1. Canada Research Chairs In 2000, the Government of Canada created a permanent program to establish 2000 research professorships—Canada Research Chairs—in eligible degree-granting institutions across the country. Communication Guidelines for Chairholders In all professional publications, presentations and conferences, we ask you to identify yourself as a Canada Research Chair and acknowledge the contribution of the program to your research.

3. ORDINARY LINES EXTRAORDINARY LINES?

4. James Joseph Sylvester Prove that it is not possible to arrange any finite number of real points so that a right line through every two of them shall pass through a third, unless they all lie in the same right line. Educational Times, March 1893 Educational Times, May 1893 H.J. Woodall, A.R.C.S. A four-line solution … containing two distinct flaws First correct solution: Tibor Gallai (1933)

5. b 5 points 10 lines 5 points 6 lines 5 points, 5 lines b 5 points, 1 line nothing between these two

6. Every set of n points in the plane determines at least n distinct lines unless all these n points lie on a single line. near-pencil

7. Every set of n points in the plane determines at least n distinct lines unless all these n points lie on a single line. This is a corollary of the Sylvester-Gallai theorem (Erdős 1943): remove this point apply induction hypothesis to the remaining n-1 points

8. On a combinatorial problem, Indag. Math. 10 (1948), 421--423 Combinatorial generalization Nicolaas de Bruijn Paul Erdős Let V be a finite set and let E be a family of of proper subsets of V such that every two distinct points of V belong to precisely one member of E. Then the size of E is at least the size of V. Furthermore, the size of E equals the size of V if and only if E is either a near-pencil or else the family of lines in a projective plane.

9. Every set of n points in the plane determines at least n distinct lines unless all these n points lie on a single line. What other icebergs could this theorem be a tip of?

10. A B C E D dist(A,B) = 1, dist(A,C) = 2, etc.

11. a b x y z a b x y z This can be taken for a definition of a line L(ab) in an arbitrary metric space Observation Line ab consists of all points x such that dist(x,a)+dist(a,b)=dist(x,b), all points y such that dist(a,y)+dist(y,b)=dist(a,b), all points z such that dist(a,b)+dist(b,z)=dist(a,z).

12. Lines in metric spaces can be exotic One line can hide another!

13. a b x y z A B C E D L(AB) = {E,A,B,C} L(AC) = {A,B,C} One line can hide another!

14. Question (Chen and C. 2006): True or false? In every metric space on n points, there are at least n distinct lines or else some line consists of all these n points.

15. Manhattan distance a b x z y lines become a b x y z Manhattan lines

16. Question (Chen and C. 2006): True or false? In every metric space on n points, there are at least n distinct lines or else some line consists of all these n points. Partial answer (Ida Kantor and Balász Patkós 2012 ): Every nondegenerate set of n points in the plane determines at least n distinct Manhattan lines or else one of its Manhattan lines consists of all these n points. “nondegenerate” means “no two points share their x-coordinate or y-coordinate”.

17. a b x z y degenerate Manhattan lines a b x y z a b x y z a a b b x x y y z z a b x z y typical Manhattan lines

18. What if degenerate sets are allowed? Theorem (Ida Kantor and Balász Patkós 2012 ): Every set of n points in the plane determines at least n/37 distinct Manhattan lines or else one of its Manhattan lines consists of all these n points.

19. Question (Chen and C. 2006): True or false? In every metric space on n points, there are at least n distinct lines or else some line consists of all these n points. Another partial answer (C. 2012 ): In every metric space on n points where all distances are 0, 1, or 2, there are at least n distinct lines or else some line consists of all these n points.

20. Another partial answer (easy exercise): In every metric space on n points induced by a connected bipartite graph, some line consists of all these n points. In every metric space on n points induced by a connected chordal graph, there are at least n distinct lines or else some line consists of all these n points. Another partial answer (Laurent Beaudou, Adrian Bondy, Xiaomin Chen, Ehsan Chiniforooshan, Maria Chudnovsky, V.C., Nicolas Fraiman, Yori Zwols 2012): Another partial answer (Pierre Aboulker and Rohan Kapadia 2014): In every metric space on n points induced by a connected distance-hereditary graph, there are at least n distinct lines or else some line consists of all these n points.

21. bipartite not chordal not distance-hereditary chordal not bipartite not distance-hereditary distance-hereditary not bipartite not chordal

22. Theorem (Pierre Aboulker, Xiaomin Chen, Guangda Huzhang, Rohan Kapadia, Cathryn Supko 2014 ): In every metric space on n points, there are at least (1/3)n 1/2 distinct lines or else some line consists of all these n points.