arXiv subjects

Roy Meshulam

Publications and source records attributed to Roy Meshulam.

At least 19 recordsLinked to original sources

arXiv2024

Topological connectivity of random permutation complexes

Let $\mathbb{S}_n$ denote the symmetric group on $[n]=\{1,\ldots,n\}$ with the uniform probability measure. For a permutation $\pi \in \mathbb{S}_n$ let $X_{\pi}$ denote the simplicial complex on the vertex set $[n]$ whose simplices are all $\{i_0,\ldots, i_m\} \subset [n]$ such that $i_0<\cdots<i_m$ and $\pi(i_0)<\cdots < \pi(i_m)$. For $r \geq 0$ let $p_r(n)$ denote the probability that $X_{\pi}$ is not topologically $r$-connected for $\pi \in \mathbb{S}_n$. It is shown that for fixed $r \geq 0$ there exist constants $0<C_r, C_r' < \infty$ such that \[ C_r \frac{(\log n)^r}{n} \leq p_r(n) \leq C_r' \frac{(\log n)^{2r}}{n}. \]

math.CO↗

arXiv2022

Maximal Generalized Rank in Graphical Matrix Spaces

In this note we prove two extensions of a recent combinatorial characterization due to Li, Qiao, Wigderson, Wigderson and Zhang (arXiv:2206.04815) of the maximal dimension of bounded rank subspaces of the graphical matrix space associated with a bipartite graph. Our first result shows that the above characterization remains valid for a wide class of generalized rank functions, including e.g. the permanental rank. Our second result extends the characterization to bounded rank subspaces of the graphical alternating matrix space associated with a general graph.

math.CO↗

arXiv2022

Random Balanced Cayley Complexes

Let $G$ be a finite group of order $n$ and for $1 \leq i \leq k+1$ let $V_i=\{i\} \times G$. Viewing each $V_i$ as a $0$-dimensional complex, let $Y_{G,k}$ denote the simplicial join $V_1*\cdots*V_{k+1}$. For $A \subset G$ let $Y_{A,k}$ be the subcomplex of $Y_{G,k}$ that contains the $(k-1)$-skeleton of $Y_{G,k}$ and whose $k$-simplices are all $\{(1,x_1),\ldots,(k+1,x_{k+1})\} \in Y_{G,k}$ such that $x_1\cdots x_{k+1} \in A$. Let $L_{k-1}$ denote the reduced $(k-1)$-th Laplacian of $Y_{A,k}$, acting on the space $C^{k-1}(Y_{A,k})$ of real valued $(k-1)$-cochains of $Y_{A,k}$. The $(k-1)$-th spectral gap $\mu_{k-1}(Y_{A,k})$ of $Y_{A,k}$ is the minimal eigenvalue of $L_{k-1}$. The following $k$-dimensional analogue of the Alon-Roichman theorem is proved: Let $k \geq 1$ and $\epsilon>0$ be fixed and let $A$ be a random subset of $G$ of size $m= \left\lceil\frac{10 k^2\log D}{\epsilon^2}\right\rceil$ where $D$ is the sum of the degrees of the complex irreducible representations of $G$. Then \[ {\rm Pr}\big[~\mu_{k-1}(Y_{A,k}) < (1-\epsilon)m~\big] =O\left(\frac{1}{n}\right). \]

math.CO↗

arXiv2021

Relative Leray numbers via spectral sequences

Let $\mathbb{F}$ be a fixed field and let $X$ be a simplicial complex on the vertex set $V$. The Leray number $L(X;\mathbb{F})$ is the minimal $d$ such that for all $i \geq d$ and $S \subset V$, the induced complex $X[S]$ satisfies $\tilde{H}_i(X[S];\mathbb{F})=0$. Leray numbers play a role in formulating and proving topological Helly type theorems. For two complexes $X,Y$ on the same vertex set $V$, define the relative Leray number $L_Y(X;\mathbb{F})$ as the minimal $d$ such that $\tilde{H}_i(X[V \setminus σ];\mathbb{F})=0$ for all $i \geq d$ and $σ\in Y$. In this paper we extend the topological colorful Helly theorem to the relative setting. Our main tool is a spectral sequence for the intersection of complexes indexed by a geometric lattice.

