Type: Seminar

Modified: Tue 30 Apr 2019 1309

Distribution: World

Expiry: 29 Apr 2020

CalTitle1: Day of Dynamics

Auth: leo@121.212.119.29 (ltzo2369) in SMS-WASM

DAY OF DYNAMICS Thanks to the International Visitor Program we are getting an influx of diverse expertise in the field of dynamical systems this month. You are all invited on Mon Feb 18th for a day of dynamical systems talks from both the pure and applied side on some of the emerging topics in the respective areas. This is an opportunity to foster interdisciplinary collaboration especially amongst early-career researchers and students. Time: Mon Feb 18th 10:30 - 5:15 Place: USYD Carslaw 373 Speakers: Nalini Joshi (USYD) Colin Guillarmou (CNRS, Paris Sud) Marco Mazzucchelli (CNRS, ENS Lyon) Nobutaka Nakazono (Aoyama Gakuin University) Milena Rodnovic (USYD) Alexander Fish (USYD) Giorgio Gubbiotti (USYD) Program: 10:30-11:15 Marco Mazzucchelli (CNRS, ENS Lyon) Min-Max Characterization of Zoll Riemannian Manifolds. 11:15 - 12:00 Milena Radnovic (USYD) Billiards within quadrics and extremal polynomials 12:00-1:00 LUNCH 1:00 - 1:45 Nalini Joshi (USYD) Dynamics of a PainlevÃ© equation in initial-value space 1:50-2:35 Colin Guillarmou (CNRS, Paris Sud) The marked length spectrum in negative curvature BREAK 2:50- 3:35 Alexander Fish (USYD) Twisted recurrence via polynomial walks 3:40- 4:25 Nobutaka Nakazono (Aoyama Gakuin University) Affine Weyl group symmetry of the discrete power function 4:30 - 5:15 Giorgio Gubbiotti (USYD) Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations Abstracts (ordered according to schedule): Marco Mazzucchelli Min-Max Characterization of Zoll Riemannian Manifolds A closed Riemannian manifold is called Zoll when its unit-speed geodesics are all periodic with the same minimal period. This class of manifolds has been thoroughly studied since the seminal work of Zoll, Bott, Samelson, Berger, and many other authors. It is conjectured that, on certain closed manifolds, a Riemannian metric is Zoll if and only if its unit-speed periodic geodesics all have the same minimal period. In this talk, I will first discuss the proof of this conjecture for the 2-sphere, which builds on the work of Lusternik and Schnirelmann. I will then show an analogous result for certain higher dimensional closed manifolds, including spheres, complex and quaternionic projective spaces: a Riemannian manifold is Zoll if and only if two suitable min-max values in a free loop space coincide. This is based on joint work with Stefan Suhr. Milena Radnovic Billiards within quadrics and extremal polynomials We will present a comprehensive study of periodic trajectories of the billiards within ellipsoids in d-dimensional space, based on a relationship established between such trajectories and the extremal polynomials on a systems of d intervals on the real line. We will also give a case study of trajectories of small periods T, d â‰¤ T â‰¤ 2d. Nalini Joshi Dynamics of a PainlevÃ© equation in initial-value space PainlevÃ© equations are nonlinear nonautonomous second-order ODEs, which appear as universal models in physics. Very little explicit information is known about their transcendental solutions. We describe a dynamical approach to deduce their global behaviors in a compactified, regularized space of initial values, which share geometric properties with elliptic surfaces. Such a space was first described by Okamoto (1979). We consider a construction suitable for an asymptotic limit, and prove properties about the complex limit sets of solutions. Joint work with Duistermaat (2011), Howes (2014) and Radnovic (2015-19). Colin Gullarmou The marked length spectrum in negative curvature We review the problem asked by Burns and Katok about determining a Riemannian metric on a closed negative manifold from the length of its closed geodesics. Alexander Fish Twisted recurrence via polynomial walks We will show how polynomial walks can be used to establish a twisted recurrence for sets of positive density in Z^d. In particular, we will demonstrate that if Î“ â‰¤ GL_d(Z) is finitely generated by unipotents and acts irreducibly on R^d, then for any set B âŠ‚ Z^d of positive density, there exists k â‰¥ 1 such that for any v âˆˆ kZ^d one can find Î³ âˆˆ Î“ with Î³v âˆˆ B âˆ’ B. Also we will show a non-linear analog of Bogolubovâ€™s theorem â€“ for any set B âŠ‚ Z^2 of positive density, and p(n) âˆˆ Z[n], p(0) = 0, deg p â‰¥ 2, there exists k â‰¥ 1 such that kZ âŠ‚ {x + p(y) | (x, y) âˆˆ B âˆ’ B}. Joint work with Kamil Bulinski. Nobutaka Nakazono Affine Weyl group symmetry of the discrete power function In this talk, we show that the discrete power function associated with circle patterns of Schramm type can be obtained from a space-filling cubic lattice, each cube has CAC property, and its affine Weyl group symmetry. Moreover, we show that this cubic lattice and its symmetry are derived form the affine Weyl group symmetry of the sixth Painlev\’e equation. This work has been done in collaboration with Profs Nalini Joshi, Kenji Kajiwara, Tetsu Masuda and Dr Yang Shi. Giorgio Gubbiotti Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations We construct the autonomous quad-equations which admit as symmetries the five-point differential-difference equations belonging to known lists found by Garifullin, Yamilov and Levi. The obtained equations are classified up to autonomous point transformations and some simple non-autonomous transformations. We discuss our results in the framework of the known literature. There are among them a few new examples of both sine-Gordon and Liouville type equations.

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