Analysis on graphs and its applications

Programme theme

Analysis on graphs and other discrete structures has been developing for quite some time, in particular due to applications to number theory, algebra, probability theory, spectral geometry, as well as to its usefulness in many practical problems. This area, however, has experienced recently a significant boost in terms of new important applications arising, new methods being developed, and new models introduced not studied before. This has happened due to numerous new applications arising in different areas of mathematics, sciences, and engineering. They swipe throughout a wide scientific landscape, which besides the fields already mentioned includes nanotechnology, microelectronics, quantum chemistry, superconductivity, optics, etc. New objects, so called quantum graphs have also emerged. These are graphs considered as one-dimensional CW-complexes and equipped with differential or pseudo-differential, rather than customary difference operators. Such graphs, besides being in many cases useful low-dimensional models of complex systems, are also used as toy models for studying difficult issues such as Anderson localisation, scattering, and quantum chaos. This has lead to interest in this research by scientists coming from different fields (physics, partial differential equations, algebra, combinatorics, number theory). The methods already used or expected to be useful in analysis on graphs come from a very wide range of topics: algebra, combinatorics, PDEs, spectral theory, micro-local and complex analysis, to name a few.

The program will assemble, essentially for the first time, a diverse group of prominent mathematicians and physicists, as well as young researchers working in or interested in entering this fast developing and fascinating area. The hope is to facilitate the process of communication and cooperation, thereby promoting accelerated progress in this burgeoning new area. Among the particular topics that will be addressed during the program are: Spectral analysis on combinatorial graphs and its applications to number theory, discrete groups, random walks, and other areas; Quantum graph models for wave propagation in thin structures and for quantum and optical switching and computing; Spectral analysis and quantum chaos on quantum graphs; Analysis on fractals.

