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• 1.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
A Gröbner basis algorithm for fast encoding of Reed-Müller codes2016Independent thesis Basic level (degree of Bachelor), 10,5 credits / 16 HE creditsStudent thesis

In this thesis the relationship between Gröbner bases and algebraic coding theory is investigated, and especially applications towards linear codes, with Reed-Müller codes as an illustrative example. We prove that each linear code can be described as a binomial ideal of a polynomial ring, and that a systematic encoding algorithm for such codes is given by the remainder of the information word computed with respect to the reduced Gröbner basis. Finally we show how to apply the representation of a code by its corresponding polynomial ring ideal to construct a class of codes containing the so called primitive Reed-Müller codes, with a few examples of this result.

• 2. Abramov, V.
Paal, E.Tallinn University of Technology.Silvestrov, Sergei D.Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.Stolin, A.Chalmers University of Techology.
Proceedings of the 3rd Baltic-Nordic Workshop “Algebra, Geometry, and Mathematical Physics”2008Conference proceedings (editor) (Refereed)
• 3.
Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
Elevers förståelse av likhetstecknet: En studie i årskurs 32016Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis

The aim of the study is to explore the understanding of the equal sign and how the teaching about the equal sign among third grade students can be varied to be as effective as possible.  The aim will be answered trough the questions: what patterns of variation can the studied students meet and what critical aspects have the students identified. Patterns of variation means that what is critical in the teaching should be varied to become visible. Critical aspects is what students need to identify to understand what should be learned. The foundation of the study is the Variation Theory, where patterns of variation and critical aspects are central concepts.

The observations were accomplished during a third grade lesson and six students were selected for interviews about the equal sign. The result of the study shows that the students met six critical aspects during the lesson. For every critical aspect there were one or several patterns of variation that was exposed to the students. The result also states that the students who were interviewed have a relational and instrumental understanding of the equal sign. The students also have understanding of a critical aspect that they did not meet in the observed lesson, namely that all numbers have to enter in a task. Four out of six students have understanding of the critical aspect that there should be equivalence in a chain of similarities. The result also show that the students understanding of the equal sign is not dependent of that they meet patterns of variation in the teaching, but that they meet the critical aspects somehow.

• 4. Aceto, Luca
Stockholm University, Faculty of Humanities, Department of Philosophy. University of Johannesburg, South Africa.
A complete classification of the expressiveness of interval logics of Allen’s relations: the general and the dense cases2016In: Acta Informatica, ISSN 0001-5903, E-ISSN 1432-0525, Vol. 53, no 3, 207-246 p.Article in journal (Refereed)

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

• 5. Aczel, Peter
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematical Logic.
Binary refinement implies discrete exponentiation2006In: Studia Logica, Vol. 84, 361-368 p.Article in journal (Refereed)
• 6.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
A zero-one law for l-colourable structures with a vectorspace pregeometry2012Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 7.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
To infinity and back: Logical limit laws and almost sure theories2014Licentiate thesis, comprehensive summary (Other academic)
• 8.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Homogenizable structures and model completeness2016In: Archive for mathematical logic, ISSN 0933-5846, E-ISSN 1432-0665, Vol. 55, no 7-8, 977-995 p.Article in journal (Refereed)

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.

• 9.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Simple structures axiomatized by almost sure theories2016In: Annals of Pure and Applied Logic, ISSN 0168-0072, E-ISSN 1873-2461, Vol. 167, no 5, 435-456 p.Article in journal (Refereed)

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.

The full text will be freely available from 2018-02-15 15:28
• 10.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
On sets with rank one in simple homogeneous structures2015In: Fundamenta Mathematicae, ISSN 0016-2736, E-ISSN 1730-6329, Vol. 228, 223-250 p.Article in journal (Refereed)

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.

• 11.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Limit laws and automorphism groups of random nonrigid structures2015In: Journal of Logic and Analysis, ISSN 1759-9008, E-ISSN 1759-9008, Vol. 7, no 2, 1-53 p., 1Article in journal (Refereed)

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.

• 12.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Limitless Analysis2014Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 13.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
A two-sample test statistic for high-dimensional multivariate data under non-normality2011Report (Other academic)

Ahmad, Ohlson, and von Rosen (2011a) present asymptotic distribution of a one-sample test statistic under non-normality, when the data are high dimensional, i.e., when the dimension of the vector, p, may exceed the sample size, n. This paper extends the case to a two-sample statistic to test the difference of mean vectors of two independent multivariate distributions, again under high-dimensional set up. Using the asymptotic theory of U-statistics, and under mild assumptions on the traces of the unknown covariance matrices, the statistic is shown to follow an approximate normal distribution when n and p are large. However, no relationship between n and p is assumed. An extension to the paired case is given, which, being essentially a one-sample statistic, supplements the asymptotic results obtained in Ahmad, Ohlson, and von Rosen (2011a).

