Problems in Geometric and Topological Combinatorics Gil Kalai Berlin, October 2011 Gil Kalai Fantasies in Geometric and Topological Combinatorics
This lecture 1. Around Tverberg’s Theorem 2. Borsuk’s problem and the combinatorics of cocycles 3. A remark about connectivity 4. The Fractional Helly Property and homology growth Gil Kalai Fantasies in Geometric and Topological Combinatorics
I. Around Tverberg’s theorem Gil Kalai Fantasies in Geometric and Topological Combinatorics
Tverberg’s theorem Tverberg’s theorem: Let X = { x 1 , x 2 , . . . , x m } be a set of m points in R d , m ≥ ( d + 1)( r − 1) + 1. Then X can be partitioned into r pairwise disjoint parts X 1 , X 2 . . . , X r such that conv ( X 1 ) ∩ conv ( X 2 ) ∩ · · · ∩ conv ( X r ) � = ∅ . Gil Kalai Fantasies in Geometric and Topological Combinatorics
Tverberg’s theorem Tverberg’s theorem: Let X = { x 1 , x 2 , . . . , x m } be a set of m points in R d , m ≥ ( d + 1)( r − 1) + 1. Then X can be partitioned into r pairwise disjoint parts X 1 , X 2 . . . , X r such that conv ( X 1 ) ∩ conv ( X 2 ) ∩ · · · ∩ conv ( X r ) � = ∅ . History: Birch (conjectured), Rado (proved a weaker result), Tverberg (proved), Tverberg (reproved), Tverberg and Vrecica (reproved), Sarkaria (reproved), Roundeff (reproved) (The easy case r = 2 is Radon’s theorem.) Gil Kalai Fantasies in Geometric and Topological Combinatorics
The Topological Tverberg’s Conjecture Topological Tverberg’s Conjecture : Let f : ∆ ( d +1)( r − 1) → R d be a continuous function from the ( d + 1)( r − 1) dimensional simplex to R d . Then there are r disjoint faces of the simplex whose images have a point in common. The topological Tverberg’s conjecture is known to hold when r is a prime power. History: B´ ar´ any and Bajm´ oczy , B´ ar´ any, Shlosman and Sz¨ ucs, ... Zivaljevic and Vrecica, Blagojevi´ c, Matschke, and Ziegler Gil Kalai Fantasies in Geometric and Topological Combinatorics
The dimensions of Tverberg’s points Let X be a set of points in R d . The Tverberg points of order r , denoted by T r ( X ), are those points that belong to the intersection of the convex hulls of r pairwise disjoint subsets of X . Gil Kalai Fantasies in Geometric and Topological Combinatorics
The dimensions of Tverberg’s points Let X be a set of points in R d . The Tverberg points of order r , denoted by T r ( X ), are those points that belong to the intersection of the convex hulls of r pairwise disjoint subsets of X . The Cascade Conjecture: | X | � dim T i ( X ) ≥ 0 . i =1 Gil Kalai Fantasies in Geometric and Topological Combinatorics
The dimensions of Tverberg’s points: a weaker conjecture The Weak Cascade Conjecture: | X | � dim conv ( T i ( X )) ≥ 0 . i =1 Gil Kalai Fantasies in Geometric and Topological Combinatorics
Why Tverberg’s conjecture follows Let X be a set of m = ( r − 1)( d + 1) + 1 points in R d . Then dim T i ( X ) ≤ d , for every i . If T r ( X ) is empty then m � dim T i ( X ) ≤ i =1 ( r − 1) d + ( − 1)(( d + 1)( r − 1) + 1 − ( r − 1)) = − 1 . Gil Kalai Fantasies in Geometric and Topological Combinatorics
An even weaker conjecture: the dimensions of the k -cores Let X be a set of points in R d . The r th core of X , denoted C r ( X ), is the set of all the points that belong to every convex hull of all but r of the points. T r ( X ) ⊂ C r ( X ). Conjecture: | X | � dim C i ( X ) ≥ 0 . i =1 I think this should be doable. Gil Kalai Fantasies in Geometric and Topological Combinatorics
An even weaker statement Let X be a set of points in R d . Denoted by A r ( X ), those points that belong to the intersection of the affine hull of r pairwise disjoint subsets of X . I think this is essentially known: | X | � (dim A i ( X )) ≥ 0 . i =1 Gil Kalai Fantasies in Geometric and Topological Combinatorics
Kadari’s theorem: Theorem: (Kadari 81-90) The cascade conjecture holds in the plane. Uses (to the best of my memory) a claim that in the plane C r ( X ) is the convex hull of T r ( X ). (Not true for d ≥ 3.) Gil Kalai Fantasies in Geometric and Topological Combinatorics
A new ∗ approach ∗∗ to topological Tverberg (Old approach) Divide your set to 3 parts (works only if 3 is a primes) (New Approach) Divide your set into two parts and divide one part again into two parts. (Something that might be needed:) If the set of Radon’s partitions is sufficiently “connected” then a Tverberg’s partition into three parts exists. ∗ not new ∗∗ not quite an approach more like a fantasy Gil Kalai Fantasies in Geometric and Topological Combinatorics
