1540-2347, Vol. 8, no 3, p. 267-298Article in journal (Refereed)
    Abstract [en]

    We prove a compactness result for holomorphic curves with boundary on an immersed Lagrangian submanifold with clean self-intersection. As an important consequence, we show that the number of intersections of such holomorphic curves with the self-intersection locus is uniformly bounded in terms of the Hofer energy.

  • 8. Coquand, Thierry
    et al.
    Palmgren, Erik
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Spitters, Bas
    Metric complements of overt closed sets2011In: Mathematical logic quarterly, ISSN 0942-5616, E-ISSN 1521-3870, Vol. 57, no 4, p. 373-378Article in journal (Refereed)
    Abstract [en]

    We show that the set of points of an overt closed subspace of a metric completion of a Bishop-locally compact metric space is located. Consequently, if the subspace is, moreover, compact, then its collection of points is Bishop-compact.

  • 9.
    Cuenca Mira, José Antonio
    et al.
    Departamento de Algebra, Geometria y Topologia, Facultad de Ciencias, Universidad de Málaga.
    Darpö, Erik
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Classification of the finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity2010In: Bulletin des Sciences Mathématiques, ISSN 0007-4497, E-ISSN 1952-4773, Vol. 134, no 3, p. 247-277Article in journal (Refereed)
    Abstract [en]

    An absolute valued algebra is a non-zero real algebra that is equipped with a multiplicative norm. We classify all finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity, up to algebra isomorphism. This completes earlier results of Ramírez Álvarez and Rochdi which, in our self-contained presentation, are recovered from the wider context of composition k-algebras with an LR-bijective idempotent.

  • 10.
    Darpo, Erik
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Herschend, Martin
    On the representation ring of the polynomial algebra over a perfect field2010In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 265, no 3, p. 601-615Article in journal (Refereed)
    Abstract [en]

    We consider the tensor product of modules over the polynomial algebra corresponding to the usual tensor product of linear operators. We present a general description of the representation ring in case the ground field k is perfect. It is made explicit in the special cases when k is real closed respectively algebraically closed. Furthermore, we discuss the generalisation of this problem to representations of quivers. In particular the representation ring of quivers of extended Dynkin type (A) over tilde is provided.

  • 11.
    Darpö, Erik
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Some modern developments in the theory of real division algebras2008Report (Other academic)
  • 12.
    Darpö, Erik
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Cuenca Mira, José Antonio
    Dieterich, Ernst
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Classification of the Finite Dimensional Absolute Valued Algebras having a Non-Zero Central Idempotent or a One-Sided Unity2009Report (Other academic)
  • 13.
    Darpö, Erik
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Hershend, Martin
    On the representation ring of the polynomial algebra over a perfect field2008Report (Other academic)
  • 14.
    Diener, Hannes
    et al.
    FB 6: Mathematik, Universität Siegen, Siegen, Germany.
    Hedin, Anton
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    The Vitali covering theorem in constructive mathematicsArticle in journal (Refereed)
    Abstract [en]

    This paper investigates the Vitali Covering Theorem from various constructive angles. A Vitali Cover of a metric space is a cover such that for every point there exists an arbitrarily small set of the cover containing this point. The VCT now states, that for any Vitali Cover one can find a finite, disjoint family of sets in the Vitali Cover that cover the entire space up to a set of a given non-zero measure. We will show, by means of a recursive counterexample, that there cannot be a fully constructive proof, but that adding a very weak semi-constructive principle suffices to give such a proof. Lastly, we will show that with an appropriate formalization in formal topology the non-constructive problems can be avoided completely.

  • 15.
    Dieterich, Ernst
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Rubinsztein, Ryszard
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    The Degree of an Eight-Dimensional Real Quadratic Division Algebra is 1, 3, or 52009Report (Other academic)
  • 16.
    Dieterich, Ernst
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Rubinsztein, Ryszard
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    The Degree of an Eight-Dimensional Real Quadratic Division Algebra is 1, 3, or 52010In: Bulletin des Sciences Mathematiques, ISSN 0007-4497, Vol. 134, no 5, p. 447-453Article in journal (Refereed)
    Abstract [en]

    A celebrated theorem of Hopf, Bott, Milnor, and Kervaire states that every finite-dimensional real division algebra has dimension 1, 2, 4, or 8. While the real division algebras of dimension 1 or 2 and the real quadratic division algebras of dimension 4 have been classified, the problem of classifying all 8-dimensional real quadratic division algebras is still open. We contribute to a solution of that problem by proving that every 8-dimensional real quadratic division algebra has degree 1, 3, or 5. This statement is sharp.