• 14.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. National University of Rwanda, Rwanda.
THE ALGEBRA OF SEMIGROUPS OF SETS2015In: Mathematica Scandinavica, ISSN 0025-5521, Vol. 116, no 2, 161-170 p.Article in journal (Refereed)

We study the algebra of semigroups of sets (i.e. families of sets closed under finite unions) and its applications. For each n greater than 1 we produce two finite nested families of pairwise different semigroups of sets consisting of subsets of R" without the Baire property.

• 15.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
On Finite-Dimensional Absolute Valued Algebras2012Licentiate thesis, comprehensive summary (Other academic)
• 16.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Corestricted Group Actions and Eight-Dimensional Absolute Valued Algebras2012Report (Other academic)

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.

• 17.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Morphisms in the Category of Finite Dimensional Absolute Valued Algebras2011Report (Other academic)

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.

• 18.
Faculté des Sciences, Université de Sfax, Tunisia .
Université de Haute Alsace, France . Lund University.
Ternary q-Virasoro-Witt Hom-Nambu-Lie algebras2010In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, Vol. 43, no 26, 265204- p.Article in journal (Refereed)

In this paper we construct ternary q-Virasoro-Witt algebras which q-deform the ternary Virasoro-Witt algebras constructed by Curtright, Fairlie and Zachos using su(1, 1) enveloping algebra techniques. The ternary Virasoro-Witt algebras constructed by Curtright, Fairlie and Zachos depend on a parameter and are not Nambu-Lie algebras for all but finitely many values of this parameter. For the parameter values for which the ternary Virasoro-Witt algebras are Nambu-Lie, the corresponding ternary q-Virasoro-Witt algebras constructed in this paper are also Hom-Nambu-Lie because they are obtained from the ternary Nambu-Lie algebras using the composition method. For other parameter values this composition method does not yield a Hom-Nambu-Lie algebra structure for q-Virasoro-Witt algebras. We show however, using a different construction, that the ternary Virasoro-Witt algebras of Curtright, Fairlie and Zachos, as well as the general ternary q-Virasoro-Witt algebras we construct, carry a structure of the ternary Hom-Nambu-Lie algebra for all values of the involved parameters.

• 19.
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematical Logic.
On second-order generalized quantifiers and finite structures2002In: Ann. Pure Appl. Logic, Vol. 115, no 1-3, 1-32 p.Article in journal (Refereed)
• 20.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Random sampling of finite graphs with constraints2014Independent thesis Advanced level (degree of Master (One Year)), 20 credits / 30 HE creditsStudent thesis
• 21.
Umeå University.
Umeå University. Umeå University.
Fast multiplication of matrices over a finitely generated semiring2008In: Information Processing Letters, ISSN 0020-0190, E-ISSN 1872-6119, Vol. 107, no 6, 230-234 p.Article in journal (Refereed)
• 22. Armstrong, Alasdair
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computing Science.
Programming and automating mathematics in the Tarski-Kleene hierarchy2014In: Journal of Logical and Algebraic Methods in Programming, ISSN 2352-2208, Vol. 83, no 2, 87-102 p.Article in journal (Refereed)

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.

• 23.
Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden .
Representation theory of C  -algebras for a higher-order class of spheres and tori2008In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 49, 053502-1-053502-13 p., 053502Article in journal (Refereed)

We construct C  -algebras for a class of surfaces that are inverse images of certain polynomials of arbitrary degree. By using the directed graph associated with a matrix, the representation theory can be understood in terms of “loop” and “string” representations, which are closely related to the dynamics of an iterated map in the plane. As a particular class of algebras, we introduce the “Hénon algebras,” for which the dynamical map is a generalized Hénon map, and give an example where irreducible representations of all dimensions exist.

