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

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

• 2.
Stockholm University, Faculty of Science, Department of Mathematics.
Universal algebraic structures on polyvector fields2014Doctoral thesis, monograph (Other academic)

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

• 3.
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, p. 3158-3180Article 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.

• 4.
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.

• 5.
Stockholm University, Faculty of Science, Department of Mathematics.
Computations in the Grothendieck Group of Stacks2012Licentiate thesis, monograph (Other academic)

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

• 6.
Stockholm University, Faculty of Science, Department of Mathematics.
Destackification and Motivic Classes of Stacks2014Doctoral thesis, comprehensive summary (Other academic)

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

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

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

• 7.
Stockholm University, Faculty of Science, Department of Mathematics.
Functorial destackification of tame stacks with abelian stabilisersManuscript (preprint) (Other academic)

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

• 8.
Stockholm University, Faculty of Science, Department of Mathematics. Max Planck Institute for Mathematics, Germany.
Motivic classes of some classifying stacks2016In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 93, no 1, p. 219-243Article in journal (Refereed)

We prove that the class of the classifying stack BPGL(n) is the multiplicative inverse of the class of the projective linear group PGL(n) in the Grothendieck ring of stacks K-0(Stack(k)) for n = 2 and n = 3 under mild conditions on the base field k. In particular, although it is known that the multiplicativity relation {T} = {S} . {PGL(n)} does not hold for all PGL(n)-torsors T -> S, it holds for the universal PGL(n)-torsors for said n.

• 9.
Stockholm University, Faculty of Science, Department of Mathematics.
The Binomial Theorem and motivic classes of universal quasi-split toriManuscript (preprint) (Other academic)

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

• 10.
Stockholm University, Faculty of Science, Department of Mathematics.
Homological perturbation theory for algebras over operads2014In: Algebraic and Geometric Topology, ISSN 1472-2747, E-ISSN 1472-2739, Vol. 14, no 5, p. 2511-2548Article in journal (Refereed)

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

• 11.
Stockholm University, Faculty of Science, Department of Mathematics.
Koszul spacesIn: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850Article in journal (Refereed)
• 12.
Stockholm University, Faculty of Science, Department of Mathematics.
Rational homotopy theory of mapping spaces via Lie theory for L-infinity algebrasArticle in journal (Refereed)
• 13.
Stockholm University, Faculty of Science, Department of Mathematics.
Rational homotopy theory of mapping spaces via Lie theory for L-infinity algebras2015In: Homology, Homotopy and Applications, ISSN 1532-0073, E-ISSN 1532-0081, Vol. 17, no 2, p. 343-369Article in journal (Refereed)

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

• 14.
Stockholm University, Faculty of Science, Department of Mathematics.
Shellability and the strong gcd-condition2009In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 16, no 2Article in journal (Refereed)

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

• 15.
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
Hirzebruch L-polynomials and multiple zeta values2018In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 372, no 1-2, p. 125-137Article in journal (Refereed)

We express the coefficients of the Hirzebruch L-polynomials in terms of certain alternating multiple zeta values. In particular, we show that every monomial in the Pontryagin classes appears with a non-zero coefficient, with the expected sign. Similar results hold for the polynomials associated to the Â-genus.

• 16.
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
Free loop space homology of highly connected manifolds2017In: Forum mathematicum, ISSN 0933-7741, E-ISSN 1435-5337, Vol. 29, no 1, p. 201-228Article in journal (Refereed)

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

• 17.
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.

• 18.
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.

• 19.
Köpenhamns universitet, Danmark.
University of Copenhagen.
Homological stability of diffeomorphism groups2013In: Pure and Applied Mathematics Quarterly, ISSN 1558-8599, E-ISSN 1558-8602, Vol. 9, no 1, p. 1-48Article in journal (Refereed)
• 20.
Stockholm University, Faculty of Science, Department of Mathematics.
Rational homotopy theory of automorphisms of highly connected manifoldsManuscript (preprint) (Other academic)

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

• 21.
Stockholm University, Faculty of Science, Department of Mathematics.
University of Copenhagen.
Rational homotopy theory of automorphisms of highly connected manifoldsArticle in journal (Refereed)
• 22.
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
A dg Lie model for relative homotopy automorphismsManuscript (preprint) (Other academic)

We construct a dg Lie model for the universal cover of the classifying space of the grouplike monoid of homotopy automorphisms of a space that fix a given subspace.

