# Workshop Programme

## for period 26 - 30 July 2010

### Analysis on Graphs and its Applications Follow-up

26 - 30 July 2010

Timetable

 Monday 26 July 08:30-08:55 Registration 08:55-09:00 Welcome from Sir David Wallace (INI Director) 09:00-10:00 Davies, B (Kings) Non-Weyl Asymptotics for Resonances of Quantum Graphs Sem 1 Consider a compact quantum graph ${\cal G}_0$ consisting of finitely many edges of finite length joined in some manner at certain vertices. Let ${\cal G}$ be obtained from ${\cal G}_0$ by attaching a finite number of semi-infinite leads to ${\cal G}_0$, possibly with more than one lead attached to some vertices. Let $H_0$ ( resp. $H$) $=-\frac{{\rm d}^2}{{\rm d} x^2}$ acting in $L^2({\cal G}_0)$ ( resp. $L^2({\cal G})$ ) subject to continuity and Kirchhoff boundary conditions at each vertex. The spectrum of $H$ is $[0,\infty)$, but unlike the normal case for Schrödinger operators $H$ may possess many $L^2$ eigenvalues corresponding to eigenfunctions that have compact support. However some eigenvalues of $H_0$ turn into resonances of $H$, and when defining the resonance counting function \[ N(r)=\#\{ \mbox{ resonances $\lambda=k^2$ of $H$ such that $|k|  Tuesday 27 July 09:00-10:00 Smilansky, U (Weizmann Institute of Science) Periodic walks on random graphs and random matrix theory Sem 1 The spectral statistics of the discrete Laplacian of d-regular graphs on V vertices are intimately connected with the distribution of the number of cycles of period t (t-cycles) on the graph. I shall discuss this connection by using a trace formula which expresses the spectral density in terms of the t-cycle counts. The trace formula will be used to write the spectral pair correlations in terms of the properly normalized variance of the t-cycle counts. Based on these results, I would like to propose a conjecture which uses Random Matrix Theory to compute the variance of the t-periodic cycle counts in the limit V,t -> infinity fixed value of with t/V. Numerical computations support this conjecture. 10:00-10:30 Morning Coffee 10:30-11:15 Harrisson, J (Baylor) Properties of zeta functions of quantum graphs Sem 1 The Ihara-Selberg zeta function plays a fundamental role in the spectral theory of combinatorial graphs. However, in contrast the spectral zeta function has remained a relatively unstudied area of analysis of quantum graphs. We consider the Laplace operator on a metric graph with general vertex matching conditions that define a self-adjoint realization of the operator. The zeta function can be constructed using a contour integral technique. In the process it is convenient to use new forms for the secular equation that extend the well known secular equation of the Neumann star graph. The zeta function is then expressed in terms of matrices defining the matching conditions at the vertices. The analysis of the zeta function allows us to obtain new results for the spectral determinant, vacuum energy and heat kernel coefficients of quantum graph which are topics of current research in their own right. The zeta function provides a unified approach which obtains general results for such spectral properties. 11:15-12:00 Terras, A (San Diego) Explicit Formulas for Zeta Functions of Graphs Sem 1 Explicit formulas for the Riemann and Dedekind zeta functions, as developed by Andre Weil, have often been compared to the Selberg trace formula. We look at analogs for the Ihara zeta function of a graph. 12:30-13:30 Sandwich Lunch at INI 14:00-14:45 Teplyaev, A (Connecticut) Uniqueness of Laplacian and Brownian motion on Sierpinski carpets Sem 1 Up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpinski carpet that is invariant with respect to the local symmetries of the carpet. Consequently for each such fractal the law of the Brownian motion is uniquely determined and the Laplacian is well defined. As a consequence, there are uniquely defined spectral and walk dimensions which determine the behavior of the natural diffusion processes by so called Einstein relation (these dimensions are not directly related to the well known Hausdorff dimension, which describes the distribution of the mass in a fractal). 14:45-15:30 Isola, T (tbc) Line integrals of one-forms on the Sierpinski gasket Sem 1 We give a definition of one-forms on the gasket and of their line integrals, and show that these are compatible with the notion of energy introduced by Kigami. We then introduce a suitable covering of the gasket (which is a projective limit of a sequence of natural finite coverings) and prove that n-exact forms have a primitive which lives on this covering. 