8,479 research outputs found
An Introduction to Harmonic Manifolds and the Lichnerowicz Conjecture
The title is self-explanatory. We aim to give an easy to read and
self-contained introduction to the field of harmonic manifolds. Only basic
knowledge of Riemannian geometry is required. After we gave the definition of
harmonicity and derived some properties, we concentrate on Z. I. Szab\'o's
proof of Lichnerowicz's conjecture in the class of compact simply connected
manifolds
Bordered Floer homology and the spectral sequence of a branched double cover I
Given a link in the three-sphere, Z. Szab\'o and the second author
constructed a spectral sequence starting at the Khovanov homology of the link
and converging to the Heegaard Floer homology of its branched double-cover. The
aim of this paper and its sequel is to explicitly calculate this spectral
sequence, using bordered Floer homology. There are two primary ingredients in
this computation: an explicit calculation of filtered bimodules associated to
Dehn twists and a pairing theorem for polygons. In this paper we give the first
ingredient, and so obtain a combinatorial spectral sequence from Khovanov
homology to Heegaard Floer homology; in the sequel we show that this spectral
sequence agrees with the previously known one.Comment: 45 pages, 16 figures. v2: Published versio
Wiener ( Ornstein-Uhlenbeck processes. A generalization of known processes
We collect, scattered through literature, as well as we prove some new
properties of two Markov processes that in many ways resemble Wiener and
Ornstein--Uhlenbeck processes. Although processes considered in this paper were
defined either in non-commutative probability context or through quadratic
harnesses we define them once more as so to say 'continuous time '
generalization of a simple, symmetric, discrete time process satisfying simple
conditions imposed on the form of its first two conditional moments. The finite
dimensional distributions of the first one (say X=(X_{t})_{t\geq0} called
q-Wiener) depends on one parameter q\in(-1,1] and of the second one (say
Y=(Y_{t})_{t\inR} called ({\alpha},q)- Ornstein--Uhlenbeck) on two parameters
({\alpha},q)\in(0,\infty)\times(-1,1]. The first one resembles Wiener process
in the sense that for q=1 it is Wiener process but also that for |q|<1 and
\foralln\geq1: t^{n/2}H_{n}(X_{t}/\surdt|q), where (H_{n})_{n\geq0} are the so
called q-Hermite polynomials, are martingales. It does not have however neither
independent increments not allows continuous sample path modification. The
second one resembles Ornstein--Uhlenbeck process. For q=1 it is a classical OU
process. For |q|<1 it is also stationary with correlation function equal to
exp(-{\alpha}|t-s|) and has many properties resembling those of its classical
version. We think that these process are fascinating objects to study posing
many interesting, open questions.Comment: 25 page
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