# Complex patterns in wave functions - drums, graphs, and disorder

Starts:

09:00

Ends:

13:00

## Location

Kavli Royal Society Centre, Chicheley Hall, Newport Pagnell, Buckinghamshire, MK16 9JJ

## Overview

Theo Murphy international scientific meeting organised by Dr Sven Gnutzmann and Professor Uzy Smilansky

### Event details

Wave functions display complex patterns which are intensively studied in many branches of Mathematics and Physics.  Their value distributions, nodal sets, extreme values, and localization properties - to cite a few examples - are investigated using diverse methods developed within a network of fields whose connectivity leaves a lot to be desired. This conference gives a unique opportunity to discuss these common questions, and present different points of view and methods, yet in a single high-level forum.

A set of world-leading researchers has been invited to lecture on their recent contributions to the field.

Biographies of the organisers and speakers are available below. Recorded audio of the presentations will be available on this page after the event and the papers will be published in a future issue of Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

### Attending this event

This is a residential conference, which allows for increased discussion and networking.  It is free to attend, however participants need to cover their accommodation and catering costs if required.

Places are limited, therefore pre-registration is essential.

### Submitting a poster

A poster session will be held throughout the meeting alongside the schedule of presentations.

Enquiries: Contact the events team.

## Schedule of talks

Chairs

3 talks Show detail Hide detail

#### Chair

Peter Kuchment, Texas A&M University, USA

#### Abstract

Peter Kuchment works in the fields of partial differential equations, integral geometry, medical imaging, mathematical physics (eg, quantum graphs and photonic crystals), and spectral theory.  He earned his PhD from Kharkov State University, Kharkov, USSR, in 1973 and his DSci from the Academy of Sciences in Kiev in 1983.  A fellow of the Institute of Physics, he has authored more than 150 research publications He also directs the Summer Mathematics Research Training (SMaRT) Camp at Texas A&M for high school students.

#### Chair

Thomas Hoffmann-Ostenhof, University of Vienna, Austria

#### Abstract

Professor Dr Thomas Hoffmann-Ostenhof was born in Vienna, Austria. In 1963 he went Zürich to the Eidgenössische Hochschule (Federal Technical University Zürich) where he received his diploma in Chemistry in 1970.  He did his PHD formally at the Univeristy of Vienna with Professor Polansky in Theoretical Chemistry, but stayed with a stipend at the Max Planck Institute for Carbon Research (Dept Radiation Chemistry) in Mülheim and der Ruhr, Germany. In 1974 he obtained his Dr Phil.  Since then he has been a member of the Institute of Theoretical Chemistry.  In 1981 he got his habilitation and the title Docent, in 1991 conferment of the title “außerdentlicher Professor".

From 1993 - 2011 he was the secretary of the Erwin Schrödinger Institute of Mathematical Physics. 1997 - 2001 he was scientist in charge in Austria for the EU project Partial Differential Equations and Quantum Mechanics: TMR grant FMRX-CT 96-0001. 2002 - 2007 he was a member of the steering committee for the program Spect ( Spectral Theory and Partial Differential Equations) from the European Science Foundation.

Since 1970 he has been married to Dr Maria Hoffmann-Ostenhof.  Almost immediately after PHD in Theoretical Chemistry he switched to Mathematical Physics. This was made easier for him since his wife is a Mathematician.  They have produced many papers together. They were in contact with Professor Thirring at the Physics Dept in Vienna and in 70s and early 80s had collaborations with, amongst others, Reinhart Ahlrichs, Ira Herbst and  Barry Simon.  They continued collaborations with Nikolai Nadirashvili, Bernard Helffer, Ari Laptev, and many other colleagues, mostly mathematicians.

Since the late 80s Professor Dr Hoffmann-Ostenhof has been  interested in qualitative properties of zero sets and nodal domains of eigenfunctions.