math.CO↗

arXiv2019

Large Simple d-Cycles in Simplicial Complexes

We show that the size of the largest simple d-cycle in a simplicial d-complex $K$ is at least a square root of $K$'s density. This generalizes a well-known classical result of Erdős and Gallai \cite{EG59} for graphs. We use methods from matroid theory applied to combinatorial simplicial complexes.

math.CO↗

arXiv2019

Near Coverings and Cosystolic Expansion -- an example of topological property testing

We study the stability of covers of simplicial complexes. Given a map $f:Y\to X$ that satisfies almost all of the local conditions of being a cover, is it close to being a genuine cover of $X$? Complexes $X$ for which this holds are called cover-stable. We show that this is equivalent to $X$ being a cosystolic expander with respect to non-abelian coefficients. This gives a new combinatorial-topological interpretation to cosystolic expansion which is a well studied notion of high dimensional expansion. As an example, we show that the $2$-dimensional spherical building $A_{3}(\mathbb{F}_q)$ is cover-stable. We view this work as a possibly first example of "topological property testing", where one is interested in studying stability of a topological notion that is naturally defined by local conditions.

math.CO↗

arXiv2019

On the topology of complexes of injective words

An injective word over a finite alphabet $V$ is a sequence $w=v_1v_2\cdots v_t$ of distinct elements of $V$. The set $\mathrm{inj}(V)$ of injective words on $V$ is partially ordered by inclusion. A complex of injective words is the order complex $Δ(W)$ of a subposet $W \subset \mathrm{inj}(V)$. Complexes of injective words arose recently in applications of algebraic topology to neuroscience, and are of independent interest in topology and combinatorics. In this article we mainly study Permutation Complexes, i.e. complexes of injective words $Δ(W)$, where $W$ is the downward closed subposet of $\mathrm{inj}(V)$ generated by a set of permutations of $V$. In particular, we determine the homotopy type of $Δ(W)$ when $W$ is generated by two permutations, and prove that any stable homotopy type is realizable by a permutation complex. We describe a homotopy decomposition for the complex of injective words $Γ(K)$ associated with a simplicial complex $K$, and point out a connection to a result of Randal-Williams and Wahl. Finally, we discuss some probabilistic aspects of random permutation complexes.

math.AT↗

arXiv2018

Pach's selection theorem does not admit a topological extension

Let $U_1,\dots, U_{d+1}$ be $n$-element sets in $R^d$ and let $\langle u_1,\ldots,u_{d+1}\rangle$ denote the convex hull of points $u_i$ in $U_i$ (for all $i$) which is a (possibly degenerate) simplex. Pach's selection theorem says that there are sets $Z_1 \subset U_1,\dots, Z_{d+1} \subset U_{d+1}$ and a point $u$ in $R^d$ such that each $|Z_i| > c_1(d)n$ and $u$ belongs to $\langle z_1,...,z_{d+1} \rangle$ for every choice of $z_1$ in $Z_1,\dots,z_{d+1}$ in $Z_{d+1}$. Here we show that this theorem does not admit a topological extension with linear size sets $Z_i$. However, there is a topological extension where each $|Z_i|$ is of order $(\log n)^(1/d)$.

math.CO↗

arXiv2018

On Lusztig-Dupont homology of flag complexes

Let $V$ be an $n$-dimensional vector space over the finite field of order $q$. The spherical building $X_V$ associated with $GL(V)$ is the order complex of the nontrivial linear subspaces of $V$. Let $\mathfrak{g}$ be the local coefficient system on $X_V$, whose value on the simplex $σ=[V_0 \subset \cdots \subset V_p] \in X_V$ is given by $\mathfrak{g}(σ)=V_0$. Following the work of Lusztig and Dupont, we study the homology module $D^k(V)=\tilde{H}_{n-k-1}(X_V;\mathfrak{g})$. Our results include a construction of an explicit basis of $D^1(V)$, and the following twisted analogue of a result of Smith and Yoshiara: For any $1 \leq k \leq n-1$, the minimal support size of a non-zero $(n-k-1)$-cycle in the twisted homology $\tilde{H}_{n-k-1}(X_V;\wedge^k \mathfrak{g})$ is $\frac{(n-k+2)!}{2}$.