• 23.
Stockholm University, Faculty of Science, Department of Mathematics.
KTH. UvA.
Siegel modular forms of degree three and the cohomology of local systems2014In: Selecta Mathematica, New Series, ISSN 1022-1824, E-ISSN 1420-9020, Vol. 20, no 1, p. 83-124Article in journal (Refereed)

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

• 24.
Stockholm University, Faculty of Science, Department of Mathematics.
Tokyo Denki University.
On the cohomology of moduli spaces of (weighted) stable rational curves2013In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 275, no 3-4, p. 1095-1108Article in journal (Refereed)

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

• 25.
Stockholm University, Faculty of Science, Department of Mathematics.
On the cohomology of the Losev-Manin moduli space2014In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 144, no 1, p. 241-252Article in journal (Refereed)

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

• 26.
Stockholm University, Faculty of Science, Department of Mathematics.
Cohomology of the moduli space of curves of genus three with level two structure2014Licentiate thesis, monograph (Other academic)

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

• 27. Campos, Ricardo
Stockholm University, Faculty of Science, Department of Mathematics.
Gravity formality2018In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 331, p. 439-483Article in journal (Refereed)

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

• 28. Carlini, Enrico
Stockholm University, Faculty of Science, Department of Mathematics. Polytechnic University of Turin, Italy.
Monomials as Sums of k-th Powers of Forms2015In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 43, no 2, p. 650-658Article in journal (Refereed)

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

• 29. Eklund, David
Stockholm University, Faculty of Science, Department of Mathematics.
A method to compute Segre classes of subschemes of projective space2013In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 12, no 2, p. 1250142-Article in journal (Refereed)

We present a method to compute the degrees of the Segre classes of a subscheme of complex projective space. The method is based on generic residuation and intersection theory. It has been implemented using the software system Macaulay2.

• 30.
Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
Abstrakta och konkreta ting i geometrilandskapet: Varför elever i årskurs 7-9 har lätt och svårt i geometriområdet samt vad läraren gör för att underlätta elevernas förståelse2019Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis

Geometry is an area in mathematics that is considered not abstract, on the contrary from other areas in mathematics. As geometry is considered an unabstracted area in mathematics, why has students around the world difficulties with geometry? TIMSS (Trends in International Mathematics and Science Study) has shown that Swedish students in 8th grade has difficulties in algebra and geometry. The study focuses on why secondary school students’ have easy to understand some parts in geometry and why they have difficulties in other parts. Furthermore the study focuses on strategies teachers use to facilitate understanding in geometry. The study is carried out by interviewing six teachers in secondary school. The interviews are recorded and transcribed for enabling thematic analysis. The result shows that teachers experiences that students have easy to understand the first dimension (length and perimeter) and easy to understand geometrical objects as for example rectangular shapes. The reason behind the easiness is that these elements in geometry is known for the students, easy for the teachers to explain and not abstract. The students have difficulties comprehending two and especially three dimensional objects, difficult geometrical objects as circular objects, objects where the height is “situated” outside the object and irregular figures, unit conversions and concepts in geometry. The reasons behind these difficulties are mainly: the elements and methods are unfamiliar and abstract to the students. The abstraction in geometry are shown as comprehending how big or small sizes are in two and three dimensions and difficulties to comprehend the big discrepancy between the numbers in unit conversion, Teachers also observe that students have difficulties in visualizing and manipulating objects. These results show that what is known and not abstract are the opposite for what students have difficulties with, i.e. unknown and abstract. The strategies the teachers use are mostly to concretize the difficulties in geometry and in that way show students why it is valid. Other strategies are concerning with building a strong foundation in geometry, to combine geometry with other subjects in school and using students prior knowledge to build new knowledge. The red articles agrees with the result from the study.

• 31. Honigs, Katrina
Stockholm University, Faculty of Science, Department of Mathematics. University of Bergen.
Derived equivalences of canonical covers of hyperelliptic and Enriques surfaces in positive characteristic2019In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823Article in journal (Refereed)

We prove that any Fourier–Mukai partner of an abelian surface over an algebraically closed field of positive characteristic is isomorphic to a moduli space of Gieseker-stable sheaves. We apply this fact to show that the set of Fourier–Mukai partners of a canonical cover of a hyperelliptic or Enriques surface over an algebraically closed field of characteristic greater than three is trivial. These results extend earlier results of Bridgeland–Maciocia and Sosna to positive characteristic.

