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• 1.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Lefschetz Properties of Monomial Ideals2018Licentiate thesis, comprehensive summary (Other academic)

This thesis concerns the study of the Lefschetz properties of artinian monomial algebras. An artinian algebra is said to satisfy the strong Lefschetz property if multiplication by all powers of a general linear form has maximal rank in every degree. If it holds for the first power it is said to have the weak Lefschetz property (WLP).

In the first paper, we study the Lefschetz properties of monomial algebras by studying their minimal free resolutions. In particular, we give an afirmative answer to an specific case of a conjecture by Eisenbud, Huneke and Ulrich for algebras having almost linear resolutions. Since many algebras are expected to have the Lefschetz properties, studying algebras failing the Lefschetz properties is of a great interest. In the second paper, we provide sharp lower bounds for the number of generators of monomial ideals failing the WLP extending a result by Mezzetti and Miró-Roig which provides upper bounds for such ideals. In the second paper, we also study the WLP of ideals generated by forms of a certain degree invariant under an action of a cyclic group. We give a complete classication of such ideals satisfying the WLP in terms of the representation of the group generalizing a result by Mezzetti and Miró-Roig.

• 2.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Lefschetz Properties of Monomial Ideals with Almost Linear ResolutionIn: Article in journal (Other academic)

We study the WLP and SLP of artinian monomial ideals in S = K[x1, . . . , xn]

via studying their minimal free resolutions. We study the Lefschetz properties of such ideals

where the minimal free resolution of S/I is linear for at least n − 2 steps. We give an

affirmative answer to a conjecture of Eisenbud, Huneke and Ulrich for artinian monomial

ideals with almost linear resolutions.

• 3.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
The Weak Lefschetz Property of Equigenerated Monomial IdealsIn: Article in journal (Other academic)

We determine the sharp lower bound for the Hilbert function in degree d of a

monomial algebra failing the WLP over a polynomial ring with n variables and generated in

degree d. We consider artinian ideals in the polynomial ring with

n variables generated by homogeneous polynomials of degree d invariant under an action of

the cyclic group Z/dZ. We give a complete classification of

such ideals in terms of the WLP depending on the action.

• 4.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Formal plethories2014In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 254, p. 497-569Article 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.

• 5.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
A note on circle maps driven by strongly expanding endomorphisms on T2018In: Dynamical systems, ISSN 1468-9367, E-ISSN 1468-9375, Vol. 33, no 2, p. 361-368Article in journal (Refereed)

We investigate the dynamics of a class of smooth maps of the two-torus T2 of the form T(x, y) = (Nx, f(x)(y)), where f(x) : T -> T is a monotone family (in x) of orientation preserving circle diffeomorphisms and N is an element of Z(+) is large. For our class of maps, we show that the dynamics essentially is the same as that of the projective action of non-uniformly hyperbolic SL(2, R)-cocycles. This generalizes a result by L.S. Young [6] to maps T outside the (projective) matrix cocycle case.

• 6.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
On the classification of fibrations2015In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 367, no 1, p. 519-557Article in journal (Refereed)

We identify the homotopy type of the moduli of maps with a given homotopy type of the base and the homotopy fiber. A new model for the space of weak equivalences and its classifying space is given.

• 7.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Hebrew University of Jerusalem. Ben-Gurion University.
Idempotent deformations of finite groupsManuscript (preprint) (Other academic)
• 8.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Classification of plethories in characteristic zero2015Licentiate thesis, comprehensive summary (Other academic)

We classify plethories over fields of characteristic zero, thus answering a question of Borger-Wieland and Bergman-Hausknecht. All plethories over characteristic zero fields are linear, in the sense that they are free plethories on a bialgebra. For the proof we need some facts from the theory of ring schemes where we extend previously known results. We also classify plethories with trivial Verschiebung over a perfect field of non-zero characteristic and indicate future work.