#### Chair

Martin Sieber, University of Bristol, UK

Session 1

3 talks Show detail Hide detail

#### Eigenfunctions and nodal sets

Steve Zelditch, Northwestern University, USA

#### Abstract

Steve Zelditch is Professor of Mathematics at Northwestern University. He works in spectral geometry, complex geometry and mathematical physics, with a particular interest in asymptotics problems. He received his bachelor’s degree from Harvard in 1975, and his PhD from UC Berkeley in 1981 under the direction of Alan Weinstein. He was Ritt Assistant Professor at Columbia (1981-1985), and went from Assistant Professor to Professor of Mathematics at Johns Hopkins University (1986-2009), before moving to Northwestern in 2010. He was an invited speaker at the ICM in Beijing (2004) and has twice been an invited speaker at the International Congress of Mathematical Physics. He gave Current Developments in Mathematics lectures at Harvard in 2009 and an invited AMS national address in 2005. He served on the Scientific Advisory Board of the Centre de Recherches Mathematiques during the years 2004-2007. He has been on the editorial boards of Annales Scientifiques de l'École Normale Supérieure, American Journal of Mathematics, Journal of Mathematical Physics and is currently on the editorial boards of the Communications in Mathematical Physics and Analysis and PDE.

### Abstract

My talk concerns the nodal (zero) sets  of analytic continuations of eigenfunctions on real analytic Riemannian manifolds to the complexification of the manifold.  The complex zero sets are simpler to study than the real zero sets.  The new result in the talk is a limit distribution of the intersection points of the complex nodal set with complexified geodesics. If the geodesic flow is ergodic, then for almost any geodesic, the intersection points condense along the real points of the geodesic and become uniformly distributed with respect to arc length.

#### Nodal count of eigenfunctions as index of instability

Gregory Berkolaiko, Texas A&M University, USA

#### Abstract

After completing his undergraduate studies in Mathematics at Voronezh State University, Russia, in 1996, Gregory Berkolaiko obtained an MPhil from University of Strathclyde, UK, in 1997 and Phd from University of Bristol, UK, in 2001.  His PhD thesis, under the supervision of Professor J P Keating, concentrated on quantum graphs.  This research was continued during the postdoc at Weizmann Institute of Science, Israel, with Professor U Smilansky.  After two years as a Lecturer at University of Strathclyde, Gregory moved to Texas A&M University as an Assistant Professor.  He was granted tenure and promoted to Associate Professor there in 2008.  Gregory continues his research on graphs as well as in other areas of mathematical physics and regularly publishes in leading international journals, such as Comm Math Phys, Phys Rev Lett, Trans AMS and others.

### Abstract

Courant's celebrated nodal bound asserts that the zero curves of the n-th eigenfunction of the Laplace operator on a compact domain partition the domain into at most n parts (which are called "nodal domains").  However, the actual number of nodal domains is usually well below the Courant's bound.

It recently transpired that the difference between the bound and the actual value can be interpreted as an index of instability of a certain energy functional with respect to suitably chosen perturbations.  The results concerning this phenomenon fall in two classes: (1) stability of the nodal partitions with respect to a perturbation of the partition boundaries and (2) stability of the eigenvalue with respect to a perturbation by magnetic field.  We will discuss examples of the available results and the connections among them.

Based on joint work with R Band, P Kuchment and U Smilansky, H Raz and T Weyand

#### 0,1,2,3,... is a tree

Ram Band, University of Bristol, UK

#### Abstract

Rami Band is currently an EPSRC postdoctoral fellow in the University of Bristol.  He works in the math department, in the mathematical physics group which was established by Professor Jon Keating. Rami completed his PhD in 2010 under the supervision of Professor Uzy Smilansky in the department of complex systems at the Weizmann Institute of Science.

A substantial part of his research concerns quantum graphs and their spectral properties.  He developed a theory for construction of isospectral objects. He works on nodal domains in various systems, such as discrete and quantum graphs and drums.

### Abstract

Sturm's oscillation theorem states that the n-th vibrational mode of a string has n-1 nodal points.  This result was also proved for a tree shaped network of strings. Namely, the n-th eigenfunction of a Schroedinger operator on a metric tree graph has exactly n-1 zeros and n nodal domains. An analogous result exists for the discrete Schroedinger operator as well. We prove an inverse theorem - if for all n, the n-th eigenfunction of a graph has n-1 zeros then the graph is a tree. The proof uses a recent result of Berkolaiko and Colin de Verdiere which connects the spectrum and the zero count via magnetic fluxes defined on the graph (see preceding talk).  Finally, we discuss the nodal count inverse problem - whether one can deduce that a graph is a tree judging from its nodal domain count.