• 32.
Stockholm University.
Stockholm University.
Coamoebas and line arrangements in dimension twoIn: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823Article in journal (Refereed)
• 33.
Stockholm University, Faculty of Science, Department of Mathematics.
A Macaulay2 package for characteristic classes and the topological Euler characteristic of complex projective schemesManuscript (preprint) (Other academic)

The Macaulay2 package CharacteristicClasses provides commands for the computation of the topological Euler characteristic, the degrees of the Chern classes and the degrees of the Segre classes of a closed subscheme of complex projective space. The computations can be done both symbolically and numerically, the latter using an interface to Bertini. We provide some background of the implementation and show how to use the package with the help of examples.

• 34.
Stockholm University, Faculty of Science, Department of Mathematics.
An algorithm for computing the topological Euler characteristic of complex projective varietiesManuscript (preprint) (Other academic)

We present an algorithm for the symbolic and numerical computation of the degrees of the Chern-Schwartz-MacPherson classes of a closed subvariety of projective space P^n. As the degree of the top Chern-Schwartz-MacPherson class is the topological Euler characteristic, this also yields a method to compute the topological Euler characteristic of projective varieties. The method is based on Aluffi's symbolic algorithm to compute degrees of Chern-Schwartz-MacPherson classes, a symbolic method to compute degrees of Segre classes, and the regenerative cascade by Hauenstein, Sommese and Wampler. The new algorithm complements the existing algorithms. We also give an example for using a theorem by Huh to compute an invariant from algebraic statistics, the maximum likelihood degree of an implicit model.

• 35.
Stockholm University, Faculty of Science, Department of Mathematics.
Globalizing L-infinity automorphisms of the Schouten algebra of polyvector fieldsIn: Differential geometry and its applications (Print), ISSN 0926-2245, E-ISSN 1872-6984Article in journal (Refereed)

Recently, Willwacher showed that the Grothendieck-Teichmuller group GRT acts by L-infinity-automorphisms on the Schouten algebra of polyvector fields T_poly(R^d) on affine space R^d. In this article, we prove that a large class of L-infinity-automorphisms on the Schouten algebra, including Willwacher's, can be globalized. That is, given an L-infinity-automorphism of T_poly(R^d) and a general smooth manifold M with the choice of a torsion-free connection, we give an explicit construction of an L-infinity-automorphism of the Schouten algebra T_poly(M) on the manifold M, depending on the chosen connection. The method we use is the Fedosov trick.

• 36.
Stockholm University, Faculty of Science, Department of Mathematics.
Topics in Computational Algebraic Geometry and Deformation Quantization2013Doctoral thesis, comprehensive summary (Other academic)

This thesis consists of two parts, a first part on computations in algebraic geometry, and a second part on deformation quantization. More specifically, it is a collection of four papers. In the papers I, II and III, we present algorithms and an implementation for the computation of degrees of characteristic classes in algebraic geometry. Paper IV is a contribution to the field of deformation quantization and actions of the Grothendieck-Teichmüller group.

In Paper I, we present an algorithm for the computation of degrees of Segre classes of closed subschemes of complex projective space. The algorithm is based on the residual intersection theorem and can be implemented both symbolically and numerically.

In Paper II, we describe an algorithm for the computation of the degrees of Chern-Schwartz-MacPherson classes and the topological Euler characteristic of closed subschemes of complex projective space, provided an algorithm for the computation of degrees of Segre classes. We also explain in detail how the algorithm in Paper I can be implemented numerically. Together this yields a symbolical and a numerical version of the algorithm.

Paper III describes the Macaulay2 package CharacteristicClasses. It implements the algorithms from papers I and II, as well as an algorithm for the computation of degrees of Chern classes.

In Paper IV, we show that L-infinity-automorphisms of the Schouten algebra T_poly(R^d) of polyvector fields on affine space R^d which satisfy certain conditions can be globalized. This means that from a given L-infinity-automorphism of T_poly(R^d) an L-infinity-automorphism of T_poly(M) can be constructed, for a general smooth manifold M. It follows that Willwacher's action of the Grothendieck-Teichmüller group on T_poly(R^d) can be globalized, i.e., the Grothendieck-Teichmüller group acts on the Schouten algebra T_poly(M) of polyvector fields on a general manifold M.