• 9.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Tensor products of affine and formal abelian groupsManuscript (preprint) (Other academic)

In this paper we study tensor products of affine abelian group schemes over a perfect field k. We first prove that the tensor product G_1 ⊗ G_2 of two affine abelian group schemes G_1,G_2  over a perfect field k exists. We then describe the multiplicative and unipotent part of the group scheme G_1 ⊗G_2. The multiplicative part is described in terms of Galois modules over the absolute Galois group of k. We describe the unipotent part of G_1 ⊗ G_2 explicitly, using Dieudonn\'e theory in positive characteristic. We relate these constructions to previously studied tensor products of formal group schemes.

• 10.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
The cohomology ring of the ring of integers of a number fieldManuscript (preprint) (Other academic)

We compute the étale cohomology ring H^*(Spec O_K,Z/nZ) where O_K is the ring of integers of a number field K. As an application, we give a non-vanishing formula for an invariant defined by Minhyong Kim.

• 11.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Tensor products of affine and formal abelian groupsManuscript (preprint) (Other academic)

In this paper we study tensor products of affine abelian group schemes over a perfect field k. We first prove that the tensor product G_1 ⊗ G_2 of two affine abelian group schemes G_1,G_2  over a perfect field k exists. We then describe the multiplicative and unipotent part of the group scheme G_1 ⊗G_2. The multiplicative part is described in terms of Galois modules over the absolute Galois group of k. We describe the unipotent part of G_1 ⊗ G_2 explicitly, using Dieudonn\'e theory in positive characteristic. We relate these constructions to previously studied tensor products of formal group schemes.

• 12.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Hebrew University of Jerusalem.
The unramified inverse Galois problem and cohomology rings of totally imaginary number fieldsManuscript (preprint) (Other academic)

We employ methods from homotopy theory to define new obstructions to solutions of embedding problems. By using these novel obstructions we study embedding problems with non-solvable kernel. We apply these obstructions to study the unramified inverse Galois problem. That is, we show that our methods can be used to determine that certain groups cannot be realized as the Galois groups of unramified extensions of certain number fields. To demonstrate the power of our methods, we give an infinite family of totally imaginary quadratic number fields such that Aut(PSL(2,q^2)) for q an odd prime power, cannot be realized as an unramified Galois group over K, but its maximal solvable quotient can. To prove this result, we determine the ring structure of the \'etale cohomology ring H^*(Spec O_K;Z/2Z) where O_K is the ring of integers of an arbitrary totally imaginary number field K.

• 13.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
GENERALIZING SERRE'S SPLITTING THEOREM AND BASS'S CANCELLATION THEOREM VIA FREE-BASIC ELEMENTS2018In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 146, no 4, p. 1417-1430Article in journal (Refereed)

We give new proofs of two results of Stafford, which generalize two famous Theorems of Serre and Bass regarding projective modules. Our techniques are inspired by the theory of basic elements. Using these methods we further generalize Serre's Splitting Theorem by imposing a condition to the splitting maps, which has an application to the case of Cartier algebras.

• 14.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Hardy inequalities for p-Laplacians with Robin boundary conditions2015In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 128, p. 365-379Article in journal (Refereed)

In this paper we study the best constant in a Hardy inequality for the p-Laplace operator on convex domains with Robin boundary conditions. We show, in particular, that the best constant equals ((p-1)/p)(p) whenever Dirichlet boundary conditions are imposed on a subset of the boundary of non-zero measure. We also discuss some generalizations to non-convex domains.

• 15.
KTH, School of Architecture and the Built Environment (ABE), Urban Planning and Environment, Geoinformatik och Geodesi.
Generalized Least Squares Adjustment of Gauss-Helmert Model2005Conference paper (Refereed)
• 16.
nstituto de Física Teórica UAM/CSIC, Universidad Autónoma de Madrid. KTH, Centres, Nordic Institute for Theoretical Physics NORDITA.
Towards the most general scalar-tensor theories of gravity: A unified approach in the language of differential forms2016In: Physical Review D, ISSN 1550-7998, E-ISSN 1550-2368, Vol. 94, no 2, article id 024005Article in journal (Refereed)

