The evolutionary history of a set of taxa is usually represented by a phylogenetic tree, and this model has greatly facilitated the discussion and testing of hypotheses. However, it is well known that more complex evolutionary scenarios are poorly described by such models. Further, even when evolution proceeds in a tree-like manner, analysis of the data may not be best served by using methods that enforce a tree structure but rather by a richer visualization of the data to evaluate its properties, at least as an essential first step. Thus, phylogenetic networks should be employed when reticulate events such as hybridization, horizontal gene transfer, recombination, or gene duplication and loss are believed to be involved, and, even in the absence of such events, phylogenetic networks have a useful role to play. This article reviews the terminology used for phylogenetic networks and covers both split networks and reticulate networks, how they are defined, and how they can be interpreted. Additionally, the article outlines the beginnings of a comprehensive statistical framework for applying split network methods. We show how split networks can represent confidence sets of trees and introduce a conservative statistical test for whether the conflicting signal in a network is treelike. Finally, this article describes a new program, SplitsTree4, an interactive and comprehensive tool for inferring different types of phylogenetic networks from sequences, distances, and trees.

https://bioinformatics.cs.vt.edu/~easychair/HusonBryant_MBE_2006.pdf

Application of Phylogenetic Networks in Evolutionary Studies

Monomial Ideals.- Squarefree monomial ideals.- Borel-fixed monomial ideals.- Three-dimensional staircases.- Cellular resolutions.- Alexander duality.- Generic monomial ideals.- Toric Algebra.- Semigroup rings.- Multigraded polynomial rings.- Syzygies of lattice ideals.- Toric varieties.- Irreducible and injective resolutions.- Ehrhart polynomials.- Local cohomology.- Determinants.- Plucker coordinates.- Matrix Schubert varieties.- Antidiagonal initial ideals.- Minors in matrix products.- Hilbert schemes of points.

https://www.springer.com/productFlyer_978-0-387-22356-8.pdf?SGWID=0-0-1297-34952457-0

Combinatorial Commutative Algebra

https://www.rand.org/content/dam/rand/pubs/reports/2021/R1056-1.pdf

On cores and indivisibility

The maximum clique problem is a classical problem in combinatorial optimization which finds important applications in different domains. In this paper we try to give a survey of results concerning algorithms, complexity, and applications of this problem, and also provide an updated bibliography. Of course, we build upon precursory works with similar goals [39, 232, 266].

The maximum clique problem

The course given during the School and Workshop “The Geometry and Topology of Singularities”, 8-26 January 2007, Cuernavaca, Mexico is based on a previous course given during the 23o Coloquio Brasileiro de Matematica (Rio de Janeiro, July 2001). It is an elementary introduction to the theory of toric varieties. This introduction does not pretend to originality but to provide examples and motivation for the study of toric varieties. The theory of toric varieties plays a prominent role in various domains of mathematics, giving explicit relations between combinatorial geometry and algebraic geometry. They provide an important field of examples and models. The Fulton’s preface of [11] explains very well the interest of these objects “Toric varieties provide a ... way to see many examples and phenomena in algebraic geometry... For example, they are rational, and, although they may be singular, the singularities are rational. Nevertheless, toric varieties have provided a remarkably fertile testing ground for general theories.” Basic references for toric varieties are [10], [11] and [15]. These references give complete proofs of the results and descriptions. They were (abusively) used for writing these notes and the reader can consult them for useful complementary references and details. Various applications of toric varieties can be found in the litterature, in particular in the book [11]. Interesting applications and suitable references are given in [7]: applications to Algebraic coding theory, Error-correcting codes, Integer programming and combinatorics, Computing resultants and solving equations, including the study of magic squares (see 8.3). Applications to Symplectic Manifolds are given in [1]. Of course this list is not exhaustive. Special thanks to Gottfried Barthel, Karl-Heinz Fieseler and Ludger Kaup: in their friendly company, I discovered the wonderful country of toric varieties and thanks to them I was able to write these notes.

