Finsler bordifications of symmetric and certain locally symmetric spaces

We give a geometric interpretation of the maximal Satake compactification of symmetric spaces $X=G/K$ of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable $G$-invariant Finsler metric on $X$. As an application, we establish the existence of natural bordifications, as orbifolds-with-corners, of locally symmetric spaces $X/\Gamma$ for arbitrary discrete subgroups $\Gamma< G$. These bordifications result from attaching $\Gamma$-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

Planet Formation by Concurrent Collapse

After reviewing the difficulties faced by the conventional theory of planet formation (based upon the aggregation of microscopic dust particles), we describe an alternative hypothesis. We propose that planets form by gravitational collapse at the same time as the star about which they orbit. This 'concurrent collapse' hypothesis avoids theoretical difficulties associated with the conventional model and suggests satisfying explanations for various poorly understood phenomena. We introduce new explanations for FU Orionis outbursts seen in young stars, the discovery of exoplanets with eccentric orbits and the existence of small rocky objects such as chondrules in the solar system.Comment: 16 pages, no figures, extended and refined version of earlier submissio

The Shape of Unlabeled Rooted Random Trees

We consider the number of nodes in the levels of unlabelled rooted random trees and show that the stochastic process given by the properly scaled level sizes weakly converges to the local time of a standard Brownian excursion. Furthermore we compute the average and the distribution of the height of such trees. These results extend existing results for conditioned Galton-Watson trees and forests to the case of unlabelled rooted trees and show that they behave in this respect essentially like a conditioned Galton-Watson process.Comment: 34 pages, 1 figur

Discrete isometry groups of symmetric spaces

This survey is based on a series of lectures that we gave at MSRI in Spring 2015 and on a series of papers, mostly written jointly with Joan Porti. Our goal here is to: 1. Describe a class of discrete subgroups $\Gamma<G$ of higher rank semisimple Lie groups, which exhibit some "rank 1 behavior". 2. Give different characterizations of the subclass of Anosov subgroups, which generalize convex-cocompact subgroups of rank 1 Lie groups, in terms of various equivalent dynamical and geometric properties (such as asymptotically embedded, RCA, Morse, URU). 3. Discuss the topological dynamics of discrete subgroups $\Gamma$ on flag manifolds associated to $G$ and Finsler compactifications of associated symmetric spaces $X=G/K$. Find domains of proper discontinuity and use them to construct natural bordifications and compactifications of the locally symmetric spaces $X/\Gamma$.Comment: 77 page

An equivariant Quillen theorem

A classical theorem due to Quillen (1969) identifies the unitary bordism ring with the Lazard ring, which classifies the universal one-dimensional commutative formal group law. We prove an equivariant generalization of this result by identifying the homotopy theoretic $\mathbb{Z}/2$-equivariant unitary bordism ring, introduced by tom Dieck (1970), with the $\mathbb{Z}/2$-equivariant Lazard ring, introduced by Cole-Greenlees-Kriz (2000). Our proof combines a computation of the homotopy theoretic $\mathbb{Z}/2$-equivariant unitary bordism ring due to Strickland (2001) with a detailed investigation of the $\mathbb{Z}/2$-equivariant Lazard ring.Comment: 19 pages; v3: Minor changes. Accepted for publication in Adv. Mat

Lexical typology through similarity semantics: Toward a semantic map of motion verbs

This paper discusses a multidimensional probabilistic semantic map of lexical motion verb stems based on data collected from parallel texts (viz. translations of the Gospel according to Mark) for 100 languages from all continents. The crosslinguistic diversity of lexical semantics in motion verbs is illustrated in detail for the domain of `go', `come', and `arrive' type contexts. It is argued that the theoretical bases underlying probabilistic semantic maps from exemplar data are the isomorphism hypothesis (given any two meanings and their corresponding forms in any particular language, more similar meanings are more likely to be expressed by the same form in any language), similarity semantics (similarity is more basic than identity), and exemplar semantics (exemplar meaning is more fundamental than abstract concepts)

Transformations of the state?

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