Boris Bukh disproved the partition conjecture! Let G be a family of subsets of a ground set X which is closed under intersection. Define t r ( G ) to be the smallest integer with the following property: Every set of t r ( G ) points from X can be divided into r parts, X 1 , X 2 , . . . , X r such that for every S 1 , S 2 , . . . , S r ∈ G with X i ⊂ S i there is a point in common to all the S ′ i s . Gil Kalai Fantasies in Geometric and Topological Combinatorics
Boris Bukh disproved the partition conjecture! Let G be a family of subsets of a ground set X which is closed under intersection. Define t r ( G ) to be the smallest integer with the following property: Every set of t r ( G ) points from X can be divided into r parts, X 1 , X 2 , . . . , X r such that for every S 1 , S 2 , . . . , S r ∈ G with X i ⊂ S i there is a point in common to all the S ′ i s . The partition conjecture (disproved by Boris Bukh): t r − 1 ≤ r ( t 2 − 1). Gil Kalai Fantasies in Geometric and Topological Combinatorics
Boris Bukh disproved the partition conjecture! Let G be a family of subsets of a ground set X which is closed under intersection. Define t r ( G ) to be the smallest integer with the following property: Every set of t r ( G ) points from X can be divided into r parts, X 1 , X 2 , . . . , X r such that for every S 1 , S 2 , . . . , S r ∈ G with X i ⊂ S i there is a point in common to all the S ′ i s . The partition conjecture (disproved by Boris Bukh): t r − 1 ≤ r ( t 2 − 1). Question: Does Tverberg’s theorem hold for oriented matroids? Gil Kalai Fantasies in Geometric and Topological Combinatorics
II. Borsuk’s problem and cocycles Gil Kalai Fantasies in Geometric and Topological Combinatorics
Borsuk’s conjecture Karol Borsuk conjectured in 1933 that every bounded set in R d can be covered by d + 1 sets of smaller diameter. Let f ( d ) be the smallest integer such that every set of diameter 1 in R d can be covered by f ( d ) sets of smaller diameter. Gil Kalai Fantasies in Geometric and Topological Combinatorics
Larman’s conjecture David Larman proposed to consider purely combinatorial special cases Conjecture : Let F be a family of subsets of { 1 , 2 , . . . , n } , and suppose that the symmetric difference between every two sets in F has at most t elements. Then F can be divided into n + 1 families such that the symmetric difference between any pair of sets in the same family is at most t − 1. To see the connection with Borsuk’s problem just consider the set of characteristic vectors of the sets in the family. Gil Kalai Fantasies in Geometric and Topological Combinatorics
Another question by Larman Problem: Does Borsuk’s conjecture hold for 2-distance sets? Gil Kalai Fantasies in Geometric and Topological Combinatorics
The cut construction The construction of Jeff Kahn and myself can (essentially) be described as follows: The cut construction: The ground set is the set of edges of the complete graph on 4 p vertices. The family F consists of all subsets of edges which represent the edge set of a complete bipartite graph. √ The cut constructions shows that f ( d ) > exp ( K d ). We would like to replace d 1 / 2 by a larger exponent. Gil Kalai Fantasies in Geometric and Topological Combinatorics
Cocycles Definition: A k -cocycle is a collection of ( k + 1)-subsets such that every ( k + 2)-set T contains an even number of sets in the collection. An alternative definition is to start with a collection G of k -sets and consider all ( k + 1)-sets that contain an odd number of members in G . It is easy to see that the two definitions are equivalent. (This equivalence expresses the fact that the k -cohomology of a simplex is zero.) Note that the symmetric difference of two cocycles is a cocycle. In other words, the set of k -cocycles form a subspace over Z/2Z, i.e., a linear binary code. Gil Kalai Fantasies in Geometric and Topological Combinatorics
Cocycles (cont.) Definition: A k -cocycle is a collection of ( k + 1)-subsets such that every ( k + 2)-set T contains an even number of sets in the collection. 1-cocycles correspond to cuts in graphs. Those were studied intensively in the combinatorics literature. 2-cocycles were studied under the name “two-graphs”. Their study was initiated by J. J. Seidel. Gil Kalai Fantasies in Geometric and Topological Combinatorics
The combinatorics of cocycles Problem: Let k be odd. What is the maximum number of simplices in a k -dimensional cocycle with n vertices? Gil Kalai Fantasies in Geometric and Topological Combinatorics
Recommend
More recommend
