Collapsibility to a subcomplex of a given dimension is NP-complete

In this paper we extend the works of Tancer and of Malgouyres and Franc\'es, showing that $(d,k)$-collapsibility is NP-complete for $d\geq k+2$ except $(2,0)$. By $(d,k)$-collapsibility we mean the following problem: determine whether a given $d$-dimensional simplicial complex can be collapsed to some $k$-dimensional subcomplex. The question of establishing the complexity status of $(d,k)$-collapsibility was asked by Tancer, who proved NP-completeness of $(d,0)$ and $(d,1)$-collapsibility (for $d\geq 3$). Our extended result, together with the known polynomial-time algorithms for $(2,0)$ and $d=k+1$, answers the question completely

Triple covers and a non-simply connected surface spanning an elongated tetrahedron and beating the cone

By using a suitable triple cover we show how to possibly model the construction of a minimal surface with positive genus spanning all six edges of a tetrahedron, working in the space of BV functions and interpreting the film as the boundary of a Caccioppoli set in the covering space. After a question raised by R. Hardt in the late 1980's, it seems common opinion that an area-minimizing surface of this sort does not exist for a regular tetrahedron, although a proof of this fact is still missing. In this paper we show that there exists a surface of positive genus spanning the boundary of an elongated tetrahedron and having area strictly less than the area of the conic surface.Comment: Expanding on the previous version with additional lower bounds, new images, corrections and improvements. Comparison with Reifenberg approac

Impossibility results on stability of phylogenetic consensus methods

We answer two questions raised by Bryant, Francis and Steel in their work on consensus methods in phylogenetics. Consensus methods apply to every practical instance where it is desired to aggregate a set of given phylogenetic trees (say, gene evolution trees) into a resulting, "consensus" tree (say, a species tree). Various stability criteria have been explored in this context, seeking to model desirable consistency properties of consensus methods as the experimental data is updated (e.g., more taxa, or more trees, are mapped). However, such stability conditions can be incompatible with some basic regularity properties that are widely accepted to be essential in any meaningful consensus method. Here, we prove that such an incompatibility does arise in the case of extension stability on binary trees and in the case of associative stability. Our methods combine general theoretical considerations with the use of computer programs tailored to the given stability requirements

Discrete Morse theory and the K(pi,1) conjecture

The aim of this thesis is to present the K(pi,1) conjecture for Artin groups, an open conjecture which goes back to the 70s, and to use the technique of discrete Morse theory to prove some results connected with it. The beginning of the study of Artin groups dates back to the introduction of braid groups in the 20s. Artin groups where defined in general by Tits and Brieskorn in the 60s, in relation with the theory of Coxeter groups and singularity theory. Deep connections with the main areas of mathematics where discovered: in addition to the theory of Coxeter groups and singularity theory, they naturally arise in the study of root systems, hyperplane arrangements, configuration spaces, combinatorics, geometric group theory (see the very recent solution of the virtually Hacken conjecture), knot theory, mapping class groups and moduli spaces of curves. There are many properties conjectured to be true for all Artin groups but proved only for some families of them, e.g. being torsion-free, having a trivial center, and having solvable word problem. Some of these problems, and also others (such as the computation of homology and cohomology), are related to an important conjecture called "K(pi,1) conjecture". Such conjecture says that a certain topological space N, constructed in a certain way from a fixed Coxeter group, is a classifying space for the corresponding Artin group. The space N admits finite CW models, therefore the K(pi,1) conjecture directly implies that Artin groups are torsion-free. A tool which is very important in our work is discrete Morse theory, introduced by Forman in the 90s. Discrete Morse theory allows to prove the homotopy equivalence of CW-complexes through elementary collapses of cells, on the basis of some combinatorial rules which can be naturally expressed with the language of graph theory. The idea of using discrete Morse theory to prove results about the K(pi,1) conjecture is present in the literature only in very recent works. This thesis is structured as follows. In the first chapter we present some of the most important known results about Coxeter groups, especially concerning their geometric and combinatorial properties. In the second chapter we do the same for Artin groups. In particular we introduce the Artin monoids, which are significantly important in the study of Artin groups. The third chapter is devoted to an introduction to the terminology and the main results of discrete Morse theory, in a version developed by Chari and Batzies after the original work of Forman. In the fourth chapter we introduce the K(pi,1) conjecture together with some of its consequences. We define a particular CW model for the space N, called Salvetti complex, and we describe its combinatorial structure. Then we give a new proof of the K(pi,1) conjecture for Artin groups of finite type (i.e. those for which the corresponding Coxeter group is finite), using discrete Morse theory. Finally, in the fifth chapter we describe some connections between the K(pi,1) conjecture and classifying space of Artin monoids. A relevant result in this direction is a theorem by Dobrinskaya published in 2006, which states that the classifying space of an Artin monoid is homotopy equivalent to the corresponding space N mentioned above. We prove that applying discrete Morse theory one can collapse the standard CW model for the classifying space of an Artin monoid and obtain the Salvetti complex. In particular, this gives an alternative prove of Dobrinskaya's theorem

An algorithm for canonical forms of finite subsets of Zd\mathbb{Z}^d up to affinities

In this paper we describe an algorithm for the computation of canonical forms of finite subsets of $\mathbb{Z}^d$, up to affinities over $\mathbb{Z}$. For fixed dimension $d$, this algorithm has worst-case asymptotic complexity $O(n \log^2 n \, s\,\mu(s))$, where $n$ is the number of points in the given subset, $s$ is an upper bound to the size of the binary representation of any of the $n$ points, and $\mu(s)$ is an upper bound to the number of operations required to multiply two $s$-bit numbers. In particular, the problem is fixed-parameter tractable with respect to the dimension $d$. This problem arises e.g. in the context of computation of invariants of finitely presented groups with abelianized group isomorphic to $\mathbb{Z}^d$. In that context one needs to decide whether two Laurent polynomials in $d$ indeterminates, considered as elements of the group ring over the abelianized group, are equivalent with respect to a change of basis

Shellability of generalized Dowling posets

A generalization of Dowling lattices was recently introduced by Bibby and Gadish, in a work on orbit configuration spaces. The authors left open the question as to whether these posets are shellable. In this paper we prove EL-shellability and use it to determine the homotopy type. Our result generalizes shellability of Dowling lattices and of posets of layers of abelian arrangements defined by root systems. We also show that subposets corresponding to invariant subarrangements are not shellable in general