The Theory of Quasiconformal Mappings in Higher Dimensions, I

We present a survey of the many and various elements of the modern higher-dimensional theory of quasiconformal mappings and their wide and varied application. It is unified (and limited) by the theme of the author's interests. Thus we will discuss the basic theory as it developed in the 1960s in the early work of F.W. Gehring and Yu G. Reshetnyak and subsequently explore the connections with geometric function theory, nonlinear partial differential equations, differential and geometric topology and dynamics as they ensued over the following decades. We give few proofs as we try to outline the major results of the area and current research themes. We do not strive to present these results in maximal generality, as to achieve this considerable technical knowledge would be necessary of the reader. We have tried to give a feel of where the area is, what are the central ideas and problems and where are the major current interactions with researchers in other areas. We have also added a bit of history here and there. We have not been able to cover the many recent advances generalising the theory to mappings of finite distortion and to degenerate elliptic Beltrami systems which connects the theory closely with the calculus of variations and nonlinear elasticity, nonlinear Hodge theory and related areas, although the reader may see shadows of this aspect in parts

Topology of Platonic Spherical Manifolds: From Homotopy to Harmonic Analysis

We carry out the harmonic analysis on four Platonic spherical three-manifolds with different topologies. Starting out from the homotopies (Everitt 2004), we convert them into deck operations, acting on the simply connected three-sphere as the cover, and obtain the corresponding variety of deck groups. For each topology, the three-sphere is tiled into copies of a fundamental domain under the corresponding deck group. We employ the point symmetry of each Platonic manifold to construct its fundamental domain as a spherical orbifold. While the three-sphere supports an~orthonormal complete basis for harmonic analysis formed by Wigner polynomials, a given spherical orbifold leads to a selection of a specific subbasis. The resulting selection rules find applications in cosmic topology, probed by the cosmic microwave background.Comment: 29 pages, 4 figure

Comparison between algebraic and topological K-theory of locally convex algebras

This paper is concerned with the algebraic K-theory of locally convex algebras stabilized by operator ideals, and its comparison with topological K-theory. We show that the obstruction for the comparison map between algebraic and topological K-theory to be an isomorphism is (absolute) algebraic cyclic homology and prove the existence of an 6-term exact sequence. We show that cyclic homology vanishes in the case when J is the ideal of compact operators and L is a Frechet algebra with bounded app. unit. This proves the generalized version of Karoubi's conjecture due to Mariusz Wodzicki and announced in his paper "Algebraic K-theory and functional analysis", First European Congress of Mathematics, Vol. II (Paris, 1992), 485--496, Progr. Math., 120, Birkh\"auser, Basel, 1994. We also consider stabilization with respect to a wider class of operator ideals, called sub-harmonic. We study the algebraic K-theory of the tensor product of a sub-harmonic ideal with an arbitrary complex algebra and prove that the obstruction for the periodicity of algebraic K-theory is again cyclic homology. The main technical tools we use are the diffeotopy invariance theorem of Cuntz and the second author (which we generalize), and the excision theorem for infinitesimal K-theory, due to the first author.Comment: Final version, to appear in Advances in Mathematic

Average-Value Tverberg Partitions via Finite Fourier Analysis

The long-standing topological Tverberg conjecture claimed, for any continuous map from the boundary of an $N(q,d):=(q-1)(d+1)$-simplex to $d$-dimensional Euclidian space, the existence of $q$ pairwise disjoint subfaces whose images have non-empty $q$-fold intersection. The affine cases, true for all $q$, constitute Tverberg's famous 1966 generalization of the classical Radon's Theorem. Although established for all prime powers in 1987 by \"Ozaydin, counterexamples to the conjecture, relying on 2014 work of Mabillard and Wagner, were first shown to exist for all non-prime-powers in 2015 by Frick. Starting with a reformulation of the topological Tverberg conjecture in terms of harmonic analysis on finite groups, we show that despite the failure of the conjecture, continuous maps \textit{below} the tight dimension $N(q,d)$ are nonetheless guaranteed $q$ pairwise disjoint subfaces -- including when $q$ is not a prime power -- which satisfy a variety of "average value" coincidences, the latter obtained as the vanishing of prescribed Fourier transforms.Comment: 9 pages; to appear in Israel J. Math. Final version eliminates some typo

Martin Boundary Theory of some Quantum Random Walks

In this paper we define a general setting for Martin boundary theory associated to quantum random walks, and prove a general representation theorem. We show that in the dual of a simply connected Lie subgroup of U(n), the extremal Martin boundary is homeomorphic to a sphere. Then, we investigate restriction of quantum random walks to Abelian subalgebras of group algebras, and establish a Ney-Spitzer theorem for an elementary random walk on the fusion algebra of SU(n), generalizing a previous result of Biane. We also consider the restriction of a quantum random walk on $SU_q(n)$ introduced by Izumi to two natural Abelian subalgebras, and relate the underlying Markov chains by classical probabilistic processes. This result generalizes a result of Biane.Comment: 29 page