Activities This Week

A geometric transition is a continuous path of geometries which abruptly changes type in the limit. The most intuitive example is to imagine blowing up a sphere so that eventually it becomes so large, it looks like a plane. This is a transition from spherical geometry to Euclidean geometry.

We will study limits of the Cartan subgroup in $SL(n,R)$. A limit group is the limit under a sequence of conjugations of the Cartan subgroup in $SL(n,R)$. We will show using the hyperreal numbers that in $SL(3,R)$ there are 5 limit groups, each determined by a degenerate triangle.

In the second part of the talk, we will show that for $n \geq 7$, there are infinitely many nonconjugate limit groups of the Cartan subgroup.

Chung and Li [Invent. Math. 2015] proved that for every expansive action of a countable polycyclic-by-finite group $\Gamma$ on a compact group $X$ by continuous group automorphisms, positive entropy implies the existence of non-diagonal asymptotic pairs. In the same paper they asked if the this holds in general for an expansive action of a countable amenable group $\Gamma$ on a compact space $X$.

In my talk I plan to explain the notions involved Chung and Li’s question and discuss a property of dynamical systems called the ``pseudo-orbit tracing property’’. R. Bowen introduced the pseudo-orbit tracing property in the 1970’s for $\mathbb{Z}$-actions while studying Axiom A maps. I will prove that Chung and Li’s question has an affirmative answer if one also assumes pseudo-orbit tracing, and explain implications for algebraic actions (automorphisms of compact abelian groups).

I will also explain why the answer to Chung and Li’s question is negative if one doesn’t assume the pseudo-orbit tracing property, even when the acting group is $\mathbb{Z}$, or when the action is algebraic (but not both).

A linearly ordered structure is weakly o-minimal if every definable set is a finite boolean combination of convex sets. A weakly o-minimal expansion of an ordered group is non-valuational if it admits no non-trivial definable convex sub-groups. By a theorem of Baizalov-Poizat if M is an o-minimal expansion of a group and N is a dense elementary substructure then the structure induced on N by all M-definable sets is weakly o-minimal non-valuational.

It is natural to ask whether all non-valuational structures are obtained in this way. We will give examples showing that this is not the case. We will show, however, that if M is non-valuational then there exists M^, an o-minimal structure embedding M densely (as an ordered set) such that M (as a pure set) extended by all M^-definable sets is precisely the structrue M. We will give a complete axiomatisation of the theory of the pair (M^,M), show that it depends only on the theory of M, and that it shares many common features with the theory of dense o-minimal pairs. In particular (M^,M) has dense open core (i.e., the reduct consisting only of definable open sets is o-minimal).

Based on joint work with E. Bar-Yehuda and Y. Peterzil.

A Ramanujan graph is a finite graph which behaves, in terms of expansion, like its universal cover (which is an infinite tree). In recent years a parallel theory has emerged for simplicial complexes of higher dimension, where the role of the tree is taken by Bruhat-Tits buildings. I will recall briefly the story of Ramanujan graphs, and then explain what are “Ramanujan complexes”, and survey some of the new results regarding their construction and their properties.