/pdf/introduction-to-toric-varieties-3d69x8qvc9.pdf

Introduction to toric varieties

We consider a continuous space which models the set of all phylogenetic trees having a fixed set of leaves. This space has a natural metric of nonpositive curvature, giving a way of measuring distance between phylogenetic trees and providing some procedures for averaging or combining several trees whose leaves are identical. This geometry also shows which trees appear within a fixed distance of a given tree and enables construction of convex hulls of a set of trees. This geometric model of tree space provides a setting in which questions that have been posed by biologists and statisticians over the last decade can be approached in a systematic fashion. For example, it provides a justification for disregarding portions of a collection of trees that agree, thus simplifying the space in which comparisons are to be made.

/pdf/geometry-of-the-space-of-phylogenetic-trees-4taz00mv8y.pdf

Geometry of the space of phylogenetic trees

We introduce the property of k-decomposability for simplicial complexes which, if satisfied by the dual simplicial complex of a convex polyhedron, implies that the diameter of the polyhedron is bounded by a polynomial of degree k + 1 in the number of facets. These properties form a hierarchy, each one implying the next. The strongest, vertex decomposability, implies the Hirsch conjecture; the weakest is equivalent to shellability. We show that several cases in which the Hirsch conjecture has been verified can be handled by these methods, which also give the shellability of a number of simplicial complexes of combinatorial interest. We conclude with a strengthened form of the property of shellability which would imply the Hirsch conjecture for polytopes.

Decompositions of Simplicial Complexes Related to Diameters of Convex Polyhedra

A proof of the sufficiency of McMullen's conditions for f-vectors of simplicial convex polytopes

This paper generalizes the Dehn-Sommerville equations for simplicial spheres to related classes of objects. The underlying motivation is to understand the combinatorial structure of arbitrary polytopes, that is, polytopes that are not necessarily simplicial. Towards this end we determine the affine span of the extended f-vectors of d-polytopes. A polytope is the convex hull of finitely many points in R e . We will generally consider a polytope of affine dimension d to be a subset of Re; this is referred to as a d-polytope. A face of a polytope is the intersection of a supporting hyperplane with the polytope. For the most part we identify a polytope P with the abstract cell complex (or lattice) realized by the boundary of P, and write a face of P as the set of vertices of P it contains. That is, we will shorten F = c o n v { v 0 . . . . . Vk} to F={Vo,.. . ,Vk} when there is no risk of confusion. By convention, the empty set is considered a ( -1)-dimensional face, and the polytope itself is a d-dimentional face; these faces will be called improper faces of P. A polytope P is called simplicial if each of its faces, except possibly P itself, is a simplex (the convex hull of affinely independent points). We will write Y (respectively, ~ ) for the set of all (respectively, all simplicial) d-polytopes. The number of/-dimensional faces (or/-faces) of a polytope P is written f , and f(P)=(fo,fl . . . . . fd-1) is called the f-vector of P. The set off -vectors of all (simplicial) polytopes is written f ( ~ e ) (f(~d)). A certain transformation on the f-vectors of simplicial polytopes has arisen in a number of different contexts, and will play an important part here. For a d-polytope P define the h-vector h(P)=(ho, hl,...,he) b y h i = i ( 1 ) i s ( d j ) s= o d i f j 1 (here we use the convention

/pdf/generalized-dehn-sommerville-relations-for-polytopes-spheres-58fbq8l6ag.pdf

Louis J. Billera

Papers

Geometry of the space of phylogenetic trees

Decompositions of Simplicial Complexes Related to Diameters of Convex Polyhedra

A proof of the sufficiency of McMullen's conditions for f-vectors of simplicial convex polytopes

Generalized Dehn-Sommerville Relations for Polytopes, Spheres and Eulerian Partially Ordered Sets

Constructions and complexity of secondary polytopes