  • 17.
    Dimitroglou Rizell, Georgios
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Knotted Legendrian surfaces with few Reeb chords2010Licentiate thesis, monograph (Other academic)
  • 18.
    Dimitroglou Rizell, Georgios
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Knotted Legendrian surfaces with few Reeb chords2011In: Algebraic and Geometric Topology, ISSN 1472-2747, E-ISSN 1472-2739, Vol. 11, no 5, p. 2903-2936Article in journal (Refereed)
    Abstract [en]

    For g > 0, we construct g + 1 Legendrian embeddings of a surface of genus g into J(1)(R-2) = R-5 which lie in pairwise distinct Legendrian isotopy classes and which all have g + 1 transverse Reeb chords (g + 1 is the conjecturally minimal number of chords). Furthermore, for g of the g + 1 embeddings the Legendrian contact homology DGA does not admit any augmentation over Z(2), and hence cannot be linearized. We also investigate these surfaces from the point of view of the theory of generating families. Finally, we consider Legendrian spheres and planes in J(1)(S-2) from a similar perspective.

  • 19. Dress, A.
    et al.
    Kåhrström, Johan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Moulton, V.
    A 'Non-Additive' Characterization of p-Adic Norms2011In: Annals of Combinatorics, ISSN 0218-0006, E-ISSN 0219-3094, Vol. 15, no 1, p. 37-50Article in journal (Refereed)
    Abstract [en]

    For F a p-adic field together with a p-adic valuation, we present a new characterization for a map p: F-n -> R boolean OR {-infinity} to be a delta-adic norm on the vector space F-n. This characterization was motivated by the concept of tight maps-maps that naturally arise within the theory of valuated matroids and tight spans. As an immediate consequence, we show that the two descriptions of the affine building of SLn(F) in terms of (i) p-adic norms given by Bruhat and Tits and (ii) tight maps given by Terhalle essentially coincide. The result suggests that similar characterizations of affine buildings of other classical groups should exist, and that the theory of affine buildings may turn out as a particular case of a yet to be developed geometric theory of valuated (and delta-valuated) matroids and their tight spans providing simply-connected G-spaces for large classes of appropriately specified groups G that could serve as a basis for an affine variant of Gromov's theory.

  • 20.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    A version of rational SFT for exact Lagrangian cobordisms in 1-jet spaces2009In: New perspectives and challenges in symplectic field theory: , Providence, RI: Amer. Math. Soc. , 2009, p. 173-199Conference paper (Refereed)
  • 21.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Etnyre, John
    Department of mathematics, Georgia Tech.
    Sabloff, Josh
    Department of mathematics, Haverford College.
    A duality exact sequence for legendrian contact homology2009In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 150, no 1, p. 1-75Article in journal (Refereed)
    Abstract [en]

    We establish a long exact sequence for Legendrian submanifolds L⊂P×R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L to P off of itself. In this sequence, the singular homology H* maps to linearized contact cohomology CH*, which maps to linearized contact homology CH*, which maps to singular homology. In particular, the sequence implies a duality between Ker(CH*→H*) and CH*/Im(H*). Furthermore, this duality is compatible with Poincaré duality in L in the following sense: the Poincaré dual of a singular class which is the image of a∈CH* maps to a class α∈CH* such that α(a)=1.

    The exact sequence generalizes the duality for Legendrian knots in R3 (see [26]) and leads to a refinement of the Arnold conjecture for double points of an exact Lagrangian admitting a Legendrian lift with linearizable contact homology, first proved in [7]

  • 22.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Takase, Masamichi
    Singular Seifert surfaces and Smale invariants for a family of 3-sphere immersions2011In: Bulletin of the London Mathematical Society, ISSN 0024-6093, E-ISSN 1469-2120, Vol. 43, no 2, p. 251-266Article in journal (Refereed)
    Abstract [en]

    A self-transverse immersion of the 2-sphere into 4-space with algebraic number of self-intersection points equal to-n induces an immersion of the circle bundle over the 2-sphere of Euler class 2n into 4-space. Precomposing these circle bundle immersions with their universal covering maps, we get for n > 0 immersions g(n) of the 3-sphere into 4-space. In this note, we compute the Smale invariants of g(n). The computation is carried out by (partially) resolving the singularities of the natural singular map of the punctured complex projective plane which extends g(n). As an application, we determine the classes represented by g(n) in the cobordism group of immersions which is naturally identified with the stable 3-stem. It follows in particular that g(n) represents a generator of the stable 3-stem if and only if n is divisible by 3.