• 24.
Institut des Hautes Études Scientifiques, Le Bois-Marie 35, route de Chartres, F-91440, Bures-sur-Yvette, France och Max Planck Institute for Gravitational Physics, Am Mühlenberg 1, D-14476, Golm, Germany .
Laboratoire de MIA, 4, rue des Frères Lumière, Université de Haute-Alsace, F-68093, Mulhouse, France . Université du Luxembourg, FSTC 162a, avenue de la Faïencerie, L-1511, Luxembourg City, Luxembourg . Department of Mathematics, KTH, S-10044, Stockholm, Sweden . Max Planck Institute for Gravitational Physics, Am Mühlenberg 1, D-14476, Golm, Germany .
Noncommutative Riemann Surfaces by Embeddings in R32009In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 288, 403-429 p.Article in journal (Refereed)

We introduce C-Algebras of compact Riemann surfaces Σ  as non-commutative analogues of the Poisson algebra of smooth functions on Σ  . Representations of these algebras give rise to sequences of matrix-algebras for which matrix-commutators converge to Poisson-brackets as N → ∞. For a particular class of surfaces, interpolating between spheres and tori, we completely characterize (even for the intermediate singular surface) all finite dimensional representations of the corresponding C-algebras

• 25.
Department of Mathematics, Royal Institute of Technology, Lindstedtsvägen 25 S-10044 Stockholm, Sweden.
Laboratoire de MIA, Université de Haute-Alsace, 4, rue des Frères Lumière, F-68093 Mulhouse, France. Department of Mathematics, Royal Institute of Technology, Lindstedtsvägen 25 S-10044 Stockholm, Sweden och Laboratoire de MIA, Université de Haute-Alsace, 4, rue des Frères Lumière, F-68093 Mulhouse, France. Department of Mathematics, Royal Institute of Technology, Lindstedtsvägen 25 S-10044 Stockholm, Sweden. Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Am Mühlenberg 1 D-14476 Golm., Germany.
Fuzzy Riemann surfaces2009In: Journal of High Energy Physics (JHEP), ISSN 1126-6708, E-ISSN 1029-8479, Vol. 2009, no 024, 1-17 p.Article in journal (Refereed)

We introduce C-Algebras (quantum analogues of compact Riemann surfaces), defined by polynomial relations in non-commutative variables and containing a real parameter that, when taken to zero, provides a classical non-linear, Poisson-bracket, obtainable from a single polynomial C(onstraint) function. For a continuous class of quartic constraints, we explicitly work out finite dimensional representations of the corresponding C-Algebras.

• 26.
Université de Haute-Alsace, Mulhouse, France. Université de Haute-Alsace, Mulhouse, France . Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Structure and Cohomology of 3-Lie Algebras Induced by Lie Algebras2014In: Springer Proceedings in Mathematics and Statistics, Berlin, Heidelberg: Springer, 2014, Vol. 85, 123-144 p.Conference paper (Refereed)

The aim of this paper is to compare the structure and the cohomology spaces of Lie algebras and induced 3-Lie algebras

• 27.
Max Planck Institute for Gravitational Physics (AEI), Am Mühlenberg 1, D-14476 Golm, Germany.
Université de Haute Alsace, Laboratoire de Mathématiques, Informatique et Applications, 4, rue des Frères Lumière F-68093 Mulhouse, France . Mälardalen University, Division of Applied Mathematics, The School of Education, Culture and Communication, Box 883, 721 23 Västerås, Sweden och Centre for Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden .
Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras2011In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 52, 123502Article in journal (Refereed)

As n-ary operations, generalizing Lie and Poisson algebras, arise in many different physical contexts, it is interesting to study general ways of constructing explicit realizations of such multilinear structures. Generically, they describe the dynamics of a physical system, and there is a need of understanding their quantization. Hom-Nambu-Lie algebras provide a framework that might be an appropriate setting in which n-Lie algebras (n-ary Nambu-Lie algebras) can be deformed, and their quantization studied. We present a procedure to construct (n + 1)-ary Hom-Nambu-Lie algebras from n-ary Hom-Nambu-Lie algebras equipped with a generalized trace function. It turns out that the implications of the compatibility conditions, that are necessary for this construction, can be understood in terms of the kernel of the trace function and the range of the twisting maps. Furthermore, we investigate the possibility of defining (n + k)-Lie algebras from n-Lie algebras and a k-form satisfying certain conditions.

• 28.
Max Planck Institute for Gravitational Physics (AEI), Am Mühlenberg 1, D-14476 Golm, Germany.
Laboratoire de Mathématiques, Informatique et Applications, Université de Haute Alsace, 4, rue des Frères Lumière, F-68093 Mulhouse, France . Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund, Sweden .
Ternary Hom–Nambu–Lie algebras induced by Hom–Lie algebras2010In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 51, 043515-1-043515-11 p., 43515Article in journal (Refereed)

The need to consider n  -ary algebraic structures, generalizing Lie and Poisson algebras, has become increasingly important in physics, and it should therefore be of interest to study the mathematical concepts related to n  -ary algebras. The purpose of this paper is to investigate ternary multiplications (as deformations of n  -Lie structures) constructed from the binary multiplication of a Hom–Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions. We show that the relation between the kernels of the twisting maps and the trace function plays an important role in this context and provide examples of Hom–Nambu–Lie algebras obtained using this construction.

