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.

Karlstad University, Faculty of Health, Science and Technology (starting 2013).

Buchberger, Igor

Karlstad University, Faculty of Health, Science and Technology (starting 2013).

Enander, Jonas

Stockholm University.

Mörtsell, Edvard

Stockholm University.

Sjörs, Stefan

Stockholm University.

Growth Histories in Bimetric Massive Gravity2012In: Journal of Cosmology and Astroparticle Physics, ISSN 1475-7516, E-ISSN 1475-7516, no 12, article id 021Article in journal (Refereed)

Abstract [en]

We perform cosmological perturbation theory in Hassan-Rosen bimetric gravity for general homogeneous and isotropic backgrounds. In the de Sitter approximation, we obtain decoupled sets of massless and massive scalar gravitational fluctuations. Matter perturbations then evolve like in Einstein gravity. We perturb the future de Sitter regime by the ratio of matter to dark energy, producing quasi-de Sitter space. In this more general setting the massive and massless fluctuations mix. We argue that in the quasi-de Sitter regime, the growth of structure in bimetric gravity differs from that of Einstein gravity.

We consider a non-linear half-space problem related to the condensation problem for the discrete Boltzmann equation and extend some known results for a single-component gas to the case when a non-condensable gas is present. The vapor is assumed to tend to an assigned Maxwellian at infinity, as the non-condensable gas tends to zero at infinity. We assume that the vapor is completely absorbed and that the non-condensable gas is diffusively reflected at the condensed phase and that the vapor molecules leaving the condensed phase are distributed according to a given distribution. The conditions, on the given distribution, needed for the existence of a unique solution of the problem are investigated. We also find exact solvability conditions and solutions for a simplified six+four-velocity model, as the given distribution is a Maxwellian at rest, and study a simplified twelve+six-velocitymodel.

It is known that for any full rational conformal field theory, the correlation functions that are obtained bythe TFT construction satisfy all locality, modular invariance and factorization conditions, and that there isa small set of fundamental correlators to which all others are related via factorization – provided that theworld sheets considered do not contain any non-trivial defect lines. In this paper we generalize both resultsto oriented world sheets with an arbitrary network of topological defect lines.

For any finite-dimensional factorizable ribbon Hopf algebra H and any ribbon automorphism of H, we establish the existence of thefollowing structure: an H-bimodule Fω and a bimodule morphism Zω from Lyubashenko’s Hopf algebra object K for the bimodulecategory to Fω. This morphism is invariant under the natural actionof the mapping class group of the one-punctured torus on thespace of bimodule morphisms from K to Fω. We further showthat the bimodule Fω can be endowed with a natural structureof a commutative symmetric Frobenius algebra in the monoidalcategory of H-bimodules, and that it is a special Frobenius algebra iff H is semisimple.The bimodules K and Fω can both be characterized as coends ofsuitable bifunctors. The morphism Zω is obtained by applying amonodromy operation to the coproduct of Fω; a similar constructionfor the product of Fω exists as well. Our results are motivated by the quest to understand the bulkstate space and the bulk partition function in two-dimensionalconformal field theories with chiral algebras that are not necessarily semisimple

The Shimura curve of discriminant 10 is uniformized by a subgroup of an arithmetic $(2,2,2,3)$ quadrilateral group. We derive the differential structure of the ring of modular forms for the Shimura curve and relate the ring generators to explicit Heun functions for the quadrilateral group. We also show that the Picard–Fuchs equation of the associated family of abelian surfaces has solutions that are modular forms. These results are used to completely describe the exceptional sets of the Heun functions, and we show how to find examples like \[ Hl\left(\frac{27}{2},\frac{7}{36}; \frac{1}{12},\frac{7}{12},\frac{2}{3},\frac{1}{2}; -\frac{96}{25}\right)=\frac{2^{1/2}5^{2/3}}{3^{4/3}}. \]

We study the differential structure of the ring of modular forms for the unit group of the quaternion algebra over Q of discriminant 6. Using these results we give an explicit formula for Taylor expansions of the modular forms at the elliptic points. Using appropriate normalizations we show that the Taylor coefficients at the elliptic points of the generators of the ring of modular forms are all rational and 6-integral. This gives a rational structure on the ring of modular forms. We give a recursive formula for computing the Taylor coefficients of modular forms at elliptic points and, as an application, give an algorithm for computing modular polynomials.

Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.

My First Meetings with Konstantin Oskolkov2013In: Recent Advances in Harmonic Analysis and Applications: In honor of Konstantin Oskolkov / [ed] Dmitriy Bilyk, Laura De Carli, Alexander Petukhov, Alexander M. Stokolos, Brett D. Wick, Springer, 2013, 1, Vol. 25, p. 27-29Chapter in book (Refereed)

Abstract [en]

This note tells about our first meetings with Konstatin Oskolkov. We discuss also optimal estimates of the rate of convergence of Fourier series obtained by Oskolkov in 1975.

Representations of finite groups2017Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

Abstract [en]

Representation theory is concerned with the ways of writing elements of abstract algebraic structures as linear transformations of vector spaces. Typical structures amenable to representation theory are groups, associative algebras and Lie algebras. In this thesis we study linear representations of finite groups. The study focuses on character theory and how character theory can be used to extract information from a group. Prior to that, concepts needed to treat character theory, and some of their ramifications, are investigated. The study is based on existing literature, with excessive use of examples to illuminate important aspects. An example treating a class of p-groups is also discussed.