Emerging Topics

Coherence and lattices

Matthew Stover
Temple University
March 27, 2019

Abstract: I will survey (in)coherence of lattices in semisimple Lie groups, with a view toward open problems and connections with the geometry of locally symmetric spaces. Particular focus will be placed on rank one lattices, where I will discuss connections with reflection groups,  "algebraic" fibrations of lattices, and analogies with classical low-dimensional topology.

 

One-relator groups, non-positive immersions and coherence

Henry Wilton
Cambridge University
March 26, 2019

Abstract: There seems to be an analogy between the classes of fundamental groups of compact 3-manifolds and of one-relator groups.  (Indeed, many 3-manifold groups are also one-relator groups.) For instance, Dehn’s Lemma for 3-manifolds (proved by Papakyriakopoulos) can be seen as analogous to Magnus’ Freiheitssatz for one-relator groups.  But the analogy is still very incomplete, and since there are deep results on each side that have no analogue on the other, there is a strong incentive to flesh it out.

Arnold diffusion and Mather theory

Ke Zhang
University of Toronto
April 11, 2018
Abstract: Arnold diffusion studies the problem of topological instability in nearly integrable Hamiltonian systems. An important contribution was made my John Mather, who announced a result in two and a half degrees of freedom and developed deep theory for its proof. We describe a recent effort to better conceptualize the proof for Arnold diffusion.

Diffusion along chains of normally hyperbolic cylinders

Marian Gidea
Yeshiva University
April 11, 2018
Abstract: We consider a geometric framework that can be applied to prove the existence of drifting orbits in the Arnold diffusion problem. The main geometric objects that we consider are 3-dimensional normally hyperbolic invariant cylinders with boundary, which admit well-defined stable and unstable manifolds. These enable us to define chains of cylinders i.e., finite, ordered families of cylinders in which each cylinder admits homoclinic connections, and any two consecutive cylinders admit heteroclinic connections.