Based on joint work with Uzy Smilansky and Idan Oren.

Session 2

4 talks Show detail Hide detail

#### Eigenfunction entanglement and random matrix theory

Professor Jon Keating FRS, University of Bristol, UK

#### Abstract

Jon Keating is based at the University of Bristol.  He gained his first degree from Oxford University and his PhD from the University of Bristol, under the supervision of Professor Sir Michael Berry FRS.  His research is centred in the areas of quantum chaos, random matrix theory and number theory.  He has held a BRIMS Research Fellowship, funded by Hewlett-Packard, and an EPSRC Senior Research Fellowship.

### Abstract

I will discuss recent joint work with Noah Linden and Huw Wells in which we consider eigenfunction entanglement for the excited states in a class of interacting spin models.  We introduce a related ensemble of random matrices and calculate the mean density of the eigenvalues and correlations in the eigenvectors.

#### The nonlinear Schroedinger equation on a star graph: standing waves and scattering of solitons

Diego Noja, Università di Milano Bicocca, Italy

#### Abstract

Diego Noja is a mathematical physicist working and teaching at the University of Milano Bicocca, Italy. He graduated in Milano with L Galgani and then gained his PhD in 1997 mainly under the influence of G Dell'Antonio. His research interests can be resumed in two main lines. The first is the rigorous description of particles and field interactions, for example between a point like particle and electromagnetic field. Point like particles create singular fields and this makes difficult to describe in a consistent way the action on the particle itself by the field. This is an old problem still unsolved.

The second more recent line of research is the dynamics of solitary waves occurring in the nonlinear Schrödinger equation in the presence of defects or non-trivial geometry, such as a junction in a ramified structure. The defect or the junction produce interesting dynamical features, among which one could mention the formation of a nonlinear energy spectrum, spontaneous symmetry breaking in ground states and nonlinear scattering of solitons.

### Abstract

Nonlinear Schroedinger equation is defined on a star graph, ie on the structure given by N halflines with a suitable  boundary condition at the common vertex. This dynamical system admits standing waves, described (explicitly) as solitons sitting at the vertex with a fixed frequency. In the case of an attractive delta vertex it is shown that these standing waves are orbitally stable, which means Lyapunov stable up to the symmetries of the system. Moreover the scattering behaviour of a fast solitary wave inpinging on different types of vertices it is described.

#### Quantum statistics on many-particle graphs

Jonathan Robbins, University of Bristol, UK

#### Abstract

Jonathan Robbins completed his undergraduate degree in Mathematics and Physics at Yale University in 1983, and his PhD, on semiclassical quantisation, at the University of California, Berkeley in 1989, where he was supervised by Robert Littlejohn.  He then moved to the Department of Physics at the University of Bristol as a NATO Postdoctoral Fellow, working with Sir Michael Berry, and in 1994 joined the Department of Mathematics. From 1994 - 2001, he was a member of BRIMS, the Basic Research Institute in the Mathematical Sciences at Hewlett-Packard Laboratories, Bristol.   His research interests include semiclassical methods in quantum mechanics, quantum chaos, geometric phases, quantum statistics and the spin-statistics relation, Hamiltonian systems, as well as liquid crystals and micromagnetics.

### Abstract

We consider quantum statistics for indistinguishable spinless particles on a one-dimensional network, or graph.  In spite of the fact that graphs are locally one-dimensional, anyon statistics emerge in a generalized form. The quantum statistics can depend on where and how the particles are exchanged, and discrete sign-valued phases appear alongside continuous anyon and Aharonov-Bohm phases.   Making use of recent results on the topology of graph configuration spaces, we obtain a complete classification of n-particle abelian statistics, which turns out to be related to the connectedness and planarity properties of the underlying graph.  The analysis is simplified by considering combinatorial rather than metric graphs.

This is joint work with Jonathan Harrison, Jon Keating and Adam Sawicki.