We use a description based on differential forms to systematically explore the space of scalar-tensor theories of gravity. Within this formalism, we propose a basis for the scalar sector at the lowest order in derivatives of the field and in any number of dimensions. This minimal basis is used to construct a finite and closed set of Lagrangians describing general scalar-tensor theories invariant under local Lorentz transformations in a pseudo-Riemannian manifold, which contains ten physically distinct elements in four spacetime dimensions. Subsequently, we compute their corresponding equations of motion and find which combinations are at most second order in derivatives in four as well as an arbitrary number of dimensions. By studying the possible exact forms (total derivatives) and algebraic relations between the basis components, we discover that there are only four Lagrangian combinations producing second-order equations, which can be associated with Horndeski's theory. In this process, we identify a new second-order Lagrangian, named kinetic Gauss-Bonnet, that was not previously considered in the literature. However, we show that its dynamics is already contained in Horndeski's theory. Finally, we provide a full classification of the relations between different second-order theories. This allows us to clarify, for instance, the connection between different covariantizations of Galileons theory. In conclusion, our formulation affords great computational simplicity with a systematic structure. As a first step, we focus on theories with second-order equations of motion. However, this new formalism aims to facilitate advances towards unveiling the most general scalar-tensor theories.

• 17.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Bounds on Hilbert Functions2013Licentiate thesis, comprehensive summary (Other academic)

This thesis is constituted of two articles, both related to Hilbert functions and h-vectors. In the first paper, we deal with h-vectorsof reduced zero-dimensional schemes in the projective plane, and, in particular, with the problem of finding the possible h-vectors for the union of two sets of points of given h-vectors. In the second paper, we generalize the Green’s Hyperplane Restriction Theorem to the case of modules over the polynomial ring.

• 18.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Bounds on Hilbert Functions and Betti Numbers of Veronese Modules2014Doctoral thesis, comprehensive summary (Other academic)

The thesis is a collection of four papers dealing with Hilbert functions and Betti numbers.In the first paper, we study the h-vectors of reduced zero-dimensional schemes in $\mathbb{P}^{2}$ . In particular we deal with the problem of findingthe possible h-vectors for the union of two sets of points of given h-vectors. To answer to this problem, we give two kinds of bounds on theh-vectors and we provide an algorithm that calculates many possibleh-vectors.In the second paper, we prove a generalization of Green’s Hyper-plane Restriction Theorem to the case of finitely generated modulesover the polynomial ring, providing an upper bound for the Hilbertfunction of the general linear restriction of a module M in a degree dby the corresponding Hilbert function of a lexicographic module.In the third paper, we study the minimal free resolution of theVeronese modules, $S_{n,d,k}=\oplus_{i\geq 0} S_{k+id}$, where  $S = \mathbb{K}[x_1 , . . . , x_n ]$ by giving a formula for the Betti numbers in terms of the reduced homology of the squarefree divisor complex. We prove that $S_{n,d,k}$ is Cohen-Macaulay if and only if k < d, and that its minimal resolutionis linear when k > d(n − 1) − n. We prove combinatorially that the resolution of $S_{2,d,k}$ is pure. We provide a formula for the Hilbert seriesof the Veronese modules. As an application, we calculate the completeBetti diagrams of the Veronese rings $S_{3,4,0} , S_{3,5,0} and S_{4,3,0}$ .In the fourth paper, we apply the same combinatorial techniques inthe study of the properties of pinched Veronese rings, in particular weshow when this ring is Cohen-Macaulay. We also study the canonicalmodule of the Veronese modules.

• 19.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Cohen-Macaulay Property of pinched Veronese Rings and Canonical Modules of Veronese  ModulesManuscript (preprint) (Other academic)

In this paper, we study the Betti numbers of pinched Veronese rings, by means of the reduced homology of the squarefree divisor complex. In particular, we study the Cohen-Macaulay property of these rings. Moreover, in the last section we compute the canonical modules of the Veronese modules.

