arXiv subjects

Matt Superdock

Publications and source records attributed to Matt Superdock.

7 recordsLinked to original sources

arXiv2025

Quantifying discontinuity

Given a compact space $X$ that does not admit an embedding (an injective continuous function) into $\mathbb{R}^d$, we study the ''degree'' of discontinuity that any injective function $X \to \mathbb{R}^d$ must have. To this end, we define a scale invariant modulus of discontinuity and obtain general lower bounds, thus obtaining quantified nonembeddability results of Haefliger--Weber type. Moreover, we establish analogous lower bounds for simplicial complexes that do not admit an almost $r$-embedding in $\mathbb{R}^d$, thus obtaining a quantified version of the topological Tverberg theorem.

math.MG↗

arXiv2022

Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes

We explore emerging relationships between the Gromov--Hausdorff distance, Borsuk--Ulam theorems, and Vietoris--Rips simplicial complexes. The Gromov--Hausdorff distance between two metric spaces $X$ and~$Y$ can be lower bounded by the distortion of (possibly discontinuous) functions between them. The more these functions must distort the metrics, the larger the Gromov--Hausdorff distance must be. Topology has few tools to obstruct the existence of discontinuous functions. However, an arbitrary function $f\colon X\to Y$ induces a continuous map between their Vietoris--Rips simplicial complexes, where the allowable choices of scale parameters depend on how much the function $f$ distorts distances. We can then use equivariant topology to obstruct the existence of certain continuous maps between Vietoris--Rips complexes. With these ideas we bound how discontinuous an odd map between spheres $S^k\to S^n$ with $k>n$ must be, generalizing a result by Dubins and Schwarz (1981), which is the case $k=n+1$. As an application, we recover or improve upon all of the lower bounds from Lim, M{\'e}moli, and Smith (2022) on the Gromov--Hausdorff distances between spheres of different dimensions. We also provide new upper bounds on the Gromov--Hausdorff distance between spheres of adjacent dimensions.

math.MG↗

arXiv2021

Simplicial complexes from finite projective planes and colored configurations

In the 7-vertex triangulation of the torus, the 14 triangles can be partitioned as $T_{1} \sqcup T_{2}$, such that each $T_{i}$ represents the lines of a copy of the Fano plane $PG(2, \mathbb{F}_{2})$. We generalize this observation by constructing, for each prime power $q$, a simplicial complex $X_{q}$ with $q^{2} + q + 1$ vertices and $2(q^{2} + q + 1)$ facets consisting of two copies of $PG(2, \mathbb{F}_{q})$. Our construction works for any colored $k$-configuration, defined as a $k$-configuration whose associated bipartite graph $G$ is connected and has a $k$-edge coloring $\chi \colon E(G) \to [k]$, such that for all $v \in V(G)$, $a, b, c \in [k]$, following edges of colors $a, b, c, a, b, c$ from $v$ brings us back to $v$. We give one-to-one correspondences between (1) Sidon sets of order 2 and size $k + 1$ in groups with order $n$, (2) linear codes with radius 1 and index $n$ in the lattice $A_{k}$, and (3) colored $(k + 1)$-configurations with $n$ points and $n$ lines. (The correspondence between (1) and (2) is known.) As a result, we suggest possible topological obstructions to the existence of Sidon sets, and in particular, planar difference sets.

math.CO↗

arXiv2021

Vertex numbers of simplicial complexes with free abelian fundamental group

We show that the minimum number of vertices of a simplicial complex with fundamental group $\mathbb{Z}^{n}$ is at most $O(n)$ and at least $\Omega(n^{3/4})$. For the upper bound, we use a result on orthogonal 1-factorizations of $K_{2n}$. For the lower bound, we use a fractional Sylvester-Gallai result. We also prove that any group presentation $\langle S | R\rangle \cong \mathbb{Z}^{n}$ whose relations are of the form $g^{a}h^{b}i^{c}$ for $g, h, i \in S$ has at least $\Omega(n^{3/2})$ generators.

math.CO↗

arXiv2019

A nonlinear Lazarev-Lieb theorem: $L^2$-orthogonality via motion planning

Lazarev and Lieb showed that finitely many integrable functions from the unit interval to $\mathbb{C}$ can be simultaneously annihilated in the $L^2$ inner product by a smooth function to the unit circle. Here we answer a question of Lazarev and Lieb proving a generalization of their result by lower bounding the equivariant topology of the space of smooth circle-valued functions with a certain $W^{1,1}$-norm bound. Our proof uses a relaxed notion of motion planning algorithm that instead of contractibility yields a lower bound for the $\mathbb{Z}/2$-coindex of a space.

math.FA↗

arXiv2019

Clean tangled clutters, simplices, and projective geometries

A clutter is \emph{clean} if it has no delta or the blocker of an extended odd hole minor, and it is \emph{tangled} if its covering number is two and every element appears in a minimum cover. Clean tangled clutters have been instrumental in progress towards several open problems on ideal clutters, including the $\tau=2$ Conjecture. Let $\mathcal{C}$ be a clean tangled clutter. It was recently proved that $\mathcal{C}$ has a fractional packing of value two. Collecting the supports of all such fractional packings, we obtain what is called the {\it core} of $\mathcal{C}$. The core is a duplication of the cuboid of a set of $0-1$ points, called the {\it setcore} of $\mathcal{C}$. In this paper, we prove three results about the setcore. First, the convex hull of the setcore is a full-dimensional polytope containing the center point of the hypercube in its interior. Secondly, this polytope is a simplex if, and only if, the setcore is the cocycle space of a projective geometry over the two-element field. Finally, if this polytope is a simplex of dimension more than three, then $\mathcal{C}$ has the clutter of the lines of the Fano plane as a minor. Our results expose a fascinating interplay between the combinatorics and the geometry of clean tangled clutters.

math.CO↗

arXiv2015

New graceful diameter-6 trees by transfers

Given a graph $G$, a labeling of $G$ is an injective function $f:V(G)\rightarrow\mathbb{Z}_{\ge 0}$. Under the labeling $f$, the label of a vertex $v$ is $f(v)$, and the induced label of an edge $uv$ is $|f(u) - f(v)|$. The labeling $f$ is graceful if the labels of the vertices are $\{0, 1, \ldots , |V(G)| - 1\}$, and the induced labels of the edges are distinct. The graph $G$ is graceful if it has a graceful labeling. The Graceful Tree Conjecture, introduced by Kotzig in the late 1960's, states that all trees are graceful. It is an open problem whether every diameter-6 tree has a graceful labeling. In this paper, we prove that if $T$ is a tree with central vertex and root $v$, such that each vertex not in the last two levels has an odd number of children, and $T$ satisfies one of the following conditions (a)-(e), then $T$ has a graceful labeling $f$ with $f(v) = 0$: (a) $T$ is a diameter-6 complete tree; (b) $T$ is a diameter-6 tree such that no two leaves of distance 2 from $v$ are siblings, and each leaf of distance 2 from $v$ has a sibling with an even number of children; (c) $T$ is a diameter-$2r$ complete tree, such that the number of vertices of distance $r - 1$ from $v$, with an even number of children, is not $3\pmod{4}$; (d) $T$ is a diameter-$2r$ tree, such that the number of vertices of distance $r - 1$ from $v$, with an even number of children, is not $3\pmod{4}$, no two leaves of distance $r - 1$ from $v$ are siblings, and each leaf of distance $r - 1$ from $v$ has a sibling with an even number of children; (e) $T$ is a diameter-6 tree, such that each internal vertex has an odd number of children. In particular, all depth-3 trees of which each internal vertex has an odd number of children are graceful.

math.CO↗