

Week 3: PhysicsPhysical aspects (June 13  17)Mind the boatDeparture from La Rochelle sunday (june 12) at 7pm, no boat on monday! ScheduleAfternoons are devoted to talks and free discussions.You can already propose a talk by email. LecturesAnuradha Jagannathan: Quantum particles on tilingsIn these lectures we will consider electrons and bosons in a quasiperiodic environment, and discuss some physical phenomena due to the highly organized spatial structure:
Renaud Leplaideur: Freezing phase transitions with support in quasicrystals Notes SlidesThe goal of this minicourse is to present new developments in ergodic theory and thermodynamic formalism. Thermodynamic formalism has been introduced into the setting of dynamical systems and ergodic theory in the 70's by Sinai, Ruelle and Bowen. It usually deals with onedimensional lattice, and dynamics is usually seen as the action of the time. The main point is that people in ergodic theory use vocabulary from Physics but it indeed turns out that the notions behind are sensibly different. In the first lecture we will explain how the thermodynamic formalism from ergodic viewpoint is related to the CurryWeiss viewpoint for Ising model or Potts model (also called meanfield interactions). We shall give a kind of dictionary that translates one notion in one setting into the other setting. The second and third lectures are devoted to recent works concerning freezing phase transitions. We will explain how we can construct examples with freezing phase transitions reaching a quasicrystal state at positive temperature. We will first recall the definitions of the objects and introduce the main tool which is the RuellePerronFrobenius operator for inducing schemes. We will then prove that a freezing phase transition occurs. For that, we shall explore how orbits (in the shift) may approach the quasicrystal. Ron Lifshitz: Symmetry, excitations, defects, and thermodynamic stability of quasicrystalsThe discovery of quasicrystals has changed our view of some of the most basic notions related to the condensed state of matter, leading to a new underlying paradigm for the notion of a crystal as a solid with longrange positional order. This paradigm shift brought about a new fundamental understanding of the basic characteristics of a crystal—its symmetry, its elementary collective excitations, and its topological defects. Before we knew about quasicrystals, it was believed that crystals break t he continuous translation and rotation symmetries of the liquidphase into a discrete lattice of translations, and a finite point group of rotations. Quasicrystals, on the other hand, possess no such symmetries—there are no translations, nor, in general, are there any rotations, leaving them invariant. Does this imply that no symmetry is left, or that the meaning of symmetry in crystallography should be revised? Without an underlying lattice, can we even describe the symmetrybreaking transition that occurs upon solidification? Can we explain the thermodynamic stability of the solid phase that is formed and possibly control its selfassembly? Can we describe the elementary excitations? Can we talk about dislocations without a lattice? If so, how can we characterize them? Afternoon talks
