Related papers: Spherical amoebae and a spherical logarithm map

We establish a rigorous geometric framework for quantum fields on a stochastic gravitational background. Starting from a master partition function that averages over metric fluctuations, we define a matter amplitude $\mathcal{K}$, whose…

Given a bounded open set $\Omega\subset \mathbb{R}^n$, we study sequences of quadratic functionals on the Sobolev space $H^1_0(\Omega)$, perturbed by sequences of bounded linear functionals. We prove that their $\Gamma$-limits, in the weak…

Let \Gamma be a lattice in G=SL(n,R) and X=G/S a homogeneous space of G, where S is a closed subgroup of G which contains a real algebraic subgroup H such that G/H is compact. We establish uniform distribution of orbits of \Gamma in X…

In this paper we derive an explicit formula for the Hirzebruch-Mumford volume of an indefinite lattice L of rank at least 3. If \Gamma is an arithmetic subgroup of the group O(L) of isometries of L and L has signature (2,n), then an…

Let $G$ be a connected semisimple Lie group with finite centre and $K$ be a maximal compact subgroup thereof. Given a function $u$ on $G$, we define $\mathcal{A} u$ to be the root mean square average over $K$, acting both on the left and…

A closed subgroup $H$ of a connected reductive group $G$ is called $\textit{spherical}$ if a Borel subgroup in $G$ has an open orbit on $G/H$. We give a combinatorial characterization for a spherical subgroup to be contained in another one…

For a simply connected (non-nilpotent) solvable Lie group $G$ with a lattice $\Gamma$ the de Rham and Dolbeault cohomologies of the solvmanifold $G/\Gamma$ are not in general isomorphic to the cohomologies of the Lie algebra $\mathfrak g$…

A general theory of matrix-spherical functions for dual Hopf algebras and right coideal subalgebras is developed. We establish their existence and define their orthogonality relations. When specialized to Kolb and Letzter's quantum…

The notion of $\Gamma$-symmetric space is a natural generalization of the classical notion of symmetric space based on $\Z_2$-grading of Lie algebras. In our case, we consider homogeneous spaces $G/H$ such that the Lie algebra $\g$ of $G$…

Let T be a free ergodic measure-preserving action of an abelian group G on (X,mu). The crossed product algebra R_T has two distinguished masas, the image C_T of L^infty(X,mu) and the algebra S_T generated by the image of G. We conjecture…

Let $\Gamma$ be a Zariski dense Anosov subgroup of a connected semisimple real algebraic group $G$. For a maximal horospherical subgroup $N$ of $G$, we show that the space of all non-trivial $NM$-invariant ergodic and $A$-quasi-invariant…

The complexity of an action of a reductive algebraic group G on an algebraic variety X is the codimension of a generic B-orbit in X, where B is a Borel subgroup of G. We classify affine homogeneous spaces G/H of complexity one. These…

We classify all harmonic maps with finite uniton number from a Riemann surface into an arbitrary compact simple Lie group $G$, whether $G$ has trivial centre or not, in terms of certain pieces of the Bruhat decomposition of the group…

On the unit sphere $\mathbb{S}$ in a real Hilbert space $\mathbf{H}$, we derive a binary operation $\odot$ such that $(\mathbb{S},\odot)$ is a power-associative Kikkawa left loop with two-sided identity $\mathbf{e}_0$, i.e., it has the left…

Let G=G(t,z) be one of the N^2-dimensional bicovariant first order differential calculi for the quantum groups GL_q(N), SL_q(N), O_q(N), or Sp_q(N), where q is a transcendental complex number and z is a regular parameter. It is shown that…

We construct the Hilbert space costratification of $G=\mathrm{SU}(2)$-quantum gauge theory on a finite spatial lattice in the Hamiltonian approach. We build on previous work where we have implemented the classical gauge orbit strata on…

We study weighted Sobolev inequalities on open convex cones endowed with $\alpha$-homogeneous weights satisfying a certain concavity condition. We establish a so-called reduction principle for these inequalities and characterize optimal…

We revisit the MIC-harmonic oscillator in flat space with monopole interaction and derive the polynomial algebra satisfied by the integrals of motion and its energy spectrum using the ad hoc recurrence approach. We introduce a…

The goal of the course was a review of results mainly due to M. Olbrich and the first author. We consider a discrete cocompact subgroup $\Gamma$ of a semisimple Lie group $G$. We relate the group cohomology of $\Gamma$ with coefficients in…

We prove an estimate for spherical functions $\varphi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian…