43 citations · 163 across the 27 of their papers we have counts for
37 papers
Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes
Henry Adams, Johnathan Bush, Nate Clause +13
We explore emerging relationships between the Gromov--Hausdorff distance, Borsuk--Ulam theorems, and Vietoris--Rips simplicial complexes. The Gromov--Hausdorff distance between two…
Unit sphere fibrations in Euclidean space
Daniel Asimov, Florian Frick, Michael Harrison +1
We show that if an open set in can be fibered by unit -spheres, then , and if , then the spheres must be pairwise linked, and $n \in \left\…
Random complexes with free involution
Florian Frick, Andrew Newman
We introduce a new model for random simplicial complexes which with high probability generates a complex that has a simply-connected double cover. Hence we develop a model for rand…
Transversal generalizations of hyperplane equipartitions
Florian Frick, Samuel Murray, Steven Simon +1
The classical Ham Sandwich theorem states that any point sets in can be simultaneously bisected by a single affine hyperplane. A generalization of Dolnikov asser…
Vietoris thickenings and complexes have isomorphic homotopy groups
Henry Adams, Florian Frick, Žiga Virk
We study the relationship between metric thickenings and simplicial complexes associated to coverings of metric spaces. Let be a cover of a separable metric space …
Colorful Words and d-Tverberg Complexes
Florian Frick, R. Amzi Jeffs
We give a complete combinatorial characterization of weakly -Tverberg complexes. These complexes record which intersection combinatorics of convex hulls necessarily arise in any…
Vertex numbers of simplicial complexes with free abelian fundamental group
Florian Frick, Matt Superdock
We show that the minimum number of vertices of a simplicial complex with fundamental group is at most and at least . For the upper bound, we use…
On inscribed trapezoids and affinely 3-regular maps
Florian Frick, Michael Harrison
We show that any embedding inscribes a trapezoid or maps three points to a line, where is the smallest power of satisfy…
Embedding dimensions of simplicial complexes on few vertices
Florian Frick, Mirabel Hu, Verity Scheel +1
We provide a simple characterization of simplicial complexes on few vertices that embed into the -sphere. Namely, a simplicial complex on vertices embeds into the -sphe…
Coupled embeddability
Florian Frick, Michael Harrison
We introduce the notion of coupled embeddability, defined for maps on products of topological spaces. We use known results for nonsingular biskew and bilinear maps to generate simp…
The topology of projective codes and the distribution of zeros of odd maps
Henry Adams, Johnathan Bush, Florian Frick
We show that the size of codes in projective space controls structural results for zeros of odd maps from spheres to Euclidean space. In fact, this relation is given through the to…
Spaces of embeddings: Nonsingular bilinear maps, chirality, and their generalizations
Florian Frick, Michael Harrison
Given a space X we study the topology of the space of embeddings of X into through the combinatorics of triangulations of X. We give a simple combinatorial formula f…
The topological Tverberg problem beyond prime powers
Florian Frick, Pablo Soberón
Tverberg-type theory aims to establish sufficient conditions for a simplicial complex such that every continuous map maps points from pairwise d…
What can you draw?
Florian Frick, Fei Peng
We address the problem of which planar sets can be drawn with a pencil and eraser. The pencil draws any union of black open unit disks in the plane . The eraser produ…
Metric Thickenings, Borsuk-Ulam Theorems, and Orbitopes
Henry Adams, Johnathan Bush, Florian Frick
Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more gen…
A nonlinear Lazarev-Lieb theorem: -orthogonality via motion planning
Florian Frick, Matt Superdock
Lazarev and Lieb showed that finitely many integrable functions from the unit interval to can be simultaneously annihilated in the inner product by a smooth func…
Borsuk-Ulam theorems for products of spheres and Stiefel manifolds revisited
Yu Hin Chan, Shujian Chen, Florian Frick +1
We give a different and possibly more accessible proof of a general Borsuk--Ulam theorem for a product of spheres, originally due to Ramos. That is, we show the non-existence of ce…
Fair splittings by independent sets in sparse graphs
Alexander Black, Umur Cetin, Florian Frick +2
Given a partition of the vertex set of a graph, we are interested in finding multiple disjoint independent sets that contain the correct fr…
Barycenters of points in polytope skeleta
Michael Gene Dobbins, Florian Frick
The first author showed that for a given point in an -polytope there are points in the -faces of , whose barycenter is . We show that we can increase the d…
Splitting loops and necklaces: Variants of the square peg problem
Jai Aslam, Shujian Chen, Florian Frick +3
Toeplitz conjectured that any simple planar loop inscribes a square. Here we prove variants of Toeplitz' square peg problem. We prove Hadwiger's 1971 conjecture that any simple loo…
Neural codes, decidability, and a new local obstruction to convexity
Aaron Chen, Florian Frick, Anne Shiu
Given an intersection pattern of arbitrary sets in Euclidean space, is there an arrangement of convex open sets in Euclidean space that exhibits the same intersections? This questi…
Chromatic numbers of stable Kneser hypergraphs via topological Tverberg-type theorems
Florian Frick
Kneser's 1955 conjecture -- proven by Lovász in 1978 -- asserts that in any partition of the -subsets of into parts, one part contains two disjoint…
Colorful coverings of polytopes and piercing numbers of colorful d-intervals
Florian Frick, Shira Zerbib
We prove a common strengthening of Bárány's colorful Carathéodory theorem and the KKMS theorem. In fact, our main result is a colorful polytopal KKMS theorem, which extends a color…
Metric reconstruction via optimal transport
Michal Adamaszek, Henry Adams, Florian Frick
Given a sample of points in a metric space and a scale , the Vietoris-Rips simplicial complex is a standard construction to attempt to recover f…