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Andreia Chapouto (Monash University)
Ian Marquette (La Trobe University)
Leo Tzou (Melbourne University)
| Time | Speaker/Event |
|---|---|
| 11:00 | Andreia Chapouto |
| 12:00 | Lunch |
| 13:30 | Ian Marquette |
| 14:30 | Afternoon Tea |
| 15:00 | Leo Tzou |
Speaker: Andreia Chapouto
Title: Gauge transform for the Korteweg-de Vries equation and well-posedness below the $H^{-1}$-scale
Abstract: In this talk, we consider the low regularity well-posedness problem for the Korteweg-de Vries equation (KdV) on the real line. Aiming to bridge the regularity gap between the scaling critical space $H^{-\frac32}$ and the known optimal well-posedness in $H^{-1}$ in $L^2$-based Sobolev spaces, we consider rough data in Fourier-Lebesgue spaces. Via infinite normal form reductions and exploiting algebraic cancellations, we introduce a new gauged KdV equation, equivalent to the original one at high regularity, but better behaved for rough solutions, below the $H^{-1}$ scale. Our method is easily adapted to other equations with quadratic derivative nonlinearities, such as the dispersion generalized BO equations, and thus it is totally independent of the KdV completely integrable structure. This talk is based on joint work with Simão Correia (IST, U. Lisboa) and João Pedro Ramos (IMPA).
Speaker: Ian Marquette
Title: The search for higher order symmetries and Hamiltonians involving Painlevé transcendents
Abstract: The six Painlevé transcendents were obtained by Painlevé, Gambier, and Fuchs in the early 1900s. They play an important role in the classification of ordinary differential equations, and they are related to several areas in physics and mathematics, such as the reduction of various nonlinear partial differential equations of mathematical physics, relativity, statistical mechanics, and quantum field theory.
The connection between Painlevé transcendents and the Schrodinger equation is much more recent. This connection was discovered has the search for symmetries of two- and higher dimensional Hamiltonians in quantum mechanics was extended beyond second order. They also appear in context of higher order Darboux transformations and other types of symmetries. This talk will be devoted to present and overview of the Hamiltonians with a potential involving Painlevé transcendents. The talk will focus on the case of superintegrability which leads to systems of PDEs which relates to the Chazy class of differential equations and their reductions to Painlevé transcendents. The talk will highlight certains of their properties and underlying algebraic structures. Examples on Euclidean and Riemannian spaces will be discussed, in particular with the fourth and sixth Painlevé transcendents.
Speaker: Leo Tzou
Title: Geodesic Levy Flight and the Foraging Hypothesis
Abstract: The Lévy Flight Foraging Hypothesis is a widely accepted dogma which asserts that animals using search strategies allowing for long jumps, also known as Lévy flights, have an evolutionary advantage over those animals using a foraging strategy based on continuous random walks modelled by Brownian motion. However, recent discoveries suggest that this popular belief may not be true in some geometric settings. In this talk we will explore some of the recent progress in this direction which combines Riemannian geometry with stochastic analysis to create a new set of properties for diffusion processes.