Click here to download the programme's final scientific report

Preprints

Title	Year	Programme
On occurrence of spectral edges for periodic operators inside the Brillouin zone Authors: ET Al, Peter Kuchment, Jon Harrison, Alex Sobolev Download PDF	2006	AGA
Modified Krein formula and analytic perturbation procedure for scattering on arbitrary junction Authors: A Yafyasov, A Adamyan, Boris Pavlov Download PDF	2006	AGA
A linear Wegner estimate for alloy type Schrödinger operators on metric graphs Authors: I Veselic, M Helm Download PDF	2006	AGA
Spectral asymptotics of percolation Hamiltonians on amenable Cayley graphs Authors: I Veselic, T Antunovic Download PDF	2006	AGA
The maximum principle for elliptic inequalities on stratified sets Authors: Oleg Penkin, SN Oshepkova Download PDF	2006	AGA
Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over ${\mathbb Z}^d$ Authors: Daniel Lenz, I Veselic, Michael Gruber Download PDF	2006	AGA
Mathematics of thermoacoustic and photoacoustic tomography Authors: L Kunyansky, Peter Kuchment Download PDF	2006	AGA
Inverse problems for quantum trees Authors: P Kurasov, Sergei Avdonin Download PDF	2006	AGA
Vacuum energy and repulsive Casimir forces in quantum star graphs Authors: L Kaplan, Justin Wilson, Stephen Fulling Download PDF	2006	AGA
An index theory for uniformly locally finite graphs Authors: Joachim von Below Download PDF	2006	AGA
The problem of fitting of the zero-range model of the tectonic plate under a localized boundary stress Authors: Boris Pavlov, L Petrova Download PDF	2006	AGA
Plasma waves in two-dimensional electron channels: propagation and trapped modes Authors: Boris Pavlov, VI Ryzhii, IA Semenikhin Download PDF	2006	AGA
Inverse spectral and scattering theory for the half-line left definite Sturm-Liouville problem Authors: Christer Bennewitz, Yaroslav Brown, Rudi Weikard Download PDF	2006	AGA
Quantum chaos on discrete graphs Authors: Uzy Smilansky Download PDF	2006	AGA
On the spectrum of the Dirichlet Laplacian in a narrow strip Authors: Michael Solomyak, Leonid Friedlander Download PDF	2006	AGA
3D scanning technology as a standard archaeological tool for pottery analysis: practice and theory Authors: Uzy Smilansky, A Karasik Download PDF	2006	AGA
Equality of Lifshitz and Van Hove exponents on amenable Cayley graphs Authors: I Veselic, T Antunovic Download PDF	2006	AGA
Vibration modes of 3n-gaskets and other fractals Authors: ET Al, Alexander Teplyaev, B Steinhurst Download PDF	2006	AGA
Crystals that nature might miss creating Authors: Toshikazu Sunada Download PDF	2006	AGA
On Bernoulli decompositions for random variables, concentration bounds, and spectral localization Authors: Michael Aizenman, F Germinet, Abel Klein, Simone Warzel Download PDF	2006	AGA
Vacuum energy in quantum graphs Authors: Justin Wilson Download PDF	2006	AGA
Sharpness of the phase transition and exponential decay of the subcritical cluster size Authors: I Veselic, T Antunovic Download PDF	2006	AGA
The modulus of continuity of Wegner estimates for random Schrödinger operators on metric graphs Authors: Michael Gruber, I Veselic Download PDF	2006	AGA
Minimizing the ground state energy of an electron in a randomly deformed lattice Authors: G Stolz, J Baker, M Loss Download PDF	2006	AGA
Index theorems for quantum graphs Authors: Justin Wilson, Peter Kuchment, Stephen Fulling Download PDF	2006	AGA
Quantum graphs where back-scattering is prohibited Authors: Jon Harrison, Uzy Smilansky, Brian Winn Download PDF	2006	AGA
On calculation of Kirchhoff coefficients for Helmholtz resonator Authors: Boris Pavlov, Jochen Brüning Download PDF	2006	AGA
Zeta functions of weighted graphs and covering graphs Authors: Harold Stark, Audrey Terras, MD Horton Download PDF	2006	AGA
Eigenvalue correlations in continuum one-dimensional Anderson models Authors: Robert Sims, G Stolz Download PDF	2006	AGA
Optimal Wegner estimates for random Schrödinger operators on metric graphs Authors: M Helm, I Veselic, Michael Gruber Download PDF	2006	AGA
Uniform existence of the integrated density of states for combinatorial and metric graphs over ${\mathbb Z}^d$ Authors: Daniel Lenz, I Veselic, Michael Gruber Download PDF	2006	AGA
Monotone unitary families Authors: Daniel Grieser Download PDF	2006	AGA
Thin tubes in mathematical physics, global analysis and spectral geometry Authors: Daniel Grieser Download PDF	2006	AGA
Spectra of graph neighborhoods and scattering Authors: Daniel Grieser Download PDF	2006	AGA

Workshops

AGAW01

Seminars

Thursday 18th January 2007
14:30 to 15:30	What is a Discrete Complex Analysis on the Equilateral Triangle Lattice?	Room 1
16:00 to 17:00	Spectra of groups	Room 1

Tuesday 23rd January 2007
14:30 to 15:30	On the K$_4$ crystal	Room 1
16:00 to 17:00	Peter Kuchment Texas A&M University Liouville theorems for equations on coverings of graphs and manifolds	Room 1

Wednesday 24th January 2007
12:00 to 14:30	AGA discussion	Discussion Room

Friday 26th January 2007
11:00 to 12:00	Asymptotic aspects of Schreier graphs of self-similar groups	Room 1

Thursday 1st February 2007
14:30 to 15:30	An approximate boundary distance representation and an approximate rec	Room 1

Wednesday 7th February 2007
16:00 to 17:00	Eigenvalue estimates for Schroedinger operators on regular metric trees	Room 1

Thursday 8th February 2007
14:30 to 15:30	Quantum chaos on graphs	Room 1

Thursday 15th February 2007
14:30 to 15:30	Scattering on a decorated star-graph as a toy model for the spectral theory of automorphic functions	Discussion Room
16:00 to 17:00	Matrix valued orthogonal polynomials and random walks	Room 1

Friday 16th February 2007
11:30 to 12:30	Leonid Gurvits Los Alamos National Laboratory Hyperbolic polynomials and lower bounds in combinatorics	Room 1

Wednesday 21st February 2007
16:00 to 17:00	Spectral edges for periodic operators - some quantum graph examples	Room 1