  • 23.
    Garner, Richard
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    A homotopy-theoretic universal property of Leinster’s operad for weak ω-categories2008Report (Other academic)
  • 24.
    Garner, Richard
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    On the strength of dependent products in the type theory of Martin-Löf2008Report (Other academic)
  • 25.
    Garner, Richard
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Two-dimensional models of type theory2008Report (Other academic)
  • 26.
    Garner, Richard
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Gambino, Nicola
    The identity type weak factorisation system2008Report (Other academic)
  • 27.
    Hedin, Anton
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Contributions to Pointfree Topology and Apartness Spaces2011Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    The work in this thesis contains some contributions to constructive point-free topology and the theory of apartness spaces. The first two papers deal with constructive domain theory using formal topology. In Paper I we focus on the notion of a domain representation of a formal space as a way to introduce generalized points of the represented space, whereas we in Paper II give a constructive and point-free treatment of the domain theoretic approach to differential calculus. The last two papers are of a slightly different nature but still concern constructive topology. In paper III we consider a measure theoretic covering theorem from various constructive angles in both point-set and point-free topology. We prove a point-free version of the theorem. In Paper IV we deal with issues of impredicativity in the theory of apartness spaces. We introduce a notion of set-presented apartness relation which enables a predicative treatment of basic constructions of point-set apartness spaces.

    List of papers
    1. Local Scott compactification
    Open this publication in new window or tab >>Local Scott compactification
    (English)Article in journal (Refereed) Submitted
    Abstract [en]

    We show how to embed certain formal topologies in locally Scott formal topologies. We call this process local Scott compactification. Examples include localic completions of metric spaces. We also prove a lifting result for morphisms. The local Scott compactification of a space corresponds to a domain representation of the space and in this way yields a space containing partial elements. In the case of the real numbers we obtain the space of partial reals, consisting of intervals where the endpoints are lower and upper reals respectively. We show that partial reals that define compact overt subspaces of the real numbers correspond precisely to intervals with real endpoints. The lifting of morphisms give sharp extensions of continuous real valued functions in the sense of interval analysis. These results are also generalized to normed vector spaces.

    Keywords
    Formal topologies, Continuous dcpos, Locally compact metric spaces
    National Category
    Algebra and Logic
    Research subject
    Mathematical Logic
    Identifiers
    urn:nbn:se:uu:diva-152064 (URN)
    Available from: 2011-04-22 Created: 2011-04-22 Last updated: 2011-06-14Bibliographically approved
    2. The domain theoretic derivative in formal topology
    Open this publication in new window or tab >>The domain theoretic derivative in formal topology
    (English)Article in journal (Refereed) Submitted
    Abstract [en]

    We investigate the possibility to develop constructively some of the theory of domain theoretical differential calculus by using formal topology. A formal point-free domain derivative of continuous functions on partial reals is defined and we prove that it is point-wise equal to the classical domain derivative.

    Keywords
    Formal topologies, Domains, Interval analysis, Differential calculus
    Research subject
    Mathematical Logic
    Identifiers
    urn:nbn:se:uu:diva-152065 (URN)
    Available from: 2011-04-22 Created: 2011-04-22 Last updated: 2011-06-21Bibliographically approved
    3. The Vitali covering theorem in constructive mathematics
    Open this publication in new window or tab >>The Vitali covering theorem in constructive mathematics
    (English)Article in journal (Refereed) Submitted
    Abstract [en]

    This paper investigates the Vitali Covering Theorem from various constructive angles. A Vitali Cover of a metric space is a cover such that for every point there exists an arbitrarily small set of the cover containing this point. The VCT now states, that for any Vitali Cover one can find a finite, disjoint family of sets in the Vitali Cover that cover the entire space up to a set of a given non-zero measure. We will show, by means of a recursive counterexample, that there cannot be a fully constructive proof, but that adding a very weak semi-constructive principle suffices to give such a proof. Lastly, we will show that with an appropriate formalization in formal topology the non-constructive problems can be avoided completely.

    Keywords
    Constructive mathematics, Reverse mathematics, Measure theory, Vitali's covering theorem, Formal topology
    Research subject
    Mathematical Logic
    Identifiers
    urn:nbn:se:uu:diva-152066 (URN)
    Available from: 2011-04-22 Created: 2011-04-22 Last updated: 2011-06-14Bibliographically approved
    4. Towards set-presentable apartness spaces
    Open this publication in new window or tab >>Towards set-presentable apartness spaces
    (English)Manuscript (preprint) (Other academic)
    Abstract [en]

    We investigate the theory of apartness spaces from a predicative point of view. We introduce a notion of set-presented apartness relation which enables a predicative treatment of basic constructions of point-set apartness spaces. Moreover we discuss notions of set presentation for set-set apartness space.

