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Washington State University
Combinatorics, Linear Algebra and Number Theory (CLaN) Seminar


Department of Mathematics
Neill Hall 5W
October 26, Monday, 4:10 - 5:00 PM


 

  Stefan Tohaneanu

Department of Mathematics
University of Idaho


Title: Three classical problems on configuration of points in the real plane

Abstract: Most of the talk is based on joint work with UI undergraduate student Ben Anzis. I will be talking about some classical problems concerning configuration of points in the real plane: Sylvester-Gallai theorem, Dirac-Motzkin conjecture (since 2013 a Theorem, due to Green and Tao), and as a bonus, the Slope problem. In the classical projective duality, the first two problems translate into a lower bound on the number of simple singularities of the line arrangement dual to the configuration of points: Sylvester-Gallai theorem is saying that if the points are not collinear, then the dual line arrangement has at least one simple singularity; Dirac-Motzkin conjecture says that if the number of points is really large (and the points still not collinear), then the number of simple singularities of the dual line arrangement is at least half the number of points. For supersolvable line arrangements, using a different, much simpler proof, we show that the Dirac-Motzkin conjecture is true with no asymptotic condition on the number of points. Over the complex numbers, the original Dirac-Motzkin conjecture is not true (Hesse configuration of points gives a counterexample), yet we conjecture that for the case of supersolvable line arrangements the bound is still true. Another interesting problem about configuration points in the real plane is the Slope Conjecture which states that any n non-collinear points determine at least n-1 different slopes. This conjecture was first proved by Ungar in 1982. We show that this conjecture (theorem) is equivalent to a conjecture we made about real supersolvable line arrangements that states that the maximum multiplicity of a singularity is bounded below by (n-1)/2.