• Corpus ID: 234742366

An Invitation to Tropical Alexandrov Curvature

@inproceedings{Amendola2021AnIT,
  title={An Invitation to Tropical Alexandrov Curvature},
  author={Carlos Am'endola and Anthea Monod},
  year={2021}
}
We study Alexandrov curvature in the tropical projective torus with respect to the tropical metric. Alexandrov curvature is a generalization of classical Riemannian sectional curvature to more general metric spaces; it is determined by a comparison of triangles in an arbitrary metric space to corresponding triangles in Euclidean space. In the polyhedral setting of tropical geometry, triangles are a combinatorial object, which adds a combinatorial dimension to our analysis. We study the effect… 

References

SHOWING 1-10 OF 65 REFERENCES
On the total curvature of tropical hypersurfaces
This paper studies the curvatures of amoebas and real amoebas (i.e. essentially logarithmic curvatures of the complex and real parts of a real algebraic hypersurface) and of tropical and real
A visual introduction to Riemannian curvatures and some discrete generalizations
We try to provide a visual introduction to some objects used in Riemannian geometry: parallel transport, sectional curvature, Ricci curvature, Bianchi identities... We then explain some of the
The tropical Grassmannian
In tropical algebraic geometry, the solution sets of polynomial equations are piecewise-linear. We introduce the tropical variety of a polynomial ideal, and we identify it with a polyhedral
A Note on Tropical Triangles in the Plane
We define transversal tropical triangles (affine and projective) and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a
Metric Spaces of Non-Positive Curvature
This book describes the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces by
Ricci curvature of Markov chains on metric spaces
Tropical optimal transport and Wasserstein distances
We study the problem of optimal transport in tropical geometry and define the Wasserstein- p distances in the continuous metric measure space setting of the tropical projective torus. We specify the
Convexity in Tree Spaces
TLDR
The geometry of metrics and convexity structures on the space of phylogenetic trees is studied, which is here realized as the tropical linear space of all ultrametrics and the tropical metric arises from the theory of orthant spaces.
CRITICAL POINTS AND CURVATURE FOR EMBEDDED POLYHEDRA
Recently a new insight into the Gauss-Bonnet Theorem and other problems in global differential geometry has come about through the connection between total curvature of embedded smooth manifolds and
Barycenters in Alexandrov spaces of curvature bounded below
We investigate barycenters of probability measures on proper Alexandrov spaces of curvature bounded below, and show that they enjoy several properties relevant to or different from those in metric
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