Nugget sessions are short 15 minute presentations, with a large (30 minute) discussion time and a strict 5-slide limit!
Random Open Dynamical Systems and Extreme Value Theory
Route to thermalization of the diatomic FPUT lattice
Open Problems in Magnetism and Billiards
Robby Marangell
Nuggets
Nugget sessions are short 15 minute presentations, with a large (30 minute) discussion time and a strict 5-slide limit!
Tue Session 1 #1
Geoff Vasil + Sidney Holden
Differential equations on large graphs
A "metric graph" is a collection of edges and vertices, but with the concept of distance between points along each segment. It is possible to define systems of differential equations along the edges and pose boundary/continuity conditions at the nodes. I will discuss ongoing work with Sid Holden regarding the behaviour of differential equations on asymptotically large graphs. In the large-graph limit, the systems resemble a hybrid between one-dimensional networks and continuous Sort-Of-Riemannian(ish) manifolds embedded in higher-dimensional space. We will discuss several puzzling observations and state a few conjectures (speculations). For example, does a dense spider web vibrate like a drum? It turns out, sort of, and sort of not; we'll explain. We are keen to consider new ideas both from a purely mathematical point of view, as well as possible novel applications.
Tue Session 1 #2
Jason Atnip
Random Open Dynamical Systems and Extreme Value Theory
An open dynamical system consists of a dynamical system and a positive measure hole in the phase space through which mass gradually escapes. In this setting one is typically interested in the rate at which mass escapes through the hole. In this talk we will discuss recent joint results with Gary Froyland, Cecilia Gonzalez-Tokman, and Sandro Vaienti in which we consider a system of random interval maps together with a random hole which depends on the random fibers. We show that for such systems there exists a unique absolutely continuous conditionally invariant measure which satisfies an exponential decay of correlations. We then turn to an application to extreme value theory, where, with the help of a recent perturbation result of Crimmins, we prove an extreme value law for similar systems.
Tue Session 2 #1
Sasha Fish
Glasner property for (semi-)group actions
It was noticed by Eli Glasner in the end of 70's that the action by multiplication of the natural numbers on the one-dimensional torus satisfies the following property: For any eps > 0 and for any infinite set A in the torus there exists n such that the set nA is eps-dense in the torus. The action of a (semi-)group G on a compact metric space X is Glasner, if for any eps > 0 and any infinite set A in X there exists g in G such that gA is eps-dense. The Glasner property is stronger than topological mixing, and intimately related to arithmetic properties of underlying dynamics. We will give an overview of state of the art of the subject and will discuss some open problems.
Tue Session 2 #2
Lauren Smith
Mesoscopic reduction of sparse oscillator networks
Synchronisation is ubiquitous to networks of coupled oscillators. In coupled oscillator networks with sparse coupling the lowest degree nodes are typically the first to desynchronise, due to their weak connection with the rest of the oscillators. Here we propose a reduced mesoscopic model based on collective coordinate reduction, which encapsulates both the microscopic dynamics of the vulnerable low degree nodes and the macroscopic dynamics of the synchronised cluster. This reduction yields properties such as the critical coupling strength for the onset of global synchronisation and depends only on the statistical properties of the full sparse network.
Wed Session 3 #1
Holger Dullin
How to fly a satellite around a space station at L1
Lagrange points are relative equilibria in the Sun-Earth-SpaceStation Newtonian gravitational problem in a rotating frame. In the 3-body problem there are 3 collinear Lagrange points called L1, L2, L3. I will review basic results and open problems concerning relative equilibria in the N-body problem. One particular case that is well understood occurs when the masses of the bodies are vastly different, as, e.g., in the case of Sun, Earth, space-station, satellite. The original Hill's problem is the description of the motion of the Moon in the field of Earth and Sun. The description of the tightly bound orbits of the satellite around the space station leads to a new type of Hill's problem which I will describe. This is joint work with Chamal Perera.
Wed Session 3 #2
Robby Marangell
The Maslov index for Non-Hamiltonian systems
This talk is primarily showing you a new hammer, and looking for some nails to hit with it. In a recent paper, we extended classical Sturm-Liouville theory to systems of ODEs. Our construction builds on the earlier work of Arnol'd from the 80's extending Sturm-Liouville theory to Hamiltonian systems via the Maslov index. We have applied our technique to some examples, but are looking for more. In principle, our technique should be applicable to 1) any system where one is interested in real eigenvalues, and 2) any other use of the Maslov index that one might want to extend.
Wed Session 4 #1
Gary Froyland
Dynamics, metrics, and spectra.
This nugget may discuss and pose some problems related to dynamics, metrics, and spectra.
Wed Session 4 #2
Guo Deng
Route to thermalization of the diatomic FPUT lattice
We study the diatomic Fermi–Pasta–Ulam-Tsingou (FPUT) system. The approach is based on resonant wave–wave interaction theory; i.e., we assume that the large time dynamics is ruled by exact resonances. After a detailed analysis of the dispersion relation, we show that there exists nontrivial resonance between the optical and acoustical modes for certain mass ratio. And we show that at resonant mass ratio the dynamics of the system is significantly different from that at non-resonant mass ratio.
Thur Session 5 #1
Georg Gottwald
This nugget poses the question if random feature maps may be good candidates for dictionaries in EDMD. We give a brief overview on both, EDMD and random feature maps.
Extended dynamic mode decomposition (EDMD) and random feature maps
Thur Session 5 #2
Sean Gasiorek
Open Problems in Magnetism and Billiards
After an initial appearance in the 1980/90's, magnetic billiards has seen a revival over the last 10 years in the context of algebraic integrability. Recent work largely lead by Bialy, Mironov, and Glutsyuk proves an algebraic integrability conjecture for magnetic billiards in the plane and on surfaces of constant curvature. It is unknown if this applies to other versions of magnetic billiards. We will also continue a common practice in the study of billiards by comparing and contrasting known (and unknown) results from standard Birkhoff billiards to its magnetic counterparts: some concepts and ideas are consistent while others are not.
Thur Session 6
Martin Wechselberger
Hidden Timescales
We shall discuss a coordinate-independent method for multiple time-scales problems that simultaneously computes parametrisations of invariant manifolds and the dynamics on them. It does so by iteratively solving a so-called conjugacy equation, yielding approximations with arbitrarily high degrees of accuracy. We highlight the emergence of hidden timescales and show how this method can uncover these surprising multiple timescale structures.