Morning lectures

• Revivals and fractalisation
  
  Lyonell Boulton (Heriot-Watt University, Edinburgh, UK)

  We will begin the course by considering the one-dimensional free Schrödinger equation with periodic boundary conditions \[ \left\{ \begin{aligned} &\partial_t u(t,x) - i \partial_{xx} u(t,x) = 0 \\ &u(t,0) = u(t,1)\\ &\partial_x u(t,0) = \partial_{x} u(t,1) \\ &u(0,x) = u_0(x)\\ \end{aligned} \right. \] for \(x\in\mathbb R\) and \(x\in [0,1]\). The solution can be found easily by separation of variables. this is one or the simplest partial differential equations, yet only about 30 years ago, Micheal Berry [1] and collaborators [2] discovered a remarkably complex behaviour which they named after the Victorian scientist William Henry Fox Talbot [3].
  The "Talbot effect" manifests in the system above through a striking pattern of rough "periodicity" in time for simple initial data \(u_0(x)\). For example, if \(u_0(x)\) is a step function, then

  • for \(t\) "rational", \(u(t,x)\) is a finite linear combination of certain copies of \(u_0(x)\), but
  • for \(t\) "irrational", \(u(t,x)\) becomes a rough continuous function of \(x\).

  We call the first phenomenon revival and the second fractalisation.

  During the course, we will examine how the classical tools of Fourier analysis and the analysis of PDEs can be used to show striking results abour revivals and fractalisation for a variety of time-evolution boundary problems. The sessions will be devoted to studying five dispersive equations (of first order in time) that support the revival/fractalisation dichotomy, either partially or coompletely. For this, we will cover subjects such as generalised Fourier expansions, spectral analysis of differential operators, the notation of fractal dimensions and perturbation methods for PDEs.

  References and further reading.
  
  [1] M. V. Berry, J. Phys. A: Math. Gen. 29 (1996) 6617.
  [2] M. V. Berry and S. Klein, J. Mod. Opt. 43 (1996) 2139-2164.
  [3] H. F. Talbot, Phil. Mag. 9 (1836) 401-407.
  [4] M. B. Erdoğan and N. Tzirakis, Dispersive Partial Differential Equations: Well-posedness and Applications. Cambridge University Press (2016).
  [5] I. Rodnianski, Contemp. Math. 255 (2000) 181-188.
  [6] L. Boulton, G. Farmakis and B. Pelloni, Proc. R. Soc. A 477 (2021) 202110241.
  [7] V. Chousionis, M. B. Erdoğan and N. Tzirakis Proc. Lond. Math. Soc. 110 (2014) 543-564.
  [8] L. Boulton, G. Farmakis and B. Pelloni, arXiv:2308.09961 (2023).
  [9] L. Boulton, P. Olver, B. Pelloni and D. Smith, Stud. Appl. Math. 147 (2021) 1209-1239.

• Geometrical aspects of spectral theory
  
  David Krejčiřik (Czech Technical University in Prague, Czechia)

  Spectral theory is an extremely rich field which has found its application in many areas of physics and mathematics. One of the reason which makes it so attractive on the formal level is that it provides a unifying framework for problems in various branches of mathematics, for example partial differential equations, calculus of variations, geometry, stochastic analysis, etc.

  The goal of the lecture is to acquaint the students with spectral methods in the theory of linear differential operators coming both from modern as well as classical physics, with a special emphasis put on geometrically induced spectral properties. We give an overview of both classical results and recent developments in the field, and we wish to always do it by providing a physical interpretation of the mathematical theorems.