• 20.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Syzygies of Veronese modulesManuscript (preprint) (Other academic)
• 21.
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
Scania. KTH, School of Industrial Engineering and Management (ITM), Machine Design (Dept.), Mechatronics. Systems Development DivisionScania AB, Södertälje, Sweden. KTH, School of Industrial Engineering and Management (ITM), Machine Design (Dept.), Mechatronics.
Deductive Functional Verification of Safety-Critical Embedded C-Code: An Experience Report2017In: Critical Systems: Formal Methods and Automated Verification, Cham: Springer, 2017, p. 3-18Conference paper (Refereed)

This paper summarizes our experiences from an exercise in deductive verification of functional properties of automotive embedded Ccode in an industrial setting. We propose a formal requirements model that supports the way C-code requirements are currently written at Scania. We describe our work, for a safety-critical module of an embedded system, on formalizing its functional requirements and verifying its C-code implementation by means of VCC, an established tool for deductive verification. We describe the obstacles we encountered, and discuss the automation of the specification and annotation effort as a prerequisite for integrating this technology into the embedded software design process.

• 22.
Department of Mathematics, Uppsala university.

This thesis consists of four papers in domain theory and a summary. The first two papers deal with the problem of defining effectivity for continuous cpos. The third and fourth paper present the new notion of an admissible domain representation, where a domain representation D of a space X is λ-admissible if, in principle, all other λ-based domain representations E of X can be reduced to X via a continuous function from E to D. In Paper I we define a cartesian closed category of effective bifinite domains. We also investigate the method of inducing effectivity onto continuous cpos via projection pairs, resulting in a cartesian closed category of projections of effective bifinite domains. In Paper II we introduce the notion of an almost algebraic basis for a continuous cpo, showing that there is a natural cartesian closed category of effective consistently complete continuous cpos with almost algebraic bases. We also generalise the notion of a complete set, used in Paper I to define the bifinite domains, and investigate what closure results that can be obtained. In Paper III we consider admissible domain representations of topological spaces. We present a characterisation theorem of exactly when a topological space has a λ-admissible and κ-based domain representation. We also show that there is a natural cartesian closed category of countably based and countably admissible domain representations. In Paper IV we consider admissible domain representations of convergence spaces, where a convergence space is a set X together with a convergence relation between nets on X and elements of X. We study in particular the new notion of weak κ-convergence spaces, which roughly means that the convergence relation satisfies a generalisation of the Kuratowski limit space axioms to cardinality κ. We show that the category of weak κ-convergence spaces is cartesian closed. We also show that the category of weak κ-convergence spaces that have a dense, λ-admissible, κ-continuous and α-based consistently complete domain representation is cartesian closed when α ≤ λ ≥ κ. As natural corollaries we obtain corresponding results for the associated category of weak convergence spaces.

• 23.
Matematiska institutionen, Uppsala universitet.
Matematiska institutionen, Uppsala universitet.
Cartesian closed categories of effective domains2001Conference paper (Refereed)
• 24.
KTH, School of Architecture and the Built Environment (ABE), Philosophy and History of Technology, Philosophy.
Decomposition of multiple AGM contraction: possibility and impossibility results2014In: Logic journal of the IGPL (Print), ISSN 1367-0751, E-ISSN 1368-9894, Vol. 22, no 4, p. 696-710Article in journal (Refereed)

Partial meet contraction, the standard operation of contraction in AGM theory, can straightforwardly be generalized to contractions with sets of sentences instead of single sentences as inputs (multiple contraction). The conditions under which multiple contraction can be reconstructed as the intersection of several contractions by single sentences are investigated. Although such reconstruction is possible in some special cases, the major result is that in the general case, full reconstruction is not possible. Therefore, multiple contraction adds to the expressive power of belief revision theory. It is concluded that this result highlights the importance of studying multiple belief change, and multiple conclusion logic in general.

• 25.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Some remarks on biequidimensionality of topological spaces and Noetherian schemesManuscript (preprint) (Other academic)

There are many examples of the fact that dimension and codimension behave somewhat counterintuitively. In EGA it is stated that a topological space is equidimensional, equicodimensional and catenary if and only if every maximal chain of irreducible closed subsets has the same length. We construct examples that show that this is not even true for the spectrum of a Noetherian ring. This gives rise to two notions of biequidimensionality, and we show how these relate to the dimension formula and the existence of a codimension function.

