Three 20 minute research talks

Three 20 minute research talks

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Yu-Wei Fan, UC, Berkeley, Surena Hozoori, Georgia Tech, Marcelo Atallah, University of Montreal

Zoom link:  :  https://umontreal.zoom.us/j/94366166514?pwd=OHBWcGluUmJwMFJyd2IwS1ROZ0FJdz09    

Yu-Wei Fan, UC Berkeley

Shifting numbers in triangulated categories

One can consider endofunctors of triangulated categories as categorical dynamical systems, and study their long term behaviors under large iterations. There are (at least) three natural invariants that one can associate to endofunctors from the dynamical perspective: Categorical entropy, and upper/lower shifting numbers. We will recall some background on categorical dynamical systems and categorical entropy, and introduce the notion of shifting numbers, which measure the asymptotic amount by which an endofunctor of a triangulated category translates inside the category. The shifting numbers are analogous to Poincare translation numbers. We additionally establish that in some examples the shifting numbers provide a quasimorphism on the group of autoequivalences. Joint work with Simion Filip.

Surena Hozoori, Georgia Tech

Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology

We give a purely contact and symplectic geometric characterization of Anosov flows in dimension 3 and set up a framework to use tools from contact and symplectic geometry and topology in the study of questions about Anosov dynamics. If time permits, we will discuss some uniqueness results for the underlying (bi-) contact structure for an Anosov flow, and/or a characterization of Anosovity based on Reeb flows.

 

Marcelo Atallah, University of Montreal

Hamiltonian no-torsion

In 2002 Polterovich notably showed that Hamiltonian diffeomorphisms of finite order, which we call Hamiltonian torsion, must be trivial on closed symplectically aspherical manifolds. We study the existence of Hamiltonian torsion and its metric rigidity properties in more general situations. First, we extend Polterovich's result to closed symplectically Calabi-Yau and closed negative monotone manifolds. Second, going beyond topological constraints, we describe how Hamiltonian torsion is related to the existence of pseudo-holomorphic spheres and answer a close variant of Problem 24 from the introductory monograph of McDuff-Salamon. Finally, we prove an analogue of Newman’s 1931 theorem for Hofer’s metric and Viterbo’s spectral metric on the Hamiltonian group of monotone symplecitc manifolds: a sufficiently small ball around the identity contains no torsion. During the talk, I shall discuss the results above and some of the key ingredients of their proofs. This talk is based on joint work with Egor Shelukhin.