Polyhedral computational geometry for averaging metric phylogenetic trees

This paper investigates the computational geometry relevant to calculations of the Frechet mean and variance for probability distributions on the phylogenetic tree space of Billera, Holmes and Vogtmann, using the theory of probability measures on spaces of nonpositive curvature developed by Sturm. We show that the combinatorics of geodesics with a specified fixed endpoint in tree space are determined by the location of the varying endpoint in a certain polyhedral subdivision of tree space. The variance function associated to a finite subset of tree space has a fixed $C^\infty$ algebraic formula within each cell of the corresponding subdivision, and is continuously differentiable in the interior of each orthant of tree space. We use this subdivision to establish two iterative methods for producing sequences that converge to the Frechet mean: one based on Sturm's Law of Large Numbers, and another based on descent algorithms for finding optima of smooth functions on convex polyhedra. We present properties and biological applications of Frechet means and extend our main results to more general globally nonpositively curved spaces composed of Euclidean orthants.Comment: 43 pages, 6 figures; v2: fixed typos, shortened Sections 1 and 5, added counter example for polyhedrality of vistal subdivision in general CAT(0) cubical complexes; v1: 43 pages, 5 figure

Cluster expansion for abstract polymer models. New bounds from an old approach

We revisit the classical approach to cluster expansions, based on tree graphs, and establish a new convergence condition that improves those by Kotecky-Preiss and Dobrushin, as we show in some examples. The two ingredients of our approach are: (i) a careful consideration of the Penrose identity for truncated functions, and (ii) the use of iterated transformations to bound tree-graph expansions.Comment: 16 pages. This new version, written en reponse to the suggestions of the referees, includes more detailed introductory sections, a proof of the generalized Penrose identity and some additional results that follow from our treatmen

Dehn filling in relatively hyperbolic groups

We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bicombings on this space. The first of these, "preferred paths", is combinatorial in nature and allows us to define the second, a relatively hyperbolic version of a construction of Mineyev. As an application, we prove a group-theoretic analog of the Gromov-Thurston 2\pi Theorem in the context of relatively hyperbolic groups.Comment: 83 pages. v2: An improved version of preferred paths is given, in which preferred triangles no longer need feet. v3: Fixed several small errors pointed out by the referee, and repaired several broken figures. v4: corrected definition 2.38. This is very close to the published versio

Saturating the random graph with an independent family of small range

Motivated by Keisler's order, a far-reaching program of understanding basic model-theoretic structure through the lens of regular ultrapowers, we prove that for a class of regular filters $D$ on $I$, $|I| = \lambda > \aleph_0$, the fact that P(I)/\de has little freedom (as measured by the fact that any maximal antichain is of size $<\lambda$, or even countable) does not prevent extending $D$ to an ultrafilter $D_1$ on $I$ which saturates ultrapowers of the random graph. "Saturates" means that M^I/\de_1 is $\lambda^+$-saturated whenever M is a model of the theory of the random graph. This was known to be true for stable theories, and false for non-simple and non-low theories. This result and the techniques introduced in the proof have catalyzed the authors' subsequent work on Keisler's order for simple unstable theories. The introduction, which includes a part written for model theorists and a part written for set theorists, discusses our current program and related results.Comment: 14 page

Still another approach to the braid ordering

We develop a new approach to the linear ordering of the braid group $B\_n$, based on investigating its restriction to the set \Div(\Delta\_n^d) of all divisors of $\Delta\_n^d$ in the monoid $B\_\infty^+$, i.e., to positive $n$-braids whose normal form has length at most $d$. In the general case, we compute several numerical parameters attached with the finite orders (\Div(\Delta\_n^d), <). In the case of 3 strands, we moreover give a complete description of the increasing enumeration of (\Div(\Delta\_3^d), <). We deduce a new and specially direct construction of the ordering on $B\_3$, and a new proof of the result that its restriction to $B\_3^+$ is a well-ordering of ordinal type $\omega^\omega$

Galois cohomology of a number field is Koszul

We prove that the Milnor ring of any (one-dimensional) local or global field K modulo a prime number l is a Koszul algebra over Z/l. Under mild assumptions that are only needed in the case l=2, we also prove various module Koszulity properties of this algebra. This provides evidence in support of Koszulity conjectures that were proposed in our previous papers. The proofs are based on the Class Field Theory and computations with quadratic commutative Groebner bases (commutative PBW-bases).Comment: LaTeX 2e, 25 pages; v.2: minor grammatic changes; v.3: classical references added, remark inserted in subsection 1.6, details of arguments added in subsections 1.4, 1.7 and sections 5 and 6; v.4: still more misprints corrected, acknowledgement updated, a sentence inserted in section 4, a reference added -- this is intended as the final versio