Time and Place: Thursday November 15th, 14:30-15:30, Mathematics
Building, Lecture Hall 2
Speaker:  Micha Sharir (Tel Aviv University)
Title: From joints to distinct distances and beyond: The dawn of an
algebraic era in combinatorial geometry
In November 2010 the earth has shaken, when Larry Guth and Nets Hawk
Katz posted a nearly complete solution to the distinct distances
problem of Erd{\H o}s, open since 1946. The excitement was twofold:
(a) The problem was one of the most famous problems, as well as one
of the hardest nuts in the area, resisiting solution in spite of many
attempts (which only produced partial improvements). (b) The proof
techniques were algebraic in nature, drastically different from
anything tried before.
The distinct distances problem is to show that any set of n points in
the plane determine Omega(n/\sqrt{\log n}) distinct distances.
(Erd{\H o}s showed that the grid attains this bound.) Guth and Katz
obtained the lower bound Omega(n/\log n).
Algebraic techniques of this nature were introduced into
combinatorial geometry in 2008, by the same pair Guth and Katz. At
that time they gave a complete solution to another (less major)
problem, the so-called joints problem, posed by myself and others
back in 1992. Since then these techniques have led to several other
developments, including an attempt, by Elekes and myself, to reduce
the distinct distances problem to an incidence problem between points
and lines in 3-space. Guth and Katz used this reduction and gave a
complete solution to the reduced problem.
One of the old-new tools that Guth and Katz bring to bear is the
Polynomial Ham Sandwich Cut, due to Stone and Tukey (1942). I will
discuss this tool, including a ``1-line'' proof thereof, and its
applications in geometry, as they are slowly emerging during the past
year and a half.
In the talk I will review all these developments, as time will
permit. Only very elementary background in algebra and geometry will
be assumed.
Light refreshments will be served after the colloquium in the faculty
lounge at 15:30.
Technion Math Net-2 (TECHMATH2)
Editor: Michael Cwikel   <techm@math.technion.ac.il> 
Announcement from: Yulia   <yulia_g@math.huji.ac.il>