    Keywords
    Constructive mathematics, General topology, Neighbourhood space, Apartness space
    Research subject
    Mathematical Logic
    Identifiers
    urn:nbn:se:uu:diva-152067 (URN)
    Available from: 2011-04-22 Created: 2011-04-22 Last updated: 2011-06-14
  • 28.
    Hedin, Anton
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Local Scott Compactification2011Report (Other academic)
  • 29.
    Hedin, Anton
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Local Scott compactificationArticle in journal (Refereed)
    Abstract [en]

    We show how to embed certain formal topologies in locally Scott formal topologies. We call this process local Scott compactification. Examples include localic completions of metric spaces. We also prove a lifting result for morphisms. The local Scott compactification of a space corresponds to a domain representation of the space and in this way yields a space containing partial elements. In the case of the real numbers we obtain the space of partial reals, consisting of intervals where the endpoints are lower and upper reals respectively. We show that partial reals that define compact overt subspaces of the real numbers correspond precisely to intervals with real endpoints. The lifting of morphisms give sharp extensions of continuous real valued functions in the sense of interval analysis. These results are also generalized to normed vector spaces.

  • 30.
    Hedin, Anton
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    The Domain Theoretic Derivative in Formal Topology2011Report (Other academic)
  • 31.
    Hedin, Anton
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    The domain theoretic derivative in formal topologyArticle in journal (Refereed)
    Abstract [en]

    We investigate the possibility to develop constructively some of the theory of domain theoretical differential calculus by using formal topology. A formal point-free domain derivative of continuous functions on partial reals is defined and we prove that it is point-wise equal to the classical domain derivative.

  • 32.
    Hedin, Anton
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Towards set-presentable apartness spacesManuscript (preprint) (Other academic)
    Abstract [en]

    We investigate the theory of apartness spaces from a predicative point of view. We introduce a notion of set-presented apartness relation which enables a predicative treatment of basic constructions of point-set apartness spaces. Moreover we discuss notions of set presentation for set-set apartness space.

  • 33.
    Hedén, Isac
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Russell’s hypersurface from a geometric point of view2011Licentiate thesis, monograph (Other academic)
  • 34.
    Jacobsson, Magnus
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Rubinsztein, Ryszard L.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Symplectic topology of SU(2)-representation varieties and link homology, I: Symplectic braid action and the first Chern class2008Report (Other academic)
  • 35.
    Johansson, Jenny
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Asymptotiska egenskaper hos vissa kombinatoriska strukturer2011Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
  • 36.
    Kastberg, Robin
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    On L-induced Tn -modules2011Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
  • 37. Khovanov, Mikhail
    et al.
    Mazorchuk, Volodymyr
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Stroppel, Catharina
    A brief review of abelian categorifications2009In: Theory and Applications of Categories, ISSN 1201-561X, E-ISSN 1201-561X, Vol. 22, no 19, p. 479-508Article in journal (Refereed)
  • 38.
    Koponen, Vera
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Entropy of formulas2009In: Archive for mathematical logic, ISSN 0933-5846, E-ISSN 1432-0665, Vol. 48, no 6, p. 515-522Article in journal (Refereed)
    Abstract [en]

    A probability distribution can be given to the set of isomorphism classes of models with universe {1, ..., n} of a sentence in first-order logic. We study the entropy of this distribution and derive a result from the 0-1 law for first-order sentences.

  • 39. Kudryavtseva, Ganna
    et al.
    Mazorchuk, Volodymyr
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Partialization of categories and inverse braid-permutation monoids2008In: International journal of algebra and computation, ISSN 0218-1967, Vol. 18, no 6, p. 989-1017Article in journal (Refereed)
    Abstract [en]

    We show how one can use the partialization functor to obtain several recently defined inverse monoids, and use this functor to define new objects, which we call the inverse braid-permutation monoids. A presentation for this monoid is obtained. Finally, we study some abstract properties of the partialization functor and its iterations. This leads to a categorification of a monoid of all order-preserving maps, and series of orthodox generalizations of the symmetric inverse semigroup.