• 29.
Max Planck Institute for Gravitational Physics (AEI), Am Mühlenberg 1, D-14476 Golm, Germany.
Université de Haute Alsace, Lab. de Mathématiques Informatique et Applications, 4, rue des Frères Lumière, F-68093 Mulhouse, France. Mälardalen University, School of Education, Culture and Communication.
Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras2011In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 52, no 12, 123502- p.Article in journal (Refereed)

As n-ary operations, generalizing Lie and Poisson algebras, arise in many different physical contexts, it is interesting to study general ways of constructing explicit realizations of such multilinear structures. Generically, they describe the dynamics of a physical system, and there is a need of understanding their quantization. Hom-Nambu-Lie algebras provide a framework that might be an appropriate setting in which n-Lie algebras (n-ary Nambu-Lie algebras) can be deformed, and their quantization studied. We present a procedure to construct (n + 1)-ary Hom-Nambu-Lie algebras from n-ary Hom-Nambu-Lie algebras equipped with a generalized trace function. It turns out that the implications of the compatibility conditions, that are necessary for this construction, can be understood in terms of the kernel of the trace function and the range of the twisting maps. Furthermore, we investigate the possibility of defining (n + k)-Lie algebras from n-Lie algebras and a k-form satisfying certain conditions. (C) 2011 American Institute of Physics. [doi:10.1063/1.3653197]

• 30.
Max Planck Institute for Gravitational Physics (AEI), Germany.
Université de Haute Alsace, France . Lund University, Sweden.
Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras2010In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 51, no 4, 043515-11 p.Article in journal (Refereed)

The need to consider n-ary algebraic structures, generalizing Lie and Poisson algebras, has become increasingly important in physics, and it should therefore be of interest to study the mathematical concepts related to n-ary algebras. The purpose of this paper is to investigate ternary multiplications (as deformations of n-Lie structures) constructed from the binary multiplication of a Hom-Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions. We show that the relation between the kernels of the twisting maps and the trace function plays an important role in this context and provide examples of Hom-Nambu-Lie algebras obtained using this construction.

• 31.
Albert Einstein Institute, Golm, Germany..
Centre for Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden .
Affine transformation crossed product type algebras and noncommutative surfaces2009In: Operator structures and dynamical systems :: July 21-25 2008, Lorentz Center, Leiden, the Netherlands, satellite conference of the fifth European Congress of Mathematics, American Mathematical Society (AMS), 2009, 503, 1-25 p.Chapter in book (Refereed)

Several classes of *-algebras associated to teh action of an affine transformation are considered, and an investigation of the interplay between the different classes is initiated. Connections are established that relate representations of *-algebras, geometry of algebraic surfaces, dynamics of affine transformations, graphs and algebras coming from a quantization procedure of Poisson structures. In particular, algebras related to surgaced being inverse images of fourth order polynomials (in $\mathbb{R}^3$) are studied in detail, and a close link between representation theory and geometric properties is established for compact as well as non-compact surfaces.

• 32.
Albert Einstein Institute, Golm, Germany.
Lund University.
Affine transformation crossed product type algebras and noncommutative surfaces2009In: Operator structures and dynamical systems: July 21-25 2008, Lorentz Center, Leiden, The Netherlands, satellite conference of the fifth European Congress of Mathematics, Amer. Math. Soc. , 2009, Vol. 503, 1-25 p.Chapter in book (Refereed)
• 33.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Linking and Morse Theory2014Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 34.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Extremal Hypergraphs2013Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 35.
Universit́e de Haute Alsace, France .
Universit́e de Haute Alsace, France . Lund University.
Generalization of n-ary Nambu algebras and beyond2009In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 50, no 8, Article number 083501- p.Article in journal (Refereed)

The aim of this paper is to introduce n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type including n-ary Nambu algebras, n-ary Nambu-Lie algebras and n-ary Lie algebras, and n-ary algebras of associative type including n-ary totally associative and n-ary partially associative algebras. We provide examples of the new structures and present some properties and construction theorems. We describe the general method allowing one to obtain an n-ary Hom-algebra structure starting from an n-ary algebra and an n-ary algebra endomorphism. Several examples are derived using this process. Also we initiate investigation of classification problems for algebraic structures introduced in the article and describe all ternary three-dimensional Hom-Nambu-Lie structures with diagonal homomorphism.

