Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Thursday, June 11, 2009, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Micha Sharir (Tel Aviv Univ., Israel)

Sharing joints, in moderation: A groundshaking clash between algebraic and combinatorial geometry

About half a year ago, Larry Guth and Nets Hawk Katz have obtained the tight upper bound $O(n^{3/2})$ on the number of joints in a set of $n$ lines in 3-space, where a joint is a point incident to at least three non-coplanar lines, thus closing the lid on a problem that has been open for nearly 20 years. While this in itself is a significant development, the groundbreaking nature of their work is the proof technique, which uses fairly simple tools from algebraic geometry, a totally new approach to combinatorial problems of this kind in discrete geometry.

In this talk I will (not have enough time to) present a simplified version of the new machinery, and the further results that we have so far obtained, by adapting and exploiting the algebraic machinery.

The first main new result is: Given a set $L$ of $n$ lines in space, and a subset of $m$ joints of $L$, the number of incidences between these joints and the lines of $L$ is $O(m^{1/3}n)$, which is worst-case tight for $m\ge n$. In fact, this holds for any sets of $m$ points and $n$ lines, provided that each point is incident to at least three lines, and no plane contains more than $O(n)$ points.

The second set of results is strongly related to the celebrated problem of Erdős on distinct distances in the plane. We reduce this problem to a problem involving incidences between points and helices (or parabolas) in 3-space, and formulate some conjectures concerning the incidence bound. Settling these conjectures in the affirmative would have almost solved Erdős's problem. So far we have several partial positive related results, interesting in their own right, which yield, among other results, that the number of distinct (mutually non-congruent) triangles determined by $s$ points in the plane is always $\Omega(s^2 / \log s)$, which is almost tight in the worst case, since the integer lattice yields an upper bound of $O(s^2)$.

Joint work with Haim Kaplan and (the late) György Elekes.

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