• 26.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
The Cohen–Macaulay space of twisted cubicsManuscript (preprint) (Other academic)

In this work, we describe the Cohen-Macaulay space CM of twisted cubics parameterizing curves $C$ together with a finite map $i: C\rightarrow \mathbb{P}^3$ that is generically a closed immersion and such that $C$ has Hilbert polynomial p(t)=3t+1 with respect to $i$. We show that CM is irreducible, smooth and birational to one component of the Hilbert scheme of twisted cubics.

• 27.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
The space of Cohen–Macaulay curvesManuscript (preprint) (Other academic)

One can consider the Hilbert scheme as a natural compactification of the space of smooth projective curves with fixed Hilbert polynomial. Here we consider a different modular compactification, namely the functor CM parameterizing curves together with a finite map to $\mathbb{P}^{n}$ that is generically a closed immersion. We prove that CM is an algebraic space by contructing a scheme W and a representable, surjective and smooth map W -> CM. Moreover, we show that CM satisfies the valuative criterion for properness.

• 28.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
The space of Cohen-Macaulay curves2012Licentiate thesis, monograph (Other academic)

In this thesis we discuss a moduli space of projective curves with a map to a given projective space. The functor CM parametrizes curves, that is, Cohen-Macaulay schemes of pure dimension 1, together with a finite map to the projective space that is an isomorphism onto its image away from a finite set of closed points.

We proof that CM is an algebraic space by contructing a scheme W and a representable, surjective and smooth map from W to CM.

• 29.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
The space of Cohen–Macaulay curves and related topics2014Doctoral thesis, comprehensive summary (Other academic)

The space of Cohen–Macaulay curves is a compactification of the space of curves that are embedded in a given projective space Pn. The idea is similar to that of the Hilbert scheme but instead of adding degenerated curves, one considers only curves without embedded or isolated points. However, the curves need not be embedded into the projective space. Instead, they come with a finite morphism to Pn that is generically a closed immersion. More precisely, the space CM of Cohen–Macaulay curves parameterizes flat families of pairs $(C,i)$ where $C$ is a curve without embedded or isolated points and $i: C\rightarrow \mathbb{P}^n$ is a finite morphism that is an isomorphism onto its image away from finitely many closed points and such that $C$ has Hilbert polynomial p(t) with respect to the map $i$.

In Paper A we show that the moduli functor CM is represented by a proper algebraic space. This is done by constructing a smooth, surjective cover $\pi: W\rightarrow CM$ and by verifying the valuative criterion for properness.

Paper B studies the moduli space CM in the particular case n = 3 and p(t) = 3t + 1, that is, the Cohen–Macaulay space of twisted cubics. We de- scribe the points of CM and show that they are in bijection with the points on the 12-dimensional component H0 of the Hilbert scheme of twisted cu- bics. Knowing the points of CM, we can then show that the moduli space is irreducible, smooth and has dimension 12.

Paper C concerns the notion of biequidimensionality of topological spaces and Noetherian schemes. In EGA it is claimed that a topological space X is equidimensional, equicodimensional and catenary if and only if all maximal chains of irreducible closed subsets in X have the same length. We construct examples of topological spaces and Noetherian schemes showing that the sec- ond property is strictly stronger. This gives rise to two different notions of biequidimensionality, and we show how they relate to the dimension formula and the existence of a codimension function.

• 30.
KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory.
KTH, School of Engineering Sciences (SCI), Physics, Condensed Matter Theory. KTH Royal Inst Technol, Dept Phys, S-10691 Stockholm, Sweden.. Univ Paris Diderot, Sorbonne Paris Cite, Sorbonne Univ, ENS,Univ PSL,CNRS,Lab Phys Ecole Normale Super, Paris, France..
Defining a bulk-edge correspondence for non-Hermitian Hamiltonians via singular-value decomposition2019In: Physical Review A: covering atomic, molecular, and optical physics and quantum information, ISSN 2469-9926, E-ISSN 2469-9934, Vol. 99, no 5, article id 052118Article in journal (Refereed)

We address the breakdown of the bulk-boundary correspondence observed in non-Hermitian systems, where open and periodic systems can have distinct phase diagrams. The correspondence can be completely restored by considering the Hamiltonian's singular-value decomposition instead of its eigendecomposition. This leads to a natural topological description in terms of a flattened singular decomposition. This description is equivalent to the usual approach for Hermitian systems and coincides with a recent proposal for the classification of non-Hermitian systems. We generalize the notion of the entanglement spectrum to non-Hermitian systems, and show that the edge physics is indeed completely captured by the periodic bulk Hamiltonian. We exemplify our approach by considering the chiral non-Hermitian Su-Schrieffer-Heger and Chern insulator models. Our work advocates a different perspective on topological non-Hermitian Hamiltonians, paving the way to a better understanding of their entanglement structure.

