Maxwell Institute Mini-symposium
in Analysis and PDEs
31 March, 2023
Maxwell Institute
for Mathematical Sciences
Edinburgh, UK
| 14:00 - 14:50 |
Nicolas Burq (Université Paris-Saclay, France)
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Title: |
Propagation of Smallness, Control and Stabilisation | |
| Abstract: |
Control and stabilization are now well understood when controls or damping act on open sets. When they act on measurable sets only, much remains to be understood. In this talk I will present results in this direction. In most cases (but not always) the results I will present will rely on the quantitative results of Logunov/Malinnikova for harmonic functions that can be translated into control or stabilisation results for heat, waves or Schrodinger equations. Most of the results presented in the talk are in collaboration with I. Moyano (Univ. Nice).
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| 15:00 - 15:50 | Mathieu Lewin (Université Paris Dauphine, France) | |
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Title: |
Derivation of nonlinear Gibbs measures from many-body quantum mechanics | |
| Abstract: |
In this talk, I will define and discuss some probability measures in infinite dimensions, which play an important role in (S)PDE, in Quantum Field Theory and for the description of Bose-Einstein condensates. Those are Gibbs measures associated with nonlinear Schrodinger-type energies. In dimensions larger than or equal to 2, the measures concentrate on distributions, and they need to be properly renormalized. After presenting the Gibbs measures, I will explain how to derive them from many-body quantum mechanics. Joint works with Phan Thanh Nam (Munich) and Nicolas Rougerie (Lyon).
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| 16:00 - 16:30 | Coffee break | |
| 16:30 - 17:20 |
Maciej Zworski (University of California at Berkeley, USA)
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Title: |
Mathematics of magic angles | |
| Abstract: |
Magic angles are a hot topic in condensed matter physics: when two sheets of graphene are twisted by those angles the resulting material is superconducting. I will present a very simple operator whose spectral properties are thought to determine which angles are magical. It comes from a 2019 PR Letter by Tarnopolsky--Kruchkov--Vishwanath. The mathematics behind this is an elementary blend of representation theory (of the Heisenberg group in characteristic three), Jacobi theta functions and spectral instability of non-self-adjoint operators (involving Hormander's bracket condition in a very simple setting). Recent mathematical progress also includes the proof of existence of generalized magic angles and computer assisted proofs of existence of real ones (Luskin--Watson, 2021). The results will be illustrated by colourful numerics which suggest many open problems (joint work with S Becker, M Embree, J Wittsten in 2020 and S Becker, T Humbert and M Hitrik in 2022).
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