On recent results for minimal spectral partitions
Bernard Helffer, Université Paris-Sud, France
Bernard Helffer is Professor at the university of Paris-Sud and is specialist in partial differential equations, spectral theory and mathematical physics. He is the author alone or in collaboration of more than 200 articles and of five books. He has been President of the french mathematical Society.
In this talk, we consider the question of minimal spectral partitions which share with nodal partitions many properties. We consider the two-dimensional case and discuss the state of the art for minimal spectral partitions with emphasis on recent results concerning the length of the boundary set.
This work has started in collaboration with T Hoffmann and has been continued with him and the following coauthors : V Bonnaillie-Noël, S Terracini, G Vial and P Bérard.
Wave packet dynamics for integrable flows
Jens Marklof, University of Bristol, UK
Jens Marklof is Professor of Mathematical Physics at the University of Bristol. He studied physics at the University of Hamburg and received his PhD from the University of Ulm in 1997. Professor Marklof held visiting positions at Princeton University, Hewlett Packard, the Newton Institute, IHES and the Université Paris-Sud. Marklof's main research objective is to explain random phenomena in nature by using tools from dynamical systems and ergodic theory. His recent research has focussed on the derivation of transport laws in the kinetic theory of gases. Professor Marklof's research is currently supported by a Leverhulme Research Fellowship, a Royal Society Wolfson Merit Award and an Advanced Grant from the European Research Council.
PT-symmetric dynamical systems
Henning Schomerus, Lancaster University, UK
Henning Schomerus is Professor in Theoretical Condensed Matter Physics at Lancaster University, UK. He graduated from Stuttgart University, Germany in 1993, spent a year as a Research Fellow at RIKEN, Japan, and obtained his doctorate in 1998 from Essen University, Germany.
After a postdoctoral position in Leiden, Netherlands (1998-2000) he became head of a Research Group at MPIPKS Dresden. In 2005 he was appointed to a Readership at Lancaster, where he led a Marie Curie Excellence team, and was awarded a personal chair in 2009. His work addresses fundamental physical problems in mesoscopic quantum transport, quantum dynamics, photonics, and quantum optics.
I describe dynamical models with balanced absorption and loss and explore the consequences of quantum-to-classical correspondence in these systems. By a mechanism paralleling the fractal Weyl law in open passive systems, the occurrence of strongly amplified states is reduced. In the context of the recently introduced PT-symmetric laser-absorbers, this phenomenon affects the number of states participating in the mode competition.
Seeing and hearing the Riemann zeros, primes, random-matrix ensembles, random waves…
Michael Berry FRS, University of Bristol, UK
After graduating from Exeter and St Andrews, Michael Berry entered Bristol University, where he has been for considerably longer than he has not. He is a physicist, focusing on the physics of the mathematics…of the physics. Applications include the geometry of singularities (caustics on large scales, vortices on fine scales) in optics and other waves, the connection between classical and quantum physics, and the physical asymptotics of divergent series. He delights in finding the arcane in the mundane – abstract and subtle concepts in familiar or dramatic phenomena:
Singularities of smooth gradient maps in rainbows and tsunamis;
The Laplace operator in oriental magic mirrors;
Elliptic integrals in the polarization pattern of the clear blue sky; Geometry of twists and turns in quantum indistinguishability;
Matrix degeneracies in overhead-projector transparencies;
Gauss sums in the light beyond a humble diffraction grating.
Two optical arrangements are envisaged in which the Riemann zeros would separate the side lobes of far-field diffraction patterns.
A counting function for the primes can be rendered as a sound signal whose harmonies are the Riemann zeros. But the individual primes cannot be discriminated as singularities in this ‘music’, because the intervals between them are too short. Conversely, if the prime singularities are detected as a series of clicks, the Riemann zeros correspond to frequencies too low to be heard. The sound generated by the Riemann zeta function itself is very different: a rising siren howl, which can be understood in detail from the Riemann-Siegel formula.
The eigenangles of random matrices in the three standard circular ensembles are rendered as sounds in several different ways. The different fluctuation properties of these ensembles can be heard, and distinguished from the extreme cases of angles that are distributed uniformly round the unit circle, and those that are Poissson-distributed. Similarly, in Gaussian random superpositions of monochromatic plane waves in 1D, 2D and 3D, the dimensions can be distinguished in sounds created from one-dimensional sections.
Wave function statistics beyond RMT: from quantum mechanics to ocean waves
Lev Kaplan, Tulane University New Orleans, USA
Lev Kaplan is Associate Professor of Physics at Tulane University in New Orleans, USA. Born in Riga, Latvia in 1971, Professor Kaplan earned a BA degree in Physics and Mathematics from the University of Pennsylvania in 1991 and a PhD in Physics from Harvard University in 1996, specializing in particle theory. Subsequently, he served as a Junior Fellow at the Harvard Society of Fellows and as national Institute for Nuclear Theory Fellow at the University of Washington. Professor Kaplan joined the Faculty of Tulane University in 2003, and serves as Associate Professor and Associate Chair of the Department of Physics and Engineering Physics. His recent research interests range from Casimir energy to quantum information, and from semiclassical methods in nanostructures to the statistics of extreme ocean waves.
As a universal theory, Random Matrix Theory (RMT) necessarily excludes all system-specific behavior associated with dynamics, boundary conditions, or interaction. This can lead to large discrepancies between the true wave function statistics for a chaotic system and the RMT prediction, even when the spectrum is well described by RMT. Several examples of strongly non-random wave function statistics in complex systems are briefly discussed. We then present a very general approach that allows known dynamical information about a chaotic system or ensemble of systems to be systematically merged with RMT. This method provides greatly improved accuracy over RMT and semiclassical methods for finite-size systems with a finite Ehrenfest time. The standard RMT predictions as well as standard semiclassical predictions are recovered in appropriate limiting cases.