• 37.
University of Gävle, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
Hilbert series of modules over Lie algebroidsArticle in journal (Refereed)

We consider modules M over Lie algebroids g_A which are of  finite type over a local noetherian ring A.  Using ideals  J\subseteq A such that g_A . J \subseteq J  and the length  l_{g_A}(M/JM)< \infty we can define in a natural way the  Hilbert series of M with respect to the defining ideal J.  This  notion is in particular studied for modules over the Lie algebroid  of k-linear derivations g_A=T_A(I) that preserve an ideal  I\subseteq A, for example when A is the ring of convergent  power series.

• 38.
Stockholm University, Faculty of Science, Department of Mathematics.
Department of mathematics, Tel-Aviv University, Israel. Institute of Nuclear Research and Nuclear Energy, Sofia, Bulgaria.
Seminar of Supersymmetries: volume 1 (edited by D. Leites)2011Book (Refereed)

Supermanifold theory is a relatively new branch of mathematics. Ideas of supersymmetry appeared to resolve several hitherto seemingly unsolvable problems of theoretical physics and quickly flourished into a rich blend of differential and algebraic geometers with own deep problems. In this book there are presented basics of linear and general algebra in superspaces, elements of algebraic and differential geometers on supermanifolds. The book is saturated by open questions of various degree of complexity and will be useful to researchers (theoretical physicists and mathematicians) as well as (research) students.

• 39. Lombardi, Luigi
The University of Uta, USA.
GV-subschemes and their embeddings in principally polarizedabelian varieties2015In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 288, no 11-12, p. 1405-1412Article in journal (Refereed)

We prove that any embedding of a GV -subscheme in a principally polarized abelian variety does not factorthrough any nontrivial isogeny. As an application, we present a new proof of a theorem of Clemens–Griffithsidentifying the intermediate Jacobian of a smooth cubic threefold to the Albanese variety of its Fano surface oflines.

• 40.
Stockholm University, Faculty of Science, Department of Mathematics.
A local duality principle for ideals of pure dimensionManuscript (preprint) (Other academic)

We prove that a certain cohomological residue associated to an ideal of pure dimension is annihilated exactly by the ideal. The cohomological residue is quite explicit and generalizes the classical local Grothendieck residue and the cohomological residue of Passare.

• 41.
Stockholm University, Faculty of Science, Department of Mathematics.
A local Grothendieck duality theorem for Cohen-Macaulay ideals2012In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 111, no 1, p. 42-52Article in journal (Refereed)

We give a new proof of a recent result due to Mats Andersson and Elizabeth Wulcan, generalizing the local Grothendieck duality theorem. It can also be seen as a generalization of a previous result by Mikael Passare. Our method does not require the use of the Hironaka desingularization theorem and it provides a semi-explicit realization of the residue that is annihilated by functions from the given ideal.

• 42.
Stockholm University, Faculty of Science, Department of Mathematics.
An effective uniform Artin-Rees lemmaManuscript (preprint) (Other academic)

We prove a global uniform Artin-Rees lemma type theorem for sections of ample line bundles over smooth projective varieties. This result is used to prove an Artin-Rees lemma for the polynomial ring with uniform degree bounds. The proof is based on multidimensional residue calculus.

• 43.
Stockholm University, Faculty of Science, Department of Mathematics.
An explicit calculation of the Ronkin functionManuscript (preprint) (Other academic)

We calculate the second order derivatives of the Ronkin function in the case of an affine linear polynomial in three variables and give an expression of them in terms of complete elliptic integrals and hypergeometric functions. This gives a semi-explicit expression of the associated Monge-Ampère measure, the Ronkin measure.

• 44.
Stockholm University, Faculty of Science, Department of Mathematics.
On Amoebas and Multidimensional Residues2012Doctoral thesis, comprehensive summary (Other academic)

This thesis consists of four papers and an introduction.

In Paper I we calculate the second order derivatives of the Ronkin function of an affine polynomial in three variables. This gives an expression for the real Monge-Ampére measure associated to the hyperplane amoeba. The measure is expressed in terms of complete elliptic integrals and hypergeometric functions.

In Paper II and III we prove that a certain semi-explicit cohomological residue associated to a Cohen-Macaulay ideal or more generally an ideal of pure dimension, respectively, is annihilated precisely by the given ideal. This is a generalization of the local duality principle for the Grothendieck residue and the cohomological residue of Passare. These results follow from residue calculus, due to Andersson and Wulcan, but the point here is that our proof is more elementary. In particular, it does not rely on the desingularization theorem of Hironaka.

In Paper IV we prove a global uniform Artin-Rees lemma for sections of ample line bundles over smooth projective varieties. We also prove an Artin-Rees lemma for the polynomial ring with uniform degree bounds. The proofs are based on multidimensional residue calculus.