#### Topological resonances in quantum graphs

Holger Schanz, University of Applied Sciences Magdeburg-Stendal, Germany

#### Abstract

Holger Schanz is a theoretical physicist with research interests in the fields of nonlinear dynamics, quantum chaos, mesoscopic systems and scattering theory.  He graduated from Technische Universität Dresden in 1992. For his Ph. D. he worked at Humboldt Universität Berlin and at the Weizmann Institute of Science in Israel. After completion of his thesis in 1996 he was Postdoc at the Max Planck Instiute for Physics of Complex Systems in Dresden. From 1998 till 2006 Holger Schanz did research at the Max Planck Institute for Dynamics and Self-Organization in Göttingen, Germany. In that period he was also assistant and associate professor at the Institute for Nonlinear Dynamics of the University of Göttingen. Following a two-year period as a consultant in mathematical finance, Holger Schanz was appointed professor at the University of Applied Sciences in Magdeburg in 2009.

### Abstract

We study a special type of narrow resonances in complex scattering systems modeled by a network (quantum graph). Due to interference the wave function is nearly trapped on a subset of the available bonds of the network. In the statistics of resonance widths, delay times or amplification factors these resonances lead to power laws with exponents that depend on the topology of the supporting subgraph rather than the number of scattering channels. Due to the strong amplification of the resonance wave function on a subgraph, topological resonances may enhance the effects of nonlinearities, eg in complex optical networks. We explain the mechanism behind the formation of topological resonances, predict their position in the spectrum and use perturbation theory to derive their distribution in the limit of small resonance width.

Session 3

4 talks Show detail Hide detail

#### The Donnelly-Fefferman growth bound on eigenfunctions - an elementary proof

Dan Mangoubi, Hebrew University of Jerusalem, Israel

#### Abstract

Dan Mangoubi is a senior lecturer at the Hebrew University of Jerusalem, completed his PhD in the Technion (Haifa) in 2006 under the direction of Leonid Polterovich and Mikhael Entov.

### Abstract

Donnelly and Fefferman proved in 1988 that eigenfunctions of eigenvalue \lambda grow like polynomials of degree \lambda^{1/2}. This result was central in the resolution of Yau's conjecture in the real analytic case and became
important in its own as it laid the basis for many analogies between harmonic functions and eigenfunctions.
The original proof used delicate Carleman type estimates, and due to the importance of the result its proof was simplified by several authors (Lin, Jerison-Lebeau, Nazarov-Polterovich-Sodin). We give a new proof which can be most easily explained

#### Random nodal portraits

Mikhail Sodin, Tel Aviv University, Israel

#### Abstract

Professor Sodin is a professor of Mathematics at Tel Aviv University since 1996, specializing in Analysis and its Applications, especially, in Mathematical Physics. Prior to his appointment at Tel Aviv he was a Senior Researcher at the Mathematical Division of the Institute for the Lower Temperature Physics at Kharkov, Ukraine.

### Abstract

We describe the progress in understanding the zero sets of smooth Gaussian random functions of several real variables. The primary examples are various ensembles of Gaussian real-valued polynomials (algebraic or trigonometric) of large degree, and smooth Gaussian functions on the Euclidean space with translation-invariant distribution.

The fundamental question is the one on the asymptotic behaviour of the number of connected components of the zero set.

In the talk, we explain how Ergodic Theorem and rudimentary harmonic analysis and multivariable calculus help to find the correct order of growth of the typical number of connected components of the zero set. We will mention a number of basic questions, which remain widely open. The talk is based on a joint work with Fedor Nazarov.

#### Universal quantum localizing transition of a partial barrier in a chaotic sea

Roland Ketzmerick, Technische Universität Dresden, Germany

#### Abstract

Roland Ketzmerick is Professor at the Technische Universität Dresden, Germany. He studied physics at the Universities of Darmstadt and Freiburg, Germany and at Purdue University, USA. He received his PhD in physics from the University of Frankfurt under the supervision of Professor Theo Geisel. After a post-doc year at the University of California Santa Barbara with Professor Walter Kohn and the Habilitation at the University of Göttingen he moved to Dresden in 2002. His research interests are in the field of quantum chaos with particular emphasis on generic systems with a mixed regular-chaotic phase space.