• 31.
Friedrich-Schiller-Univesität, JGermany.
Computation of Poincare-Betti series for monomial rings2005In: Rendiconti dell'Istituto di Matematica dell'Università di Trieste, ISSN 0049-4704, E-ISSN 2464-8728, Vol. 37, no 1-2, p. 85-94Article in journal (Refereed)
• 32.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Matrix Riemann-Hilbert problems with jumps across Carleson contours2018In: Monatshefte für Mathematik (Print), ISSN 0026-9255, E-ISSN 1436-5081, Vol. 186, no 1, p. 111-152Article in journal (Refereed)

We develop a theory of n x n-matrix Riemann-Hilbert problems for a class of jump contours and jump matrices of low regularity. Our basic assumption is that the contour Gamma is a finite union of simple closed Carleson curves in the Riemann sphere. In particular, unbounded contours with cusps, corners, and nontransversal intersections are allowed. We introduce a notion of L-p-Riemann-Hilbert problem and establish basic uniqueness results and Fredholm properties. We also investigate the implications of Fredholmness for the unique solvability and prove a theorem on contour deformation.

• 33.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Computing Seshardi constants on smooth toric surfacesManuscript (preprint) (Other academic)

In this paper we compute the Seshadri constants at the general point on many smooth polarized toric surfaces. We consider the case when the degree of jet separation is small or the core of the associated polygon is a line segment. Our main result is that in this case the Seshadri constant at the general point can often be determined in terms of easily computable invariants of the surfaces at hand. Lastly we consider the case that the core of the associated polygon is a point for a smooth polarized toric surface (X, L ). We show that in this case X can be constructed via consecutive equivariant blow-ups of either P^2 or P^1 x P^1.

• 34.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Fachbereich Mathematik und Informatik, Philipps-Universität Marburg. Department of Mathematics, University of Nebraska-Lincoln. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago. Instytut Matematyki, UP. Instytut Matematyki UP.
Bounded Negativity and Arrangements of Lines2015In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, Vol. 2015, no 19, p. 9456-9471Article in journal (Refereed)

The Bounded Negativity Conjecture predicts that for any smooth complex surface X there exists a lower bound for the selfintersection of reduced divisors on X. This conjecture is open. It is also not known if the existence of such a lower bound is invariant in the birational equivalence class of X. In the present note, we introduce certain constants H(X) which measure in effect the variance of the lower bounds in the birational equivalence class of X. We focus on rational surfaces and relate the value of H(ℙ^2) to certain line arrangements. Our main result is Theorem 3.3 and the main open challenge is Problem 3.10.

• 35.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Department of mathematics & statistics, University of Michigan-Dearborn.
A note on higher order Gauss MapsManuscript (preprint) (Other academic)

We study Gauss maps of order k, associated to a projective variety X embedded in projective space via a line bundle L. We show that if X is a smooth, complete complex variety and L is a k-jet spanned line bundle on X, with k > 1, then the Gauss map of order k has finite fibers, unless X = P^n is embedded by the Veronese embedding of order k. In the case where X is a toric variety, we give a combinatorial description of the Gauss maps of order k, its image and the general fibers.

• 36.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
LatticePolytopes: A package for computations with lattice polytopes in Macaulay2Manuscript (preprint) (Other academic)

We introduce the package LatticePolytopes for Macaulay2. The package provides methods for computations related to Cayley structures, local positivity and smoothness for lattice polytopes.

