Related papers: Stability, fragility, and Rota's Conjecture

We prove one direction of a recently posed conjecture by Gan-Gross-Prasad, which predicts the branching laws that govern restriction from p-adic $GL_n$ to $GL_{n-1}$ of irreducible smooth representations within the Arthur-type class. We…

This article is studying the roots of the reliability polynomials of linear consecutive-\textit{k}-out-of-\textit{n}:\textit{F} systems. We are able to prove that these roots are unbounded in the complex plane, for any fixed $k\ge2$. In the…

An $n$ dimensional minimal submanifold $\Sigma$ of $\R^{n+m}$ is called non-parametric if $\Sigma$ can be represented as the graph of a vector-valued function $f:D\subset \R^n \mapsto \R^m$. This note provides a sufficient condition for the…

Fix a weakly minimal (i.e., superstable $U$-rank $1$) structure $\mathcal{M}$. Let $\mathcal{M}^*$ be an expansion by constants for an elementary substructure, and let $A$ be an arbitrary subset of the universe $M$. We show that all…

We generalise a key result of one-relator group theory, namely Magnus's Freiheitssatz, to partially commutative groups, under sufficiently strong conditions on the relator. The main theorem shows that under our conditions, on an element $r$…

Combinatorial rigidity theory seeks to describe the rigidity or flexibility of bar-joint frameworks in R^d in terms of the structure of the underlying graph G. The goal of this article is to broaden the foundations of combinatorial rigidity…

Let $N\subset GL(n,R)$ be the group of upper triangular matrices with $1$s on the diagonal, equipped with the standard Carnot group structure. We show that quasiconformal homeomorphisms between open subsets of $N$, and more generally…

We study the problem of realizing families of subgroups as the set of stabilizers of configurations from a subshift of finite type (SFT). This problem generalizes both the existence of strongly and weakly aperiodic SFTs. We show that a…

The notion of branch-depth for matroids was introduced by DeVos, Kwon and Oum as the matroid analogue of the tree-depth of graphs. The contraction-deletion-depth, another tree-depth like parameter of matroids, is the number of recursive…

We prove that the extension complexity of the independence polytope of every regular matroid on $n$ elements is $O(n^6)$. Past results of Wong and Martin on extended formulations of the spanning tree polytope of a graph imply a $O(n^2)$…

Enochs' conjecture asserts that each covering class of modules (over any fixed ring) has to be closed under direct limits. Although various special cases of the conjecture have been verified, the conjecture remains open in its full…

Choose any oriented link type X and closed braid representatives X[+], X[-] of X, where X[-] has minimal braid index among all closed braid representatives of X. The main result of this paper is a `Markov theorem without stabilization'. It…

Bicircular lift matroids are a class of matroids defined on the edge set of a graph. For a given graph $G$, the circuits of its bicircular lift matroid $L(G)$ are the edge sets of those subgraphs of $G$ that contain at least two cycles, and…

The results obtained in this paper grew from an attempt to generalize the main theorem of [1]. There it was shown that any circuit injection (a 1-1 onto edge map f such that if C is a circuit then f(C) is a circuit) from a 3-connected, not…

Mayhew and Royle (2008) showed that there are 564 excluded minors for the class of GF(5)-representable matroids having at most 9 elements. We enumerate the excluded minors for GF(5)-representable matroids having 10 elements: there are…

The category of quasi frames (or qframes) is introduced and studied. In the context of qframes we can jointly study problems related to the L-Surjunctivity and Stable Finiteness Conjectures. As a consequences of our main results, we can…

Let $S$ be a closed Shimura variety uniformized by the complex $n$-ball. The Hodge conjecture predicts that every Hodge class in $H^{2k} (S, \Q)$, $k=0, \ldots, n$, is algebraic. We show that this holds for all degree $k$ away from the…

We prove that for every proper minor-closed class $M$ of matroids representable over a prime field, there exists a constant-competitive matroid secretary algorithm for the matroids in $M$. This result relies on the extremely powerful…

We address a special case of a conjecture of M. Talagrand relating two notions of "threshold" for an increasing family $\mathcal F$ of subsets of a finite set $V$. The full conjecture implies equivalence of the "Fractional…

Split matroids form a minor-closed class of matroids, and are defined by placing conditions on the system of split hyperplanes in the matroid base polytope. They can equivalently be defined in terms of structural properties involving cyclic…