For a compact space X we consider extending endomorphisms of the algebra C(X) to be endomorphisms of Arens-Hoffman and Cole extensions of C(X). Given a non-linear, monic polynomial p ∈ C(X)[t], with C(X)[t]/pC(X)[t] semi-simple, we show that if an endomorphism of C(X) extends to the Arens-Hoffman extension with respect to p, then it also extends to the simple Cole extension with respect to p. W...

Let k be an algebraically closed field and let T be the k-linear algebraic triangulated category generated by a w-spherical object for an integer w. For certain values of w this category is classical. For instance, if w = 0 then it is the compact derived category of the dual numbers over k. As main results of the paper we show that for w ≤ 0, the category T has no non-trivial t-structures, but ...

We define a motivic analogue of the Haar measure for groups of the form G(k((t))), where k is an algebraically closed field of characteristic zero, and G is a reductive algebraic group defined over k. A classical Haar measure on such groups does not exist since they are not locally compact. We use the theory of motivic integration introduced by M. Kontsevich to define an additive function on a ...

We study polynomials with coefficients in a field L as dynamical systems where L is any algebraically closed and complete ultrametric field with dense valuation group and characteristic zero residual field. We give a complete description of the dynamical and parameter space of cubic polynomials. In particular we characterize cubic polynomials with compact Julia sets. Also, we prove that any inf...

We give a concise introduction to the notion of algebraic integrability. Our exposition is based on examples and phenomena, rather than on detailed proofs of abstract theorems. We mainly focus on algebraic integrability in the sense of Adler-van Moerbeke, where the fibres of the momentum map are affine parts of Abelian varieties; as it turns out, most examples from classical mechanics are of th...

We algebraically analysis the quantum Hall effect of a system of particles living on the disc B1 in the presence of an uniform magnetic field B. For this, we identify the non-compact disc with the coset space SU(1, 1)/U(1). This allows us to use the geometric quantization in order to get the wavefunctions as the Wigner D-functions satisfying a suitable constraint. We show that the corresponding...

Let F be a non-Archimedean locally compact field with residual characteristic p, let G be an inner form of GLn(F), n 1 and let R be an algebraically closed field of characteristic different from p. When R has characteristic > 0, the image of an irreducible smooth R-representation π of G by the Aubert involution need not be irreducible. We prove that this image (in the Grothendieck group of G) c...

Flatband networks are characterized by the coexistence of dispersive and flatbands. Flatbands (FBs) are generated by compact localized eigenstates (CLSs) with local network symmetries, based on destructive interference. Correlated disorder and quasiperiodic potentials hybridize CLSs without additional renormalization, yet with surprising consequences: (i) states are expelled from the FB energy ...