• 36.
Department of Philosophy and Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A..
Department of Mathematics, University of Pittsburgh, Pennsylvania, U.S.A.. Institute for Advanced Study, Princeton, New Jersey, U.S.A..
Homotopy limits in type theory2015In: Mathematical Structures in Computer Science, ISSN 0960-1295, E-ISSN 1469-8072, Vol. 25, no 5, 1040-1070 p.Article in journal (Refereed)

Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant. We discuss some of the challenges posed by this approach to formalizing homotopy-theoretic material. We also compare our constructions with the more classical approach to homotopy limits via fibration categories.

• 37.
Stockholm University, Faculty of Science, Department of Mathematics. Carnegie Mellon University, USA.
A cubical model of homotopy type theory2016Report (Other academic)
• 38. Awodey, Steve
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Ultrasheaves and double negation2004In: Notre Dame Journal of Formal Logic, ISSN 0029-4527, Vol. 45, no 4, 235-245 p.Article in journal (Refereed)

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.

• 39.
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematical Logic.
Ultrasheaves and double negation2002Report (Other scientific)
• 40.
Champlain College.
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.
Quaternionic modular forms and exceptional sets of hypergeometric functions2015In: International Journal of Number Theory, ISSN 1793-0421, Vol. 11, no 2, 631-643 p.Article in journal (Refereed)

We determine the exceptional sets of hypergeometric functions corresponding to the(2, 4, 6) triangle group by relating them to values of certain quaternionic modular formsat CM points. We prove a result on the number fields generated by exceptional values, and by using modular polynomials we explicitly compute some examples.

• 41.
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
On a class of power ideals2015In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 219, no 8, 3158-3180 p.Article in journal (Refereed)

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

• 42.
Stockholm University, Faculty of Science, Department of Mathematics.
Configuration spaces, props and wheel-free deformation quantization2016Doctoral thesis, monograph (Other academic)

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

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

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

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

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

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

• 43.
Stockholm University, Faculty of Science, Department of Mathematics.
Connectivity through bounds for the Castelnuovo–Mumford regularity2017In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 147, 46-54 p.Article in journal (Refereed)

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

• 44.
Linnaeus University, Faculty of Technology, Department of Mathematics.
Diagonal Ranks of Semigroups2013Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

We introduce the notion of diagonal ranks of semigroups,which are numerical characteristics of semigroups. Some base propertiesof diagonal ranks are obtained. A new criterion for a monoidbeing a group is obtained using diagonal ranks.For some semigroup classes we investigate whether their diagonal acts are finitely generatedor not. For the semigroups of full transformations, partial transformations andbinary relations we find the general form of the generating pairs.

• 45.
University of Ljubljana.
Massachusetts Institute of Technology. Stockholm University, Faculty of Science, Department of Mathematics. University of San Diego. INRIA Paris. Aarhus University.
The HoTT Library: A formalization of homotopy type theory in Coq2017In: CPP 2017: Proceedings of the 6th ACM SIGPLAN Conference on Certified Programs and Proofs, New York, NY, USA: Association for Computing Machinery (ACM), 2017Conference paper (Refereed)

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

• 46.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Formal plethories2014In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 254, 497-569 p.Article in journal (Refereed)

Unstable operations in a generalized cohomology theory E give rise to a functor from the category of algebras over E* to itself which is a colimit of representable functors and a comonoid with respect to composition of such functors. In this paper I set up a framework for studying the algebra of such functors, which I call formal plethories, in the case where E* is a Prüfer ring. I show that the "logarithmic" functors of primitives and indecomposables give linear approximations of formal plethories by bimonoids in the 2-monoidal category of bimodules over a ring.

• 47.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Om talet Π2016Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 48.
Stockholm University, Faculty of Science, Department of Mathematics.
Homological perturbation theory for algebras over operads2014In: Algebraic and Geometric Topology, ISSN 1472-2747, E-ISSN 1472-2739, Vol. 14, no 5, 2511-2548 p.Article in journal (Refereed)

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

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

• 49.
Stockholm University, Faculty of Science, Department of Mathematics.
Homotopical Morita theory for coringsManuscript (preprint) (Other academic)

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

• 50.
Stockholm University, Faculty of Science, Department of Mathematics.
Homotopic Hopf-Galois extensions revisitedManuscript (preprint) (Other academic)

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

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