Date:

NOTE UNUSUAL DAY

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

[1] and [2], 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 [1] 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

of arithmetic lattices in G adapts to giving additional approximate lattices.

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 [1] 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

[1] arXiv:2011.12009

[2] arXiv:1911.01129

--------------------------------------------------------------------------------------------------------------------------------

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.

**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

[1] and [2], 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 [1] 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

of arithmetic lattices in G adapts to giving additional approximate lattices.

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 [1] 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

[1] arXiv:2011.12009

[2] arXiv:1911.01129

--------------------------------------------------------------------------------------------------------------------------------

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.