• 45.
Stockholm University, Faculty of Science, Department of Mathematics.
Isomorphism classes of abelian varieties over finite fields2016Licentiate thesis, monograph (Other academic)

Deligne and Howe described polarized abelian varieties over finite fields in terms of finitely generated free Z-modules satisfying a list of easy to state axioms. In this thesis we address the problem of developing an effective algorithm to compute isomorphism classes of (principally) polarized abelian varieties over a finite field, together with their automorphism groups. Performing such computations requires the knowledge of the ideal classes (both invertible and non-invertible) of certain orders in number fields. Hence we describe a method to compute the ideal class monoid of an order and we produce concrete computations in dimension 2, 3 and 4.

• 46.
Stockholm University, Faculty of Science, Department of Mathematics.
Introduction to the Ekedahl invariantsManuscript (preprint) (Other academic)
• 47.
Stockholm University, Faculty of Science, Department of Mathematics.
The Ekedahl Invariants for finite groups2013Licentiate thesis, monograph (Other academic)
• 48.
Stockholm University, Faculty of Science, Department of Mathematics.
The Ekedahl invariants for finite groupsManuscript (preprint) (Other academic)
• 49.
Stockholm University, Faculty of Science, Department of Mathematics.
Power ideals, Fröberg conjecture and Waring problems2014Licentiate thesis, monograph (Other academic)

This thesis is divided into two chapters. First, we want to study particularclasses of power ideals, with particular attention to their relation with the Fröberg conjecture on the Hilbert series of generic ideals. In the second part,we study a generalization (introduced by Fröberg, Ottaviani, and Shapiro in 2012)of the classical Waring problem for polynomials about writing homogeneouspolynomials as sums of powers. We see also how the theories of fat points andsecant varieties of Veronese varieties play a crucial role in the relation betweenthose chapters and in providing tools to nd an answer to our questions.

The main results are the computation of the Hilbert series of particularclasses of power ideals, which in particular give us a proof of the Fröberg conjecturefor generic ideals generated by eight homogeneous polynomials of thesame degree in four variables, and the solution of the generalized Waring problemin the case of sums of squares in three and four variables. We also beginthe study of the generalized Waring problem for monomials.

• 50.
Stockholm University, Faculty of Science, Department of Mathematics.
Waring-type problems for polynomials: Algebra meets Geometry2016Doctoral thesis, monograph (Other academic)

In the present thesis we analyze different types of additive decompositions of homogeneous polynomials. These problems are usually called Waring-type problems and their story go back to the mid-19th century and, recently, they received the attention of a large community of mathematicians and engineers due to several applications. At the same time, they are related to branches of Commutative Algebra and Algebraic Geometry.

The classical Waring problem investigates decompositions of homogeneous polynomials as sums of powers of linear forms. Via Apolarity Theory, the study of these decompositions for a given polynomial F is related to the study of configuration of points apolar to F, namely, configurations of points whose defining ideal is contained in the perp'' ideal associated to F. In particular, we analyze which kind of minimal set of points can be apolar to some given polynomial in cases with small degrees and small number of variables. This let us introduce the concept of Waring loci of homogeneous polynomials.

From a geometric point of view, questions about additive decompositions of polynomials can be described in terms of secant varieties of projective varieties. In particular, we are interested in the dimensions of such varieties. By using an old result due to Terracini, we can compute these dimensions by looking at the Hilbert series of homogeneous ideal.

Hilbert series are very important algebraic invariants associated to homogeneous ideals. In the case of classical Waring problem, we have to look at power ideals, i.e., ideals generated by powers of linear forms. Via Apolarity Theory, their Hilbert series are related to Hilbert series of ideals of fat points, i.e., ideals of configurations of points with some multiplicity. In this thesis, we consider some special configuration of fat points. In general, Hilbert series of ideals of fat points is a very active field of research. We explain how it is related to the famous Fröberg's conjecture about Hilbert series of generic ideals.

Moreover, we use Fröberg's conjecture to deduce the dimensions of several secant varieties of particular projective varieties and, then, to deduce results regarding some particular Waring-type problems for polynomials.

In this thesis, we mostly work over the complex numbers. However, we also analyze the case of classical Waring decompositions for monomials over the real numbers. In particular, we classify for which monomials the minimal length of a decomposition in sum of powers of linear forms is independent from choosing the ground field as the field of complex or real numbers.

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