Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Generalizations of the Joints Problem
Fourth Annual MIT PRIMES Conference
Background
Determining the
Constant
A Conjecture
Joseph Zurier
Mentor: Ben Yang
Problem by: Larry Guth
The General
Problem
Another
Generalization
Further Research
Acknowledgements
May 17, 2014
Joints
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
Definition
Let S be a collection of lines in R3 . We say that the
point p ∈ R3 is a joint if there exist lines `1 , `2 , `3 ∈ S
such that `1 ∩ `2 ∩ `3 = p and `1 , `2 , `3 do not lie in a
common plane.
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Joints
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
The Joints Problem
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
Given n lines, how many joints can we make?
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
The Joints Problem
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
Given n lines, how many joints can we make?
First asked in a 1992 paper by Chazelle et al.
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
The Joints Problem
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
Given n lines, how many joints can we make?
First asked in a 1992
by Chazelle et al.
paper
2
Obvious bound: n2 ≈ n2
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
The Joints Problem
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
Given n lines, how many joints can we make?
First asked in a 1992
by Chazelle et al.
paper
2
Obvious bound: n2 ≈ n2
7
Their bound: cn 4
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Generalizations of
the Joints Problem
Recent Work
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
3
2
2009: Guth and Katz proved that J ≤ cn .
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Generalizations of
the Joints Problem
Recent Work
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
3
2
2009: Guth and Katz proved that J ≤ cn .
Method of proof: the polynomial method
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Generalizations of
the Joints Problem
Recent Work
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
3
2
Determining the
Constant
2009: Guth and Katz proved that J ≤ cn .
Method of proof: the polynomial method
Lemma
A Conjecture
The General
Problem
1
There exists a nonzero polynomial P of degree d ≤ (n!k) n
that vanishes on k points in Rn .
Another
Generalization
Further Research
Acknowledgements
Generalizations of
the Joints Problem
Recent Work
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
3
2
Determining the
Constant
2009: Guth and Katz proved that J ≤ cn .
Method of proof: the polynomial method
Lemma
A Conjecture
The General
Problem
1
There exists a nonzero polynomial P of degree d ≤ (n!k) n
that vanishes on k points in Rn .
What about c?
Another
Generalization
Further Research
Acknowledgements
Joints Problem, generalized
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
What happens if we change the parameters of the problem?
Can we bound these cases as well?
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Lower Bounds
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
Consider a k × k × k grid.
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Lower Bounds
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Lower Bounds
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Generalizations of
the Joints Problem
Lower Bounds
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
3
3
3k 2 lines make k 3 joints, so J = ( 13 ) 2 n 2
Lower Bounds
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
Consider k planes in general position.
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Lower Bounds
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Generalizations of
the Joints Problem
Lower Bounds
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
k
2
lines make
k
3
√
joints, so J =
2 32
3 n
Generalizations of
the Joints Problem
The Constant
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Weak upper bound: 10n
3
2
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Generalizations of
the Joints Problem
The Constant
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Weak upper bound: 10n
3
New upper bound: 34 n 2
3
2
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Generalizations of
the Joints Problem
The Constant
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Weak upper bound: 10n
3
New upper bound: 34 n 2
An observation:
3
2
Determining the
Constant
A Conjecture
The General
Problem
4 n3
2
3
2
√
=
2 3
n2
3
Another
Generalization
Further Research
Acknowledgements
Line Removal Conjecture
The polynomial method proof includes a step where each
line is removed as part of an inductive argument.
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Line Removal Conjecture
The polynomial method proof includes a step where each
line is removed as part of an inductive argument.
The following suffices to determine the constant:
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Line Removal Conjecture
The polynomial method proof includes a step where each
line is removed as part of an inductive argument.
The following suffices to determine the constant:
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Conjecture
Suppose we have a set S of n lines {`i } in R3 . Given
any such set S, let f (S) be the number of joints formed
by lines in S. Also, let g (`0 , S) be the number of joints
formed by `0 and two members of S. Then there exists
a sequence {ai , i ≤ k} with the following properties:
1. k ≤
n
2
2. ∀ 0 ≤ x ≤ k − 1 g (`ax+1 , S\{`ai , i ≤ x}) ≤
1
(6f (S\{`ai , i ≤ x})) 3
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Generalizations of
the Joints Problem
Line Removal Conjecture
The polynomial method proof includes a step where each
line is removed as part of an inductive argument.
