The seminar takes place every Thursday from 13:00 to 14:30 at AA-5340.
Organizers:
Changjie Chen, email: changjie.chen@umontreal.ca
Jérémy Perazzelli, email: jeremy.perazzelli@umontreal.ca
November 6: Connor Sell (UQAM)
Title: Cusp cross-sections of arithmetic hyperbolic manifolds
Abstract: The ends of finite-volume hyperbolic (n+1)-manifolds are cusps of the form B x R+ for some compact, flat n-manifold B. In 2009, McReynolds built on work on Long and Reid to prove that every flat n-manifold arises as the cusp cross section of some hyperbolic (n+1)-manifold using an arithmetic construction. A natural further question to ask is under what conditions each cross-section can arise. In this talk, we discuss a condition that describes exactly when a given flat manifold arises as a cusp cross-section in a commensurability class of cusped, arithmetic, hyperbolic manifolds. This is joint work with Duncan McCoy.
November 20: Dustin Connery-Grigg (IMJ-PRG, Sorbonne Université)
Title: The geometry and topology of Hamiltonian Floer theory on surfaces
Abstract: Given a Hamiltonian dynamical system on a symplectic manifold, what is the relationship between the dynamical features which the system exhibits, and the topology of the underlying manifold? In 1989, Andreas Floer introduced a way to do relative Morse theory with the Hamiltonian action functional, and thereby answered a long-standing conjecture due to Arnol’d on relating the number of necessary fixed points to the underlying topology. Floer theory has since become ubiquitous in modern symplectic topology. Unfortunately, it is generally very difficult to understand how the Floer theory of a given Hamiltonian relates to the system’s broader qualitative dynamics. In this talk, I will give a brief introduction to Floer’s theory, and discuss some results in low-dimensions which provide links between the qualitative dynamics of low-dimensional Hamiltonian systems and their associated Floer theory.
November 27: Jean Pierre Mutanguha (McGill)
Title: Canonical decompositions of free-by-cyclic groups
Abstract: Free-by-cyclic groups can be defined as mapping tori of free group automorphisms. I will discuss a dynamical decomposition of automorphisms that produces a canonical decomposition of the corresponding free-by-cyclic groups. This will involve the partial order on attracting laminations for an automorphism. The decomposition of free-by-cyclic groups can be considered an extension of the JSJ decomposition of 3-manifolds. This talk is on joint work with Spencer Dowdall, Yassine Geurch, Radhika Gupta, and Caglar Uyanik.
December 4: Christopher Karpinski (McGill)
Title: The conjugacy problem for Dehn twist automorphisms of groups
Abstract: A Dehn twist automorphism of an arbitrary group is an algebraic generalization of a Dehn twist homeomorphism of a surface, defined in terms of a graph-of-groups decomposition of the group. Cohen and Lustig developed an algorithm to solve the conjugacy problem for Dehn twists of non-abelian free groups. We generalize the results of Cohen-Lustig to free products of finitely generated free abelian groups. Our main tool is the construction of a canonical (acylindrical) graph-of-groups decomposition of the mapping torus of the Dehn twist automorphism. This is joint work with Bratati Som and Amir Weiss.