Section: New Results

Non-linear Perron-Frobenius theory, nonexpansive mappings and metric geometry

Order isomorphisms and antimorphisms on cones

Participant : Cormac Walsh.

We have been studying non-linear operators on open cones, particularly ones that preserve or reverse the order structure associated to the cone. A bijective map that preserves the order in both directions is called an order isomorphism. Those that reverse the order in both directions are order antimorphisms. These are closely related to the isometries of the Hilbert and Thompson metrics on the cone.

Previously, we have shown  [133] that if there exists an antimorphism on a finite-dimensional open cone that is homogeneous of degree $-1$, then the cone must be a symmetric cone, that is, have a transitive group of linear automorphisms and be self-dual. This result was improved in [44], where we showed that the homogeneity assumption is not actually necessary: every antimorphism on a cone is automatically homogeneous of degree $-1$.

The study of the order isomorphisms of a cone goes back to Alexandrov and Zeeman, who considered maps preserving the light cone that arises in special relativity. This work was extended to more general cones by Rothaus; Noll and Schäffer; and Artstein-Avidan and Slomka. It was shown, in the finite-dimensional case, that all isomorphisms are linear if the cone has no one-dimensional factors. There are also some results in infinite dimension—however these are unsatisfactory because of the strong assumptions that must be made in order to get the finite-dimensional techniques to work. For example, a typical assumption is that the cone is the convex hull of its extreme rays, which is overly restrictive in infinite dimension. Using different techniques more suited to infinite dimension, we have been developing a necessary and sufficient criterion on the geometry of a cone for all its isomorphisms to be linear.

Horofunction compactifications of symmetric spaces

Participant : Cormac Walsh.

This work is in collaboration with Thomas Haettel (Montpellier), Anna-Sofie Schilling (Heidelberg), Anna Wienhard (Heidelberg).

The symmetric spaces form a fascinating class of geometrical space. These are the spaces in which there is a point reflection through every point. An example is the space $Pos(ℂ,n)$ of positive definite $n\times n$ Hermitian matrices.

The interesting metrics on such spaces are the ones that are invariant under all the symmetries, in particular the invariant Finsler metrics. When the symmetric space is non-compact, as in the example just referred to, it is profitable to study the horofunction boundary of such metrics.

An important technique in trying to understand symmetric spaces is to look at their flats. These are subspaces that are, as their name suggests, flat in some sense. Because of the abundance of symmetries, there are many flats; indeed, every pair of points lies in a flat. Furthermore, given any two flats, there is a symmetry taking one to the other, and so they are all alike. It turns out that the restriction of an invariant Finsler metric to a single flat determines the metric everywhere, and gives the flat the geometry of a normed space.

Symmetric spaces can be compactified by means of the Satake compactification. In fact, there are several such compactifications, one associated to each irreducible faithful representation of the invariance group of the space. In [41], we show that each Satake compactification can be constructed as a horofunction compactification by choosing an appropriate invariant Finsler metric. In fact, the metrics we construct have polyhedral balls on the flat.

An important step in the proof is to show that the closure of a flat in the horofunction compactification of the symmetric space is the same as the horofunction compactification of the flat viewed as a metric space in its own right. This is not true for every metric space, since in general one might not be able to distinguish horofunctions by looking at a subspace.

The set of minimal upper bounds of two matrices in the Loewner order

Participant : Nikolas Stott.

A classical theorem of Kadison shows that the space of symmetric matrices equipped with the Loewner order is an anti-lattice, meaning that two matrices have a least upper bound if and only if they are comparable. In [24], we refined this theorem by characterizing the set of minimal upper bounds: we showed that it is homeomorphic to the quotient space $O\left(p\right)\setminus O(p,q)/O\left(q\right)$, where $O(p,q)$ denotes the orthogonal group associated to the quadratic form with signature $(p,q)$, and $O\left(p\right)$ denotes the standard $p$th orthogonal group.

Generalization of the Hellinger distance

Participant : Stéphane Gaubert.

In  [64] (joint work with Rajendra Bhatia of Ashoka University and Tanvi Jain, Indian Statistic Institute, New Delhi), we study some generalizations of the Hellinger distance to the space of positive definite matrices.

Spectral inequalities for nonnegative tensors and their tropical analogues

Participant : Stéphane Gaubert.

In [39] (joint work with Shmuel Friedland, University of Illinois at Chicago) we extend some characterizations and inequalities for the eigenvalues of nonnegative matrices, such as Donsker-Varadhan, Friedland-Karlin, Karlin-Ost inequalities, to nonnegative tensors. These inequalities are related to a correspondence between nonnegative tensors and ergodic control: the logarithm of the spectral radius of a tensor is given by the value of an ergodic problem in which instantaneous payments are given by a relative entropy. Some of these inequalities involve the tropical spectral radius, a limit of the spectral radius which we characterize combinatorially as the value of an ergodic Markov decision process.