1,320 research outputs found
Homotopy theory with bornological coarse spaces
We propose an axiomatic characterization of coarse homology theories defined
on the category of bornological coarse spaces. We construct a category of
motivic coarse spectra. Our focus is the classification of coarse homology
theories and the construction of examples. We show that if a transformation
between coarse homology theories induces an equivalence on all discrete
bornological coarse spaces, then it is an equivalence on bornological coarse
spaces of finite asymptotic dimension. The example of coarse K-homology will be
discussed in detail.Comment: 220 pages (complete revision
Inquisitive bisimulation
Inquisitive modal logic InqML is a generalisation of standard Kripke-style
modal logic. In its epistemic incarnation, it extends standard epistemic logic
to capture not just the information that agents have, but also the questions
that they are interested in. Technically, InqML fits within the family of
logics based on team semantics. From a model-theoretic perspective, it takes us
a step in the direction of monadic second-order logic, as inquisitive modal
operators involve quantification over sets of worlds. We introduce and
investigate the natural notion of bisimulation equivalence in the setting of
InqML. We compare the expressiveness of InqML and first-order logic in the
context of relational structures with two sorts, one for worlds and one for
information states. We characterise inquisitive modal logic, as well as its
multi-agent epistemic S5-like variant, as the bisimulation invariant fragment
of first-order logic over various natural classes of two-sorted structures.
These results crucially require non-classical methods in studying bisimulation
and first-order expressiveness over non-elementary classes of structures,
irrespective of whether we aim for characterisations in the sense of classical
or of finite model theory
A classification of separable Rosenthal compacta and its applications
The present work consists of three parts. In the first one we determine the
prototypes of separable Rosenthal compacta and we provide a classification
theorem. The second part concerns an extension of a theorem of S. Todorcevic.
The last one is devoted to applications.Comment: 55 pages, no figure
Compactness of higher-order Sobolev embeddings
We study higher-order compact Sobolev embeddings on a domain endowed with a probability measure and satisfying
certain isoperimetric inequality. Given , we present a
condition on a pair of rearrangement-invariant spaces and
which suffices to guarantee a compact embedding of the Sobolev
space into . The condition is given in terms
of compactness of certain one-dimensional operator depending on the
isoperimetric function of . We then apply this result to the
characterization of higher-order compact Sobolev embeddings on concrete measure
spaces, including John domains, Maz'ya classes of Euclidean domains and product
probability spaces, whose standard example is the Gauss space
A Thurston boundary for infinite-dimensional Teichm\"uller spaces
For a compact surface , Thurston introduced a compactification of its
Teichm\"uller space by completing it with a boundary
consisting of projective measured geodesic laminations. We
introduce a similar bordification for the Teichm\"uller space
of a noncompact Riemann surface , using the technical tool of geodesic
currents. The lack of compactness requires the introduction of certain
uniformity conditions which were unnecessary for compact surfaces. A technical
step, providing a convergence result for earthquake paths in ,
may be of independent interest.Comment: 42 pages, 3 figure
Finsler bordifications of symmetric and certain locally symmetric spaces
We give a geometric interpretation of the maximal Satake compactification of
symmetric spaces of noncompact type, showing that it arises by
attaching the horofunction boundary for a suitable -invariant Finsler metric
on . As an application, we establish the existence of natural
bordifications, as orbifolds-with-corners, of locally symmetric spaces
for arbitrary discrete subgroups . These bordifications
result from attaching -quotients of suitable domains of proper
discontinuity at infinity. We further prove that such bordifications are
compactifications in the case of Anosov subgroups. We show, conversely, that
Anosov subgroups are characterized by the existence of such compactifications
among uniformly regular subgroups. Along the way, we give a positive answer, in
the torsion free case, to a question of Ha\"issinsky and Tukia on convergence
groups regarding the cocompactness of their actions on the domains of
discontinuity.Comment: 88 page
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