33,697 research outputs found
K-Stability for Fano Manifolds with Torus Action of Complexity One
We consider Fano manifolds admitting an algebraic torus action with general
orbit of codimension one. Using a recent result of Datar and Szekelyhidi, we
effectively determine the existence of Kahler-Ricci solitons for those
manifolds via the notion of equivariant K-stability. This allows us to give new
examples of Kahler-Einstein Fano threefolds, and Fano threefolds admitting a
non-trivial Kahler-Ricci soliton.Comment: 19 pages, 5 figures, changed to a more precise titl
A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups
We give a simple combinatorial criterion, in terms of an action on a
hyperbolic simplicial complex, for a group to be hierarchically hyperbolic. We
apply this to show that quotients of mapping class groups by large powers of
Dehn twists are hierarchically hyperbolic (and even relatively hyperbolic in
the genus 2 case). Under residual finiteness assumptions, we construct many
non-elementary hyperbolic quotients of mapping class groups. Using these
quotients, we reduce questions of Reid and Bridson-Reid-Wilton about finite
quotients of mapping class groups to residual finiteness of specific hyperbolic
groups.Comment: Revised according to comments from reader
On the frontiers of polynomial computations in tropical geometry
We study some basic algorithmic problems concerning the intersection of
tropical hypersurfaces in general dimension: deciding whether this intersection
is nonempty, whether it is a tropical variety, and whether it is connected, as
well as counting the number of connected components. We characterize the
borderline between tractable and hard computations by proving
-hardness and #-hardness results under various
strong restrictions of the input data, as well as providing polynomial time
algorithms for various other restrictions.Comment: 17 pages, 5 figures. To appear in Journal of Symbolic Computatio
Systolic geometry and simplicial complexity for groups
Twenty years ago Gromov asked about how large is the set of isomorphism
classes of groups whose systolic area is bounded from above. This article
introduces a new combinatorial invariant for finitely presentable groups called
{\it simplicial complexity} that allows to obtain a quite satisfactory answer
to his question. Using this new complexity, we also derive new results on
systolic area for groups that specify its topological behaviour.Comment: 35 pages, 9 figure
Acylindrical hyperbolicity of cubical small-cancellation groups
We provide an analogue of Strebel's classification of geodesic triangles in
classical groups for groups given by Wise's cubical presentations
satisfying sufficiently strong metric cubical small cancellation conditions.
Using our classification, we prove that, except in specific degenerate cases,
such groups are acylindrically hyperbolic.Comment: Added figures. Exposition improved in Section 3,
correction/simplification in Section 5, background added and citations
updated in Section
Dynamics of a birth-death process based on combinatorial innovation
A feature of human creativity is the ability to take a subset of existing
items (e.g. objects, ideas, or techniques) and combine them in various ways to
give rise to new items, which, in turn, fuel further growth. Occasionally, some
of these items may also disappear (extinction). We model this process by a
simple stochastic birth--death model, with non-linear combinatorial terms in
the growth coefficients to capture the propensity of subsets of items to give
rise to new items. In its simplest form, this model involves just two
parameters . This process exhibits a characteristic 'hockey-stick'
behaviour: a long period of relatively little growth followed by a relatively
sudden 'explosive' increase. We provide exact expressions for the mean and
variance of this time to explosion and compare the results with simulations. We
then generalise our results to allow for more general parameter assignments,
and consider possible applications to data involving human productivity and
creativity.Comment: 21 pages, 4 figure
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