The following suffices to determine the constant:
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Conjecture
Suppose we have a set S of n lines {`i } in R3 . Given
any such set S, let f (S) be the number of joints formed
by lines in S. Also, let g (`0 , S) be the number of joints
formed by `0 and two members of S. Then there exists
a sequence {ai , i ≤ k} with the following properties:
1. k ≤
n
2
2. ∀ 0 ≤ x ≤ k − 1 g (`ax+1 , S\{`ai , i ≤ x}) ≤
1
(6f (S\{`ai , i ≤ x})) 3
Generalized to Rm , the numbers work out if we use
instead of 12
1
m−1
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Line Removal Conjecture
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Line Removal Conjecture
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Line Removal Conjecture
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Line Removal Conjecture
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Line Removal Conjecture
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Line Removal Conjecture
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Line Removal Conjecture
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
The General Problem
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
The parameters are:
I
The dimension of the space Rn
Determining the
Constant
A Conjecture
The General
Problem
I
The dimension of the objects Pa that are intersecting
I
The dimension of their intersection Pb
Further Research
I
The number k of Pa that must intersect to make a joint
Acknowledgements
Another
Generalization
The General Problem
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
The parameters are:
I
The dimension of the space Rn
Determining the
Constant
A Conjecture
The General
Problem
I
The dimension of the objects Pa that are intersecting
I
The dimension of their intersection Pb
Further Research
I
The number k of Pa that must intersect to make a joint
Acknowledgements
Need n = b + k(a − b)
Another
Generalization
Generalizations of
the Joints Problem
Lower Bound
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
1
k
(k − 1)! k−1 k−1
x
k
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Joints Redefined
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Suppose we have a collection of p planes and ` lines in R4 .
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Joints Redefined
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Suppose we have a collection of p planes and ` lines in R4 .
Whenever two lines and one plane intersect at a common
point such that their tangent vectors span R4 , we call this
point a joint.
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Joints Redefined
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Suppose we have a collection of p planes and ` lines in R4 .
Whenever two lines and one plane intersect at a common
point such that their tangent vectors span R4 , we call this
point a joint.
Letting p + ` = n, let’s bound the number of joints as a
function of n.
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Upper Bound
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
Suppose the lines are held in some set of containing planes
such that each line is contained in exactly one containing
plane.
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Upper Bound
Joints can be made in two ways:
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Upper Bound
Joints can be made in two ways:
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Upper Bound
Joints can be made in two ways:
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Generalizations of
the Joints Problem
Upper Bound
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Determining the
Constant
A Conjecture
Theorem
The General
Problem
3
The number of joints is ≤ kn 2 .
Another
Generalization
Further Research
Acknowledgements
Significance
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Lemma
Given a set of S lines in R3 , there exists a set K ⊂ S with
|K | ≤ 13 |S| such that given any three lines in S intersecting
in a joint, exactly one is in K .
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Significance
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Lemma
Given a set of S lines in R3 , there exists a set K ⊂ S with
|K | ≤ 13 |S| such that given any three lines in S intersecting
in a joint, exactly one is in K .
We can use this lemma to give a new proof of the joints
3
theorem J ≤ kn 2 .
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Further Directions
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
This problem is far from resolved, and there are a few
distinct ways to proceed.
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Further Directions
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
This problem is far from resolved, and there are a few
distinct ways to proceed.
I
Conjecture in
R3
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Further Directions
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
This problem is far from resolved, and there are a few
distinct ways to proceed.
R3
I
Conjecture in
I
Generalization with homogeneous dimension of
intersecting objects
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Further Directions
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
This problem is far from resolved, and there are a few
distinct ways to proceed.
R3
I
Conjecture in
I
Generalization with homogeneous dimension of
intersecting objects
I
Generalization of the idea in R4 , with objects of
different dimensions determining joints
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Acknowledgements
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Thanks to Ben Yang for helping me throughout the research
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Acknowledgements
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Thanks to Ben Yang for helping me throughout the research
Thanks to Dr. Larry Guth for suggesting this problem
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Acknowledgements
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Thanks to Ben Yang for helping me throughout the research
Thanks to Dr. Larry Guth for suggesting this problem
This project could not have been possible without the
PRIMES program itself - a big thank you to Dr. Gerovitch
and Dr. Khovanova
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements
Acknowledgements
Generalizations of
the Joints Problem
Joseph Zurier
, Mentor: Ben
Yang
, Problem by:
Larry Guth
Background
Thanks to Ben Yang for helping me throughout the research
Thanks to Dr. Larry Guth for suggesting this problem
This project could not have been possible without the
PRIMES program itself - a big thank you to Dr. Gerovitch
and Dr. Khovanova
Thank you for listening!
Determining the
Constant
A Conjecture
The General
Problem
Another
Generalization
Further Research
Acknowledgements