This volume contains the proceedings of the CIEM workshop on Tropical Geometry, held December 12-16, 2011, at the International Centre for Mathematical Meetings (CIEM), Castro Urdiales, Spain. ...Tropical geometry is a new and rapidly developing field of mathematics which has deep connections with various areas of mathematics and physics, such as algebraic geometry, symplectic geometry, complex analysis, dynamical systems, combinatorics, statistical physics, and string theory. As reflected by the content of this volume, this meeting was mainly focused on the geometric side of the tropical world with an emphasis on relations between tropical geometry, algebraic geometry, and combinatorics. This volume provides an overview of current trends concerning algebraic and combinatorial aspects of tropical geometry through eleven papers combining expository parts and development of modern techniques and tools.
This article develops a method to compute the action induced on the fundamental group by an automorphism of a real rational surface. Then, it uses this method to compute the induced actions for a ...certain family of basic quadratic real automorphisms. By utilizing an invariant set in the fundamental group, we introduce a method to estimate a lower bound of the action’s growth rate on the fundamental group. The growth rate of this induced action provides a lower bound for the entropy of the real surface automorphism. These calculations are carried out for an important family of real surface automorphisms, and new lower bounds are obtained for the entropy of these automorphisms.
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Inspired by results of Eskin and Mirzakhani (J Mod Dyn 5(1):71–105, 2011) counting closed geodesics of length
≤
L
in the moduli space of a fixed closed surface, we consider a similar question in the
...O
u
t
(
F
r
)
setting. The Eskin-Mirzakhani result can be equivalently stated in terms of counting the number of conjugacy classes (within the mapping class group) of pseudo-Anosovs whose dilatations have natural logarithm
≤
L
. Let
N
r
(
L
)
denote the number of
O
u
t
(
F
r
)
-conjugacy classes of fully irreducibles satisfying that the natural logarithm of their dilatation is
≤
L
. We prove for
r
≥
3
that as
L
→
∞
, the number
N
r
(
L
)
has double exponential (in
L
) lower and upper bounds. These bounds reveal behavior not present in the surface setting or in classical hyperbolic dynamical systems.
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For a stratified group
G
, we construct a class of Lie groups endowed with a left-invariant distribution locally diffeomorphic to the flat distribution of
G
. Vice versa, we show that all Lie groups ...with a left-invariant distribution that is locally diffeomorphic to the flat distribution of
G
belong to the class we constructed, if the Lie algebra of
G
has finite Tanaka prolongation.
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Let
M
be an oriented three-dimensional Riemannian manifold of constant sectional curvature
k
=
0
,
1
,
-
1
and let
S
O
M
be its direct orthonormal frame bundle (direct refers to positive ...orientation), which may be thought of as the set of all positions of a small body in
M
. Given
λ
∈
R
, there is a three-dimensional distribution
D
λ
on
S
O
M
accounting for infinitesimal rototranslations of constant pitch
λ
. When
λ
≠
k
2
, there is a canonical sub-Riemannian structure on
D
λ
. We present a geometric characterization of its geodesics, using a previous Lie theoretical description. For
k
=
0
,
-
1
, we compute the sub-Riemannian length spectrum of
S
O
M
,
D
λ
in terms of the complex length spectrum of
M
(given by the lengths and the holonomies of the periodic geodesics) when
M
has positive injectivity radius. In particular, for two complex length isospectral closed hyperbolic 3-manifolds (even if they are not isometric), the associated sub-Riemannian metrics on their direct orthonormal bundles are length isospectral.
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We prove that a three-dimensional smooth complete intersection of two quadrics over a field k is k-rational if and only if it contains a line defined over k. To do so, we develop a theory of ...intermediate Jacobians for geometrically rational threefolds over arbitrary, not necessarily perfect, fields. As a consequence, we obtain the first examples of smooth projective varieties over a field k which have a k-point, and are rational over a purely inseparable field extension of k, but not over k.
In this paper we give a new proof of a theorem by Alexandrov on the Gauss curvature prescription of Euclidean convex sets. This proof is based on the duality theory of convex sets and on optimal mass ...transport. A noteworthy property of this proof is that it does not rely neither on the theory of convex polyhedra nor on P.D.E. methods (which appeared in all the previous proofs of this result).
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We prove some injectivity results: that a Coxeter monoid
Z
-algebra (or 0-Hecke algebra) injects in the incidence
Z
-algebra of the corresponding Bruhat poset, for any Coxeter group; that the Hecke ...algebra of a right-angled Coxeter group injects in the Coxeter monoid
Z
q
,
q
-
1
-algebra (and then in the incidence
Z
q
,
q
-
1
-algebra of the corresponding Bruhat poset); that a right-angled Artin group injects in the group of invertible elements of the Hecke algebra of the corresponding Coxeter group (and then in the group of invertible elements of a Coxeter monoid algebra and in the one of an incidence algebra).
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In this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of
free curves
. We construct families of arrangements which are nearly free and ...possess interesting geometric properties. More generally, we study 3-syzygy curve arrangements and we present examples that admit unexpected curves.
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