Roland Ketzmerick has been a Fulbright fellow, received the Otto-Klung-Prize, and is currently a Max Planck Fellow at the local Max Planck Institute for the Physics of Complex Systems.

### Abstract

Generic 2D Hamiltonian systems possess partial barriers in their chaotic phase space that restrict classical transport. Quantum mechanically the transport is suppressed if Planck's constant h is large compared to the classical flux, such that wave packets and states are localized. In the opposite limit classical transport is mimicked.  Designing a quantum map with an isolated partial barrier of controllable flux is the key to investigating the transition from this form of quantum localization to mimicking classical transport. It is observed that quantum transport follows a universal transition curve as a function of the expected scaling parameter flux/h. We find this curve to be symmetric to flux/h=1, having a width of two orders of magnitude, and exhibiting no quantized steps. We establish the relevance of random matrix models with local channel coupling, improving on previous models relying on global coupling. It turns out that a phenomenological 2x2-model gives an accurate analytical description of the transition curve.

#### Nodal length fluctuations for arithmetic random waves

Igor Wigman, Cardiff University, UK

#### Abstract

Igor Wigman did his PhD in Tel-Aviv University in Number Theory under the direction of Professor Zeev Rudnick. After concluding the PhD studies in 2006, he was a postdoctoral fellow in CRM Montreal in 2006-2009 and KTH Stockholm in 2009-2010. He is currently a lecturer in Pure Mathematics in the Cardiff University. His research interests include Number Theory, Probability and Random Fields and their applications in Mathematical Physics, especially Quantum Chaology.

### Abstract

Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus (“arithmetic random waves”). We study the distribution of the nodal length of random eigenfunctions for large eigenvalues, and our primary result is that the asymptotics for the variance is non-universal, and is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy.

This work is joint with Manjunath Krishnapur and Par Kurlberg.

Session 4

4 talks Show detail Hide detail

#### Fluctuations and extreme values in multifractal patterns

Yan Fyodorov, Queen Mary University of London, UK

#### Abstract

Yan Fyodorov is the Professor of Applied Probability & Mathematical Physics in the School of Mathematical Sciences at Queen Mary, University of London. After graduating from St. Petersburg Polytechnical University (Russia) he received his PhD degree in Mathematical and Theoretical Physics from the Theory Division of the Petersburg Institute of Nuclear Physics.

After spending several years there as a research fellow he left Russia for Germany in the year 1991 as a Humboldt Research Fellow, and then spent another year as a postdoctoral researcher at the Weizmann institute (Israel).

After that he worked a few more years at the University of Essen as a research assistant before getting a professorship in the UK (2000)where he has stayed ever since.

His research interests are centred around Physical Mathematics of random matrices.

It includes applications of Random Matrix Theory to physics of disordered systems, both classical and quantum, among others to areas of Anderson Localization, Quantum Chaotic Scattering, Mesoscopics, and Statistical Mechanics.

The focus of his current research supported by EPSRC grant is mainly on exploring statistical properties of random landscapes and the extreme value statistics of strongly correlated random processes and fields, in particular 1/f noises.

The latter show fascinating relations to random matrices, Burgers turbulence, multifractal measures, and Riemann zeta-function.

Awards and Fellowships: Leverhulme  Research Fellowship (2008); Bessel Research Award (2006);  Institute Henri Poincare/Gauthier-Villars prize in Theoretical Physics (1999)

### Abstract

To understand the sample-to-sample fluctuations in disorder-generated multifractal patterns we investigate analytically as well as numerically the statistics of high values of the simplest model - the ideal periodic $1/f$ Gaussian noise. By employing the thermodynamic formalism we predict the characteristic scale and the precise scaling form of the distribution of number of points above a given level. We demonstrate that the powerlaw forward tail of the probability density, with exponent controlled by the level, results in an important difference between the mean and the typical values of the counting function. This can be further used to determine the typical threshold $x_m$ of extreme values in the pattern which turns out to be given by $x_m^{(typ)}=2-c\ln{\ln{M}}/\ln{M}$ with $c=3/2$. Such observation provides a rather compelling explanation of the mechanism behind universality of $c$. Revealed mechanisms are conjectured to retain their qualitative validity for a broad class of disorder-generated multifractal fields. In particular, we predict that the typical value of the maximum $p_{max}$ of intensity is to be given by $-\ln{p_{max}} = \alpha_{-}\ln{M} + \frac{3}{2f'(\alpha_{-})}\ln{\ln{M}} + O(1)$, where $f(\alpha)$ is the corresponding singularity spectrum vanishing at $\alpha=\alpha_{-}>0$. For the $1/f$ noise we also derive exact as well as well-controlled approximate formulas for the mean and the variance of the counting function without recourse to the thermodynamic formalism. The presentation will be based on the joint work with Pierre Le Doussal and Alberto Rosso: arXiv:1207.4614v1