Tuesday 27th February 2007
14:30 to 15:30	Is there an interesting index theory for quantum graphs?	Room 1

Thursday 1st March 2007
16:00 to 17:00	Another Martini for quantum graphs	Room 1

Tuesday 6th March 2007
16:00 to 17:00	Audrey Terras University of California, San Diego Zeta functions of weighted graphs and covering graphs	Room 1

Wednesday 7th March 2007
11:30 to 12:30	Discussion / Open problems session	Discussion Room
16:00 to 17:00	Zeta and L-functions of graphs	Room 1

Tuesday 13th March 2007
14:30 to 15:30	Michael Plum A computer-assisted existence proof for photonic band gaps	Room 1
16:00 to 17:00	Validated computation tool for Perron-Frobenius eigenvalue using graph theory	Room 1

Thursday 15th March 2007
14:30 to 15:30	Justin Wilson University of Maryland, College Park The Method of Images on a Quantum Graph	Room 1

Tuesday 20th March 2007
11:30 to 12:30	Carlos Berenstein University of Maryland, College Park Internet tomography	Room 1

Thursday 22nd March 2007
11:30 to 12:30	Quantum ergodicity on graphs	Room 1
14:30 to 15:30	Vadim Kostrykin Johannes Gutenberg-Universität Mainz On the negative spectrum of quantum graphs	Room 1

Tuesday 27th March 2007
11:30 to 12:30	Quantum Chaos on Discrete Graphs	Room 2
15:30 to 16:30	Controllability of Dynamical Systems on Graphs	Room 2

Thursday 29th March 2007
10:30 to 11:30	On Approximation of the Eigenvalues of Perturbed Periodic Schrodinger Operators	Room 2
14:00 to 15:00	Solvability of the Dirichlet problem on stratified sets	Room 2

Monday 16th April 2007
14:30 to 15:30	Multiplicities of eigenvalues and topology	Room 1

Wednesday 18th April 2007
14:30 to 15:30	Lower bounds of the entropy of quantum limits (after N. Anantharaman and S. Nonnemacher)	Room 1

Thursday 19th April 2007
14:30 to 15:30	Travelling fronts on networks for reaction-diffusion-equations	Room 1

Tuesday 24th April 2007
14:30 to 15:30	Kirchhoff Constants for Helmholtz Resonator	Room 1

Thursday 26th April 2007
14:30 to 15:30	On Inverse Problems for Trees	Room 1
16:00 to 17:00	Can one hear the shape of a network?	Room 1

Tuesday 1st May 2007
16:00 to 17:00	Spectral inequalities for a class of hypo-elliptic operators	Room 1

Tuesday 15th May 2007
14:30 to 15:30	Alexander Teplyaev University of Connecticut Schur complement, Drichlet-to-Neumann map, and eigenfunctions on self-similar graphs	Room 1

Thursday 17th May 2007
14:30 to 15:30	Differential operators with singularities and generalised Nevanlinna functions	Room 1

Wednesday 23rd May 2007
11:00 to 12:00	Michael Levitin University of Reading; University College London Pollution-free methods for finding eigenvalues in the gaps of the continuous spectrum	Room 1
16:00 to 17:00	Shi Zhou University College London Structural constraints in complex networks	Room 1

Thursday 24th May 2007
16:00 to 17:00	Volodymyr Nekrashevych Texas A&M University Self-similar graphs, algebra and fractals	Room 1

Tuesday 5th June 2007
14:30 to 15:30	Wave propagation in networks of thin fibres: small diameter asymptotics	Room 1

Thursday 7th June 2007
14:30 to 15:30	Asymptotical properties of random trees on manifolds	Room 1

Friday 8th June 2007
11:00 to 12:00	Transition density of Markov processes on fractal-like spaces	Room 1

Tuesday 12th June 2007
14:30 to 15:30	Stanislav Molchanov University of North Carolina On one hydrostatic problem	Room 1

Wednesday 13th June 2007
10:30 to 11:30	Self-similar groups and Schur complement	Room 1
16:00 to 17:00	Applications of computer methods and mathematical ideas in archaeology	Room 1