• 37.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
A topological approach to data visualization2015Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

Barcoding is a mathematical tool, to analyze data, which is based on the theory of persistent homology. In this thesis both Hierarchical Clustering and Barcoding are defined and analyzed according to three criterion: Continuity, Computability and Visualizability. It is also presented how the two methods, barcoding and hierarchical clustering, are connected and why barcoding, in some cases, is a generalized method of hierarchical clustering. Lastly some more question of interest, for better understanding barcoding, are stated.

• 38.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
A remark on Getzler's semi-classical approximation2012In: Geometry And Arithmetic, EMS Publishing House , 2012, p. 309-316Conference paper (Refereed)

Ezra Getzler notes in the proof of the main theorem of [Get98] that ”A proof of the theorem could no doubt be given using [a combinatorial interpretation in terms of a sum over necklaces]; however, we prefer to derive it directly from Theorem 2.2”. In this note we give such a direct combinatorial proof using wreath product symmetric functions.

• 39.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

This thesis contains three articles related to operads and moduli spaces of admissible covers of curves. In Paper A we isolate cohomology classes coming from modular forms inside a certain space of admissible covers, thereby showing that this moduli space can be used as a substitute for a Kuga–Sato variety. Paper B contains a combinatorial proof of Ezra Getzler’s semiclassical approximation for modular operads, and a proof of a formula needed in Paper A. In Paper C we explain in what sense spaces of admissible covers form a modular operad, by introducing the notion of an operad colored by a groupoid.

• 40.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Cohomology of local systems on loci of d-elliptic Abelian surfaces2013In: The Michigan mathematical journal, ISSN 0026-2285, E-ISSN 1945-2365, Vol. 62, no 4, p. 705-720Article in journal (Refereed)

We consider the loci of d-elliptic curves in $M_2$, and corresponding loci of d-elliptic surfaces in $A_2$. We show how a description of these loci as quotients of a product of modular curves can be used to calculate cohomology of natural local systems on them, both as mixed Hodge structures and $\ell$-adic Galois representations. We study in particular the case d=2, and compute the Euler characteristic of the moduli space of n-pointed bi-elliptic genus 2 curves in the Grothendieck group of Hodge structures.

• 41.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Cusp form motives and admissible G-covers2012In: Algebra & Number Theory, ISSN 1937-0652, E-ISSN 1944-7833, Vol. 6, no 6, p. 1199-1221Article in journal (Refereed)

There is a natural S-n-action on the moduli space (M) over bar (1,n) (B(Z/mZ)(2)) of twisted stable maps into the stack B(Z/mZ)(2), and so its cohomology may be decomposed into irreducible S-n-representations. Working over Spec Z [1/m] we show that the alternating part of the cohomology of one of its connected components is exactly the cohomology associated to cusp forms for Gamma(m). In particular this offers an alternative to Scholl's construction of the Chow motive associated to such cusp forms. This answers in the affirmative a question of Manin on whether one can replace the Kuga-Sato varieties used by Scholl with some moduli space of pointed stable curves.

• 42.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

We describe the modular operad structure on the moduli spaces of pointed stable curves equipped with an admissible G-cover. To do this we are forced to introduce the notion of an operad colored not by a set but by the objects of a groupoid. This construction interpolates in some sense between “framed” and “colored” versions of operads; we hope that it will be of independent interest. An algebra over this operad is the same thing as a G-equivariant CohFT. Our main theorem is an extension of the symmetric function formalism for modular operads to this setting; we prove an analogue of the formula of Getzler and Kapranov describing the effect of the “free modular operad” functor on the level of symmetric functions.

• 43.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Noetherian approximation of algebraic spaces and stacks2015In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 422, p. 105-147Article in journal (Refereed)

We show that every scheme (resp. algebraic space, resp. algebraic stack) that is quasi-compact with quasi-finite diagonal can be approximated by a noetherian scheme (resp. algebraic space, resp. stack). More generally, we show that any stack which is etale-locally a global quotient stack can be approximated. Examples of applications are generalizations of Chevalley's, Serre's and Zariski's theorems and Chow's lemma to the non-noetherian setting. We also show that every quasi-compact algebraic stack with quasi-finite diagonal has a finite generically flat cover by a scheme.

