Seminar on metric and measurable limits in model theory, Spring 2011.
Tuesdays, 18:00-19:45 at 209 Einstein.
NOTE CHANGE OF ROOM!
Our first goal will be a model-theoretic view of
Note: On Tuesday, March 29 class will end at 18:45; we will reschedule the hour
Gromov's asymptotic cone.
Given a group $G$ with a finite set of generators, define the distance of $g$ to $h$ to be the length of
the shortest word $w$ in these generators, such that $gw=h$. This makes $G$ into a (discrete) metric space. Gromov defines a way to "look at the group from afar" (depending on an ultrafilter $D$)
and see another, connected, metric space, the asymptotic cone. For a similar cone on Lie groups, he write:
"it seems that these groups G have pretty looking finite dimensional cones $Con_D(G)$ which are (essentially) independent of the choice of the ultrafilter D.". It turns out that
model theory can aptly describe the cone, give a precise meaning to "essentially", and prove the statement. The same method also prove a rigidity conjecture of Margulis, proved by Kleiner-Leeb: for a
general class of homogeneous metric spaces, quasi-isometries are close to actual isometries.
We will follow a text by Kramer-Tent, complementing it when necessary. See:
[Kramer-Tent 02] http://front.math.ucdavis.edu/0209.5122
[Kramer-Tent 04] http://front.math.ucdavis.edu/0311.5101
The proofs use basic ideas from a number of subjects usually covered in basic notions courses, including commutative algebra (valuations), algebraic topology (a little homology), differential geometry, and LIe groups,
but all mostly at the level of basic definitions only. We will not assume knowledge of these subjects, and will develop them as needed. The only area where really detailed knowledge is needed is of Lie groups;
here we will try to understand the statements, but will give proofs only for $SL_n(\Rr)$ where they are usually accessible.
I expect the talks will take half the session, and the second half will be used for filling in gaps and background and discussions.
arXiv:1002.4456 A Model Theoretic Proof of Szemerédi's Theorem. Henry Towsner.
Measures under NIP
Baldwin, John; Benedikt, Michael Stability theory, permutations of indiscernibles, and embedded finite models. Trans. Amer. Math. Soc. 352 (2000), no. 11, 4937–4969
Model theory prerequisites: this part of the seminar at least requires little previous knowledge of model theory; a clear understanding of the completeness
and compactness theorems of Logic 1 will suffice, if you are willing to fill in learn a few basic facts about saturation yourself.
Some class notes by Elad / Assaf: