Abstract not found

ABSTRACT. We give a rigorous definition of tropical fans (the “local building blocks for tropical varieties”) and their morphisms. For such a morphism of tropical fans of the same dimension we show that the number of inverse images (counted with suitable tropical multiplicities) of a point in the target does not depend on the chosen point — a statement that can be viewed as the beginning of a tropical intersection theory. As an application we consider the moduli spaces of rational tropical curves (both abstract and in some Rr) together with the evaluation and forgetful morphisms. Using our results this gives new, easy, and unified proofs of various tropical independence statements, e.g. of the fact that the numbers of rational tropical curves (in any Rr) through given points are independent of the points. 1.

This article is based on the Clay Mathematics Senior Scholar Lecture that was delivered

Lecture 1. Basic definitions, results, and examples 5 1.1. Order complexes and face posets 5

Abstract. Let m12, m13,..., mn−1,n be the slopes of the ( n) lines con- 2 necting n points in general position in the plane. The ideal In of all algebraic relations among the mij defines a configuration space called the slope variety of the complete graph. We prove that In is reduced and Cohen-Macaulay, give an explicit Gröbner basis for it, and compute its Hilbert series combinatorially. We proceed chiefly by studying the associated Stanley-Reisner simplicial complex, which has an intricate recursive structure. In addition, we are able to answer many questions about the geometry of the slope variety by translating them into purely combinatorial problems concerning enumeration of trees. 1.

A phylogenetic tree represents historical evolutionary relationships between different species or organisms. The space of possible phylogenetic trees is both complex and exponentially large. Here we study combinatorial features of neighbourhoods within this space, with respect to four standard tree metrics. We focus on the splits of a tree: the bipartitions induced by removing a single edge from the tree. We characterize those splits appearing in trees that are within a given distance of the original tree, demonstrating close connections between these splits, the Whitney number of a tree, and the binary characters with a given parsimony length.

Abstract. Natural Dowling analogues of the complex of phylogenetic trees are studied. Using discrete Morse theory, we find their homotopy types. In the process, the homotopy types of certain subposets of Dowling lattices are determined. 1.

When the null or alternative hypothesis of a statistical testing problem is a composite of regions of varying dimensionality, the likelihood ratio test is statistically inappropriate. Its inappropriateness is revealed not by its performance under the Neyman-Pearson criterion but by the fact that it yields incorrect inferences in certain regions of the sample space due to its inability to adapt to the diering dimensions in the composite hypothesis. Maximum likelihood estimators and associated model selection procedures also are inappropriate for such composite models. Tests and estimators based on the p-values associated with the various regions that determine the composite model are more appropriate for this geometry. Similar issues arise when the boundary of the null or alternative hypothesis is a composite of regions of varying dimensionality. Corresponding author: Michael D. Perlman, Department of Statistics, University of Washington, Seattle, WA 98195. email: michael@ms.washin...

Abstract. Phylogenetic algebraic geometry is concerned with certain complex projective algebraic varieties derived from finite trees. Real positive points on these varieties represent probabilistic models of evolution. For small trees, we recover classical geometric objects, such as toric and determinantal varieties and their secant varieties, but larger trees lead to new and largely unexplored territory. This paper gives a self-contained introduction to this subject and offers numerous open problems for algebraic geometers. 1.

This article argues for an affirmative answer to the question in the title. In future interactions between mathematics and biology, both fields will contribute to each other, and, in particular, research in the life sciences will inspire new theorems in “pure” mathematics. This point is illustrated by a snapshot of four recent contributions from biology to geometry, combinatorics and algebra.