#### Pseudo-integrable systems

Zeev Rudnick, Tel Aviv University, Israel

#### Abstract

Zeev Rudnick has been professor of Mathematics at Tel Aviv University since 1995, specializing in Number Theory and Quantum Chaos.

Prior to his appointment at Tel Aviv he was an Assistant Professor at Princeton and in Stanford.

### Abstract

The Laplacian perturbed by a delta-potential, also known as a point scatterer,  is a popular model used to study the transition between integrability and chaos in quantum mechanics.

I will discuss eigenvalue and eigenfunction statistics for this model (results joint with Henrik Uebersacher).

#### Ordered spectral statistics for one-dimensional disordered Hamiltonians

Christophe Texier, Université Paris-Sud, France

#### Abstract

From 1995-1999 Christophe Texier was at University Paris 6 Pierre et Marie Curie where he gained his PhD.  He was Postdoctoral assistant at University of Geneva from 1999 - 2000, and from 2000 until present has been Assistant professor at University Paris-Sud.

### Abstract

I will discuss the ordered spectral statistics problem for one-dimensional disordered quantum Hamiltonians (determination of the distribution of the n-th energy level).

When eigenstates are all strongly localised, these distributions coincide with Gumbel laws, reflecting the absence of correlations between energy levels.

I will focus on a supersymmetric Hamiltonian presenting delocalisation at the bottom of the spectrum. As a consequence the
distribution of the n-th eigenvalue differs from the universal distributions obtained for uncorrelated variables.

These results may be used in order to study the finite size effect on the low energy density of states, near the delocalisation point. This provides a picture on the structure of the low energy wave functions and the localization properties of this model.

Reference: C T  & C Hagendorf, J Phys A43 (2010)

#### Wave function multifractality in simple models

Eugene Bogomolny, Université Paris-Sud, France

#### Abstract

Eugene Bogomolny is a Research Director in the Laboratory of Theoretical Physics and Statistical Models in the University Paris-Sud, Orsay, France. He has been graduated from the Moscow Institute of Physics and Technology, Russia. He received his PhD in physics from the Institute of Experimental and Theoretical Physics in Moscow in 1975 under the supervision of L B Okun and defended the second thesis in physics in 1985.

Till 1991 he worked as a senior researcher in the L D Landau Institute of Theoretical Physics, Chernogolovka, Russia.

His research interests include high-energy physics, theory of solitons, higher orders of perturbation series in quantum mechanics, field theories, and classical mechanics. From '90 he works mainly on different problems of quantum chaos, trace formulas, random matrix theory and its relations to number theory.

### Abstract

Different aspects of multifractality of wave functions in various systems are shortly discussed. Special attention is given to  multifractality of ground state wave functions of one-dimensional spin chain models.

Session 5

5 talks Show detail Hide detail

#### On recent results for minimal spectral partitions

Bernard Helffer, Université Paris-Sud, France

#### Abstract

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.

### Abstract

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

#### Abstract

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

#### Abstract

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.

### Abstract

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

#### Abstract

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.
• ### Abstract

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

#### Abstract

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.

### Abstract

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.

Complex patterns in wave functions - drums, graphs, and disorder Scientific discussion meeting organised by Professor Guy Poppy FSB, Professor Paul Jepson, Professor John Pickett CBE FRS and Dr Michael Birkett FRSC. Kavli Royal Society Centre, Chicheley Hall Newport Pagnell Buckinghamshire MK16 9JJ