Tuesday 19th June 2007
14:30 to 15:30	The spectral theory of some directed graphs	Room 2
16:00 to 17:00	On the random analytic functions	Room 2

Thursday 21st June 2007
11:00 to 12:00	Ian Wood University of Kent Some spectral problems for non-selfadjoint operators	Room 1
14:30 to 15:30	Bernoulli decompositions for random variables and applications	Room 1

Friday 22nd June 2007
14:30 to 15:30	Analysis on Graphs and its Applications	Room 1

Tuesday 26th June 2007
14:30 to 15:30	Diffusion in narrow randomly perturbed tubes	Room 1
16:00 to 17:00	The spectral minimum of the random displacement model	Room 1

Saturday 21st March 2015
11:00 to 12:00	Longitude Found	Room 1

Organisers

Jeroen Jeneson Technische Universität Eindhoven
Alexander McNeil Heriot-Watt University
Hans Ringstroem KTH - Royal Institute of Technology
M Roeckner Aalborg Universitet
G Roehrle Aalborg Universitet
Alexander Sobolev University College London
Jean-Marc Vanden-Broeck Aalborg Universitet

Speakers

Carlos Berenstein University of Maryland, College Park
Leonid Gurvits Los Alamos National Laboratory
Vadim Kostrykin Johannes Gutenberg-Universität Mainz
Peter Kuchment Texas A&M University
Michael Levitin University of Reading; University College London
Stanislav Molchanov University of North Carolina
Volodymyr Nekrashevych Texas A&M University
Michael Plum
Alexander Teplyaev University of Connecticut
Audrey Terras University of California, San Diego
Justin Wilson University of Maryland, College Park
Ian Wood University of Kent
Shi Zhou University College London

Show All

Analysis on graphs and its applications

Programme theme

Preprints

On occurrence of spectral edges for periodic operators inside the Brillouin zone

Modified Krein formula and analytic perturbation procedure for scattering on arbitrary junction

A linear Wegner estimate for alloy type Schrödinger operators on metric graphs

Spectral asymptotics of percolation Hamiltonians on amenable Cayley graphs

The maximum principle for elliptic inequalities on stratified sets

Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over ${\mathbb Z}^d$

Mathematics of thermoacoustic and photoacoustic tomography

Inverse problems for quantum trees

Vacuum energy and repulsive Casimir forces in quantum star graphs

An index theory for uniformly locally finite graphs

The problem of fitting of the zero-range model of the tectonic plate under a localized boundary stress

Plasma waves in two-dimensional electron channels: propagation and trapped modes

Inverse spectral and scattering theory for the half-line left definite Sturm-Liouville problem

Quantum chaos on discrete graphs

On the spectrum of the Dirichlet Laplacian in a narrow strip

3D scanning technology as a standard archaeological tool for pottery analysis: practice and theory

Equality of Lifshitz and Van Hove exponents on amenable Cayley graphs

Vibration modes of 3n-gaskets and other fractals

Crystals that nature might miss creating

On Bernoulli decompositions for random variables, concentration bounds, and spectral localization

Vacuum energy in quantum graphs

Sharpness of the phase transition and exponential decay of the subcritical cluster size

The modulus of continuity of Wegner estimates for random Schrödinger operators on metric graphs

Minimizing the ground state energy of an electron in a randomly deformed lattice

Index theorems for quantum graphs

Quantum graphs where back-scattering is prohibited

On calculation of Kirchhoff coefficients for Helmholtz resonator

Zeta functions of weighted graphs and covering graphs

Eigenvalue correlations in continuum one-dimensional Anderson models

Optimal Wegner estimates for random Schrödinger operators on metric graphs

Uniform existence of the integrated density of states for combinatorial and metric graphs over ${\mathbb Z}^d$

Monotone unitary families

Thin tubes in mathematical physics, global analysis and spectral geometry

Spectra of graph neighborhoods and scattering

Workshops

Anaylsis on graphs and other discrete structures

Quantum graphs, their spectra and applications

Graph models of mesoscopic systems, wave-guides and nano-structures

Analysis on graphs and fractuals

Analysis on graphs and its applications follow-up

Seminars

Organisers

Speakers

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