Kazhdan seminar: Ehud Hrushovski and Itay Kaplan - Topics in Model Theory

Date:

NOTE UNUSUAL DAY

Link: Topics in Model Theory (Teams)

0.1. Provisional syllabus. This course is intended to cover three main results.

They neighbour a large number of existing theorems and concepts that there is
no room to detail here, and I will try to learn and explain some connections. See
 and , and references there for adjacent material. The proof of (3) depends
logically on (2), and (2) on (1); but we may cover them in reverse order.

1) The Hausdorff core of a theory.

This is a construction of a compact topological structure J canonically associated
with a given theory T, along with a compact automorphism group G acting
on it. One view way to view G is as the group of self-interpretations of a canonical
minimal expansion of the theory enjoying a structural Ramsey property. Thus
for instance the proof of Ramsey’s original theorem requires the use of a linear
ordering <; it shows that the theory of a structureless infinite set becomes Ramsey
upon adding < to the language. Here G is the two-element group, reflecting
the symmetry broken by choosing < over >. The core and its automorphism group G become locally compact if T is presented locally; this locally compact group is the starting point of the analysis of approximate subgroups in (2).

2) Approximate subgroups.

An approximate subgroup of a group G is a symmetric subset X G, such
that the product set XX is commensurable to X, i.e. is contained in a finite
union of translates of X. Homomorphisms from f from a subgroup H of G into a Lie group L provide one source of approximate subgroups; namely f^{-1}(U) where U is a compact
neighborhood of the identity in L. Here L can be taken to be connected with no
compact normal subgroups, and is then uniquely determined by the corresponding
commensurability class of approximate subgroups.

Quasimorphisms g : H\to R also give rise, by pullback of a bounded set, to
approximate subgroups. These only arise for non-amenable G.
Theorem 5.16 (or 5.19) in  shows that an approximate subgroup of any group
decomposes into the above two special classes, belonging to Lie theory in the first
case and bounded cohomology in the second.

3) Approximate lattices.

An approximate subgroup X of a 2nd countable locally compact group G is
called an approximate lattice if it is discrete (and closed), and has finite covolume
in the sense that there exists a Borel subset B of finite volume in G, with XB = G.
Assume G is an algebraic group over a local field. The classical adelic constrution
The simplest example is the approximate lattice {a/p^n\in Z[1/p] : |a|\leq  p^n} in
(Q_p; +), analogous to the lattice Z in R; both can be described as the set of  rationals in a completion of Q, whose norm in every other completion is bounded by 1. Theorem 8.4 of  shows that when G is semisimple, this arithmetic construction is the only additional source of approximate lattices. For instance any number field contained in R gives rise to an approximate lattice in SL_2(R), and all approximate lattices of SL_2(R) are either commensurable to one of these or
to a lattice.

References

 arXiv:2011.12009
 arXiv:1911.01129

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Dear all,

Now the text, handwritten notes and video are all available here, and will be updated:

http://people.maths.ox.ac.uk/hrushovski/autumn2021.html

The video at the moment is only of lecture 2, thanks to Arturo, but I think this will be updated soon too.