• 44.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Multidimensional Persistence and Noise2017In: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383, Vol. 17, no 6, p. 1367-1406Article in journal (Refereed)

In this paper, we study multidimensional persistence modules (Carlsson and Zomorodian in Discrete Comput Geom 42(1):71–93, 2009; Lesnick in Found Comput Math 15(3):613–650, 2015) via what we call tame functors and noise systems. A noise system leads to a pseudometric topology on the category of tame functors. We show how this pseudometric can be used to identify persistent features of compact multidimensional persistence modules. To count such features, we introduce the feature counting invariant and prove that assigning this invariant to compact tame functors is a 1-Lipschitz operation. For one-dimensional persistence, we explain how, by choosing an appropriate noise system, the feature counting invariant identifies the same persistent features as the classical barcode construction.

• 45.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Weil restriction and the Quot scheme2015In: Algebraic Geometry, ISSN 2313-1691, E-ISSN 2214-2584, Vol. 2, no 4, p. 514-534Article in journal (Refereed)

We introduce a concept that we call module restriction, which generalizes the classical Weil restriction. After having established some fundamental properties as existence and étaleness, we apply our results to show that the Quot functor Quotn FX/S of Grothendieck is representable by an algebraic space for any quasi-coherent sheaf FX on any separated algebraic space X → S.

• 46.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
An intrinsic definition of the Rees algebra of a moduleIn: Proceedings of the Edinburgh Mathematical Society, ISSN 0013-0915, E-ISSN 1464-3839Article in journal (Refereed)

This paper concerns a generalization of the Rees algebra of ideals due to Eisenbud, Huneke and Ulrich that works for any finitely generated module over a noetherian ring. Their definition is in terms of maps to free modules. We give an intrinsic definition using divided powers.

• 47.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Good Hilbert functorsManuscript (preprint) (Other academic)

We introduce the good Hilbert functor and prove that it is algebraic. This functor generalizes various versions of the Hilbert moduli problem, such as the multigraded Hilbert scheme and the invariant Hilbert scheme. Moreover, we generalize a result concerning formal GAGA for good moduli spaces.

• 48.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Gotzmann's persistence theorem for finite modulesManuscript (preprint) (Other academic)

We prove a generalization of Gotzmann's persistence theorem in the case of modules with constant Hilbert polynomial. As a consequence, we show that the defining equations that give the embedding of a Quot scheme of points into a Grassmannian are given by a single Fitting ideal.

• 49.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Hilbert schemes and Rees algebras2016Doctoral thesis, comprehensive summary (Other academic)

The topic of this thesis is algebraic geometry, which is the mathematical subject that connects polynomial equations with geometric objects. Modern algebraic geometry has extended this framework by replacing polynomials with elements from a general commutative ring, and studies the geometry of abstract algebra. The thesis consists of six papers relating to some different topics of this field.

The first three papers concern the Rees algebra. Given an ideal of a commutative ring, the corresponding Rees algebra is the coordinate ring of a blow-up in the subscheme defined by the ideal. We study a generalization of this concept where we replace the ideal with a module. In Paper A we give an intrinsic definition of the Rees algebra of a module in terms of divided powers. In Paper B we show that features of the Rees algebra can be explained by the theory of coherent functors. In Paper C we consider the geometry of the Rees algebra of a module, and characterize it by a universal property.

The other three papers concern various moduli spaces. In Paper D we prove a partial generalization of Gotzmann’s persistence theorem to modules, and give explicit equations for the embedding of a Quot scheme inside a Grassmannian. In Paper E we expand on a result of Paper D, concerning the structure of certain Fitting ideals, to describe projective embeddings of open affine subschemes of a Hilbert scheme. Finally, in Paper F we introduce the good Hilbert functor parametrizing closed substacks with proper good moduli spaces of an algebraic stack, and we show that this functor is algebraic under certain conditions on the stack.

• 50.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Rees algebras of modules and coherent functorsManuscript (preprint) (Other academic)

We show that several properties of the theory of Rees algebras of modules become more transparent using the category of coherent functors rather than working directly with modules. In particular, we show that the Rees algebra is induced by a canonical map of coherent functors.

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