- Series
- School of Mathematics Colloquium
- Time
- Friday, December 5, 2014 - 4:00pm for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Danny Calegari – University of Chicago
- Organizer
- John Etnyre

**Please Note:** Kick-off of the Tech Topology Conference, December 5-7, 2014

In 1985, Barnsley and Harrington defined a "Mandelbrot Set" M
for pairs of similarities -- this is the set of complex numbers z
with norm less than 1 for which the limit set of the semigroup
generated by the similarities x -> zx and x -> z(x-1)+1 is
connected. Equivalently, M is the closure of the set of roots of
polynomials with coefficients in {-1,0,1}. Barnsley and Harrington
already noted the (numerically apparent) existence of infinitely
many small "holes" in M, and conjectured that these holes were
genuine. These holes are very interesting, since they are "exotic"
components of the space of (2 generator) Schottky semigroups. The
existence of at least one hole was rigorously confirmed by Bandt in
2002, but his methods were not strong enough to show the existence
of infinitely many holes; one difficulty with his approach was that
he was not able to understand the interior points of M, and on the
basis of numerical evidence he conjectured that the interior points
are dense away from the real axis. We introduce the technique of
traps to construct and certify interior points of M, and use
them to prove Bandt's Conjecture. Furthermore, our techniques let
us certify the existence of infinitely many holes in M. This is
joint work with Sarah Koch and Alden Walker.