Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe topological Tverberg problem beyond prime powers
135
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Tverberg-type theory aims to establish sufficient conditions for a simplicial complex $Sigma$ such that every continuous map $fcolon Sigma to mathbb{R}^d$ maps $q$ points from pairwise disjoint faces to the same point in $mathbb{R}^d$. Such results are plentiful for $q$ a power of a prime. However, for $q$ with at least two distinct prime divisors, results that guarantee the existence of $q$-fold points of coincidence are non-existent---aside from immediate corollaries of the prime power case. Here we present a general method that yields such results beyond the case of prime powers. In particular, we prove previously conjectured upper bounds for the topological Tverberg problem for all $q$.
rate research
Read More
A seminal theorem of Tverberg states that any set of $T(r,d)=(r-1)(d+1)+1$ points in $mathbb{R}^d$ can be partitioned into $r$ subsets whose convex hulls have non-empty $r$-fold intersection. Almost any collection of fewer points in $mathbb{R}^d$ cannot be so divided, and in these cases we ask if the set can nonetheless be $P(r,d)$--partitioned, i.e., split into $r$ subsets so that there exist $r$ points, one from each resulting convex hull, which form the vertex set of a prescribed convex $d$--polytope $P(r,d)$. Our main theorem shows that this is the case for any generic $T(r,2)-2$ points in the plane and any $rgeq 3$ when $P(r,2)=P_r$ is a regular $r$--gon, and moreover that $T(r,2)-2$ is tight. For higher dimensional polytopes and $r=r_1cdots r_k$, $r_i geq 3$, this generalizes to $T(r,2k)-2k$ generic points in $mathbb{R}^{2k}$ and orthogonal products $P(r,2k)=P_{r_1}times cdots times P_{r_k}$ of regular polygons, and likewise to $T(2r,2k+1)-(2k+1)$ points in $mathbb{R}^{2k+1}$ and the product polytopes $P(2r,2k+1)=P_{r_1}times cdots times P_{r_k} times P_2$. As with Tverbergs original theorem, our results admit topological generalizations when $r$ is a prime power, and, using the constraint method of Blagojevic, Frick, and Ziegler, allow for dimensionally restrict
In 2017 Bar-Natan and the first author showed that solutions to the Kashiwara--Vergne equations are in bijection with certain knot invariants: homomorphic expansions of welded foams. Welded foams are a class of knotted tubes in $mathbb{R}^4$, which can be finitely presented algebraically as a circuit algebra, or, equivalently, a wheeled prop. In this paper we describe the Kashiwara-Vergne groups $mathsf{KV}$ and $mathsf{KRV}$ -- the symmetry groups of Kashiwara-Vergne solutions -- as automorphisms of the completed circuit algebras of welded foams, and their associated graded circuit algebra of arrow diagrams, respectively. Finally, we provide a description of the graded Grothendieck-Teichmuller group $mathsf{GRT}_1$ as automorphisms of arrow diagrams.
For a graph whose vertex set is a finite set of points in $mathbb R^d$, consider the closed (open) balls with diameters induced by its edges. The graph is called a (an open) Tverberg graph if these closed (open) balls intersect. Using the idea of halving lines, we show that (i) for any finite set of points in the plane, there exists a Hamiltonian cycle that is a Tverberg graph; (ii) for any $n$ red and $n$ blue points in the plane, there exists a perfect red-blue matching that is a Tverberg graph. Using the idea of infinite descent, we prove that (iii) for any even set of points in $mathbb R^d$, there exists a perfect matching that is an open Tverberg graph; (iv) for any $n$ red and $n$ blue points in $ mathbb R^d $, there exists a perfect red-blue matching that is a Tverberg graph.
Let $mathbb{F}_p$ be a prime field, and ${mathcal E}$ a set in $mathbb{F}_p^2$. Let $Delta({mathcal E})={||x-y||: x,y in {mathcal E} }$, the distance set of ${mathcal E}$. In this paper, we provide a quantitative connection between the distance set $Delta({mathcal E})$ and the set of rectangles determined by points in ${mathcal E}$. As a consequence, we obtain a new lower bound on the size of $Delta({mathcal E})$ when ${mathcal E}$ is not too large, improving a previous estimate due to Lund and Petridis and establishing an approach that should lead to significant further improvements.
We present a motivated exposition of the proof of the following Tverberg Theorem: For every integers $d,r$ any $(d+1)(r-1)+1$ points in $mathbb R^d$ can be decomposed into $r$ groups such that all the $r$ convex hulls of the groups have a common point. The proof is by well-known reduction to the Barany Theorem. However, our exposition is easier to grasp because additional constructions (of an embedding $mathbb R^dsubsetmathbb R^{d+1}$, of vectors $varphi_{j,i}$ and statement of the Barany Theorem) are not introduced in advance in a non-motivated way, but naturally appear in an attempt to construct the required decomposition. This attempt is based on rewriting several equalities between vectors as one equality between vectors of higher dimension.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