math.CO↗

arXiv2018

Graph codes and local systems

It is shown that the good expander codes introduced by Sipser and Spielman, can be realized as the first homology of a graph with respect to a certain twisted coefficient system.

math.CO↗

arXiv2018

Quantitative aspects of acyclicity

We study several aspects of the $k$-th Cheeger constant of a complex X, a parameter that quantifies the distance of $X$ from a complex $Y$ with nontrivial $k$-th cohomology over $\mathbb{Z}_2$. Our results include general methods for bounding the cosystolic norm of a cochain and for bounding the Cheeger constant of a complex, a discussion of expansion of pseudomanifolds and geometric lattices, probabilistic upper bounds on Cheeger constants, and application of non-Abelian expansion to random complexes.

math.CO↗

arXiv2018

Spectral expansion of random sum complexes

Let $G$ be a finite abelian group of order $n$ and let $Δ_{n-1}$ denote the $(n-1)$-simplex on the vertex set $G$. The sum complex $X_{A,k}$ associated to a subset $A \subset G$ and $k < n$, is the $k$-dimensional simplicial complex obtained by taking the full $(k-1)$-skeleton of $Δ_{n-1}$ together with all $(k+1)$-subsets $σ\subset G$ that satisfy $\sum_{x \in σ} x \in A$. Let $C^{k-1}(X_{A,k})$ denote the space of complex valued $(k-1)$-cochains of $X_{A,k}$. Let $L_{k-1}:C^{k-1}(X_{A,k}) \rightarrow C^{k-1}(X_{A,k})$ denote the reduced $(k-1)$-th Laplacian of $X_{A,k}$, and let $μ_{k-1}(X_{A,k})$ be the minimal eigenvalue of $L_{k-1}$. It is shown that for any $k \geq 1$ and $ε>0$ there exists a constant $c(k,ε)$ such that if $A$ is a random subset of $G$ of size $m=\lceil c(k,ε) \log n \rceil$, then $μ_{k-1}(X_{A,k}) > (1-ε)m$ asymptotically almost surely.

math.CO↗

arXiv2017

Maximal rank in matrix spaces via graph matchings

We study the maximal rank in affine subspaces of symmetric or alternating matrices, in terms of the matching numbers of certain associated graphs. Applications include simple proofs of upper bounds on the dimension of such subspaces in terms of their maximal rank.

math.CO↗

arXiv2017

Betti numbers of complexes with highly Connected links

Let X be a k-dimensional simplicial complex such that the (k-j-2)-dimensional homology of the links of all j-dimensional simplices in X vanishes. An upper bound is given on the (k-1)-th Betti number of X. Examples based on sum complexes show that this bound is asymptotically sharp for all fixed j<k.

math.CO↗

arXiv2016

A Tverberg type theorem for matroids

Let b(M) denote the maximal number of disjoint bases in a matroid M. It is shown that if M is a matroid of rank d+1, then for any continuous map f from the matroidal complex M into the d-dimensional Euclidean space there exist t \geq \sqrt{b(M)}/4 disjoint independent sets σ_1,\ldots,σ_t \in M such that \bigcap_{i=1}^t f(σ_i) \neq \emptyset.

math.CO↗

arXiv2016

Homology of spaces of directed paths in Euclidean pattern spaces

The paper addresses certain topological aspects of Dijkstra's PV-model for parallel computations in concurrency theory. The main result is a computation of the homology of the trace space associated with PV-programs in which access and release of every resource happen without time delay.

math.CO↗

arXiv2014

Expansion of Building-Like Complexes

Following Gromov, the coboundary expansion of building-like complexes is studied. In particular, it is shown that for any $n \geq 1$, there exists a constant $ε(n)>0$ such that for any $0 \leq k <n$ the $k$-th coboundary expansion constant of any $n$-dimensional spherical building is at least $ε(n)$.

math.CO↗

arXiv2013

Bounded quotients of the fundamental group of a random 2-complex

Let D denote the (n-1)-dimensional simplex. Let Y be a random 2-dimensional subcomplex of D obtained by starting with the full 1-skeleton of D and then adding each 2-simplex independently with probability p. For a fixed c>0 it is shown that if p=\frac{(6+7c) \log n}{n} then a.a.s. the fundamental group π(Y) does not have a nontrivial quotient of order at most n^c.

math.CO↗