  • 40.
    Kåhrström, Johan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Kostant’s problem and parabolic subgroups2008Report (Other academic)
  • 41.
    Kåhrström, Johan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Kostant's problem and parabolic subgroups2010In: Glasgow Mathematical Journal, ISSN 0017-0895, E-ISSN 1469-509X, Vol. 52, p. 19-32Article in journal (Refereed)
    Abstract [en]

    Let g be a finite dimensional complex semi-simple Lie algebra with Weyl group W and simple reflections S. For I subset of S let g(I) be the corresponding semi-simple subalgebra of g. Denote by W-I the Weyl group of g(I) and let w(o) and w(o)(I) be the longest elements of W and W-I, respectively In this paper we show that the answer to Kostant's problem, i.e. whether the Universal enveloping algebra subjects onto the space of all ad-finite linear transformations of a given module, is the same for the simple highest weight g(I)-module L-I(x) of highest weight x . 0, x is an element of W-I, as the answer for the simple highest weight g-module L(xw(o)(l)w(o)) of highest weight xw(o)(I)w(o). 0. We also give a new description Of the unique quasi-simple quotient of the Verma module Delta(e) with the same annihilator as L(y), y is an element of W.

  • 42.
    Kåhrström, Johan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Tensoring with infinite-dimensional modules in O_02010In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 13, no 5, p. 561-587Article in journal (Refereed)
    Abstract [en]

    We show that the principal block O-0 of the BGG category O for a semi-simple Lie algebra g acts faithfully on itself via exact endofunctors which preserve tilting modules, via right exact endofunctors which preserve projective modules and via left exact endofunctors which preserve injective modules. The origin of all these functors is tensoring with arbitrary (not necessarily finite-dimensional) modules in the category O. We study such functors, describe their adjoints and show that they give rise to a natural (co) monad structure on O-0. Furthermore, all this generalises to parabolic subcategories of O-0. As an example, we present some explicit computations for the algebra sl(3).

  • 43.
    Kåhrström, Johan
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Mazorchuk, Volodymyr
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    A new approach to Kostant's problem2010In: Algebra & number theory, ISSN 1937-0652, Vol. 4, no 3, p. 231-254Article in journal (Refereed)
    Abstract [en]

    For every involution w of the symmetric group S-n we establish, in terms of a special canonical quotient of the dominant Verma module associated with w, an effective criterion to verify whether the universal enveloping algebra U(sl(n)) surjects onto the space of all ad-finite linear transformations of the simple highest weight module L(w). An easy sufficient condition derived from this criterion admits a straightforward computational check (using a computer, for example). All this is applied to get some old and many new results, which answer the classical question of Kostant in special cases; in particular we give a complete answer for simple highest weight modules in the regular block of sl(n), n <= 5.

  • 44.
    Larsson, Daniel
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Burchnall-Chaundy Theory, Ore Extensionsand σ-Differential Operators2008Report (Other academic)
  • 45.
    Larsson, Daniel
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Global and Arithmetic Hom-Lie Algebras2008Report (Other academic)
  • 46.
    Lenngren, Nils
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    k-uniform tilings by regular polygons2009Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
  • 47.
    Mazorchuk, Volodymyr
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Classification of simple q_2-supermodules2010In: Tohoku mathematical journal, ISSN 0040-8735, Vol. 62, no 3, p. 401-426Article in journal (Refereed)
    Abstract [en]

    We classify all simple supermodules over the queer Lie superalgebra q 2 up to classification of equivalence classes of irreducible elements in a certain Euclidean ring.

  • 48.
    Mazorchuk, Volodymyr
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Koszul duality for stratified algebras I. Balanced quasi-hereditary algebras2010In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 131, no 1-2, p. 1-10Article in journal (Refereed)
    Abstract [en]

    We give a complete picture of the interaction between Koszul and Ringel dualities for quasi-hereditary algebras admitting linear tilting (co)resolutions of standard and costandard modules. We show that such algebras are Koszul, that the class of these algebras is closed with respect to both dualities and that on this class these two dualities commute. All arguments reduce to short computations in the bounded derived category of graded modules.

  • 49.
    Mazorchuk, Volodymyr
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    KOSZUL DUALITY FOR STRATIFIED ALGEBRAS II. STANDARDLY STRATIFIED ALGEBRAS2010In: Journal of the Australian Mathematical Society, ISSN 1446-7887, E-ISSN 1446-8107, Vol. 89, no 1, p. 23-49Article in journal (Refereed)
    Abstract [en]

    We give a complete picture of the interaction between the Koszul and Ringel dualities for graded standardly stratified algebras (in the sense of Cline, Parshall and Scott) admitting linear tilting (co)resolutions of standard and proper costandard modules. We single out a certain class of graded standardly stratified algebras, imposing the condition that standard filtrations of projective modules are finite, and develop a tilting theory for such algebras. Under the assumption on existence of linear tilting (co)resolutions we show that algebras from this class are Koszul, that both the Ringel and Koszul duals belong to the same class, and that these two dualities on this class commute.

  • 50.
    Mazorchuk, Volodymyr
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Lectures on algebraic categorification2012Book (Refereed)
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