15:30-16:00 Afternoon Coffee 16:00-16:45 Guido, D (Roma) Some (noncommutative) geometrical aspects of the Sierpisnki gasket Sem 1 We present here a 2-parameter family of spectral triples for the Sierpinski gasket, based on spectral triples for the circle. Any hole (lacuna) of the gasket is suitably identified with a circle, and the triple for the gasket is defined as the direct sum of the triples for the lacunas. The first parameter is a scaling parameter for the correspondence between circles and lacunas, the second describes the metric on the circle, which is, roughly, a power of the euclidean metric. We study for which parameters the following features of the gasket can be recovered by the corresponding triple: the integration on the gasket (w.r.t. the Hausdorff measure), a non-trivial distance on the gasket, a non-trivial Dirichlet form (the Kigami energy). 16:45-17:30 Winn, B (Loughborough) Relationship between scattering matrix and spectrum of quantum graphs Sem 1 We investigate the equivalence between spectral characteristics of the Laplace operator on a metric graph, and the associated unitary scattering operator. We prove that the statistics of level spacings, and moments of observations in the eigenbases coincide in the limit that all bond lengths approach a positive constant value. 17:30-17:45 Chernyshev, V (BMSTU) Statistical properties of semiclasssical solutions of the non-stationary Schrödinger equation on metric graphs Sem 1 The talk is devoted to the development of the semiclassical theory on quantum graphs. For the non-stationary Schrödinger equation, propagation of the Gaussian packets initially localized in one point on an edge of the graph is described. Emphasis is placed on statistics behavior of asymptotic solutions with increasing time. It is proven that determination of the number of quantum packets on the graph is associated with a well-known number-theoretical problem of counting the number of integer points in an expanding polyhedron. An explicit formula for the leading term of the asymptotics is presented. It is proven that for almost all incommensurable passing times Gaussian packets are distributed asymptotically uniformly in the time of passage of edges on a finite compact graph. Distribution of the energy on infinite regular trees is also studied. The presentation is based on the joint work with A.I. Shafarevich. 17:45-18:00 Petrillo, T (UCSD) On the Splitting of Primes in Coverings Sem 1 An explicit formula for zeta functions on graphs will be introduced. The result will be extended to L-functions on graphs. An example and possible extensions will be discussed.  Wednesday 28 July 09:00-10:00 Milton, G (Utah) Complete characterization and synthesis of the response function of elastodynamic networks Sem 1 In order to characterize what exotic properties elastodynamic composite materials with high contrast constituents can have in the continuum it makes sense to first understand what behaviors discrete networks of springs and masses can exhibit. The response function of a network of springs and masses, an elastodynamic network, is the matrix valued function W(omega), depending on the frequency omega, mapping the displacements of some accessible or terminal nodes to the net forces at the terminals. We give necessary and sufficient conditions for a given function W(omega) to be the response function of an elastodynamic network assuming there is no damping. In particular we construct an elastodynamic network that can mimic any achievable response in the frequency or time domain. It builds upon work of Camar-Eddine and Seppecher, who characterized the possible response matrices of static three-dimensional spring networks. Authors: F. Guevara Vasquez (University of Utah), G.W. Milton (University of Utah), D.Onofrei (University of Utah) 10:00-10:30 Morning Coffee 10:30-11:15 Lenz, D (Jena) Unbounded Laplacians on graphs: Basic spectral properties and the heat equation Sem 1 We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus on phenomena related to unboundedness of the Laplacians. In particular, we provide criteria for essential selfadjointness, empty essential spectrum and stochastic incompleteness. (Joint work with Matthias Keller). 11:15-12:00 Kovarik, H (Torino) Heat kernel estimates for Laplace operators on metric trees Sem 1 We consider the integral kernel of the semigroup generated by a differential Laplace operator on certain class of infinite metric trees. We will show how the time decay of the heat kernel depends on the geometry of the tree. (This is a joint work with Rupert Frank). 12:00-12:45 Laptev, A (Imperial College London) On some sharp spectral inequalities for Schrödinger operators on graphs Sem 1 12:45-13:00 Onofrei, D (Utah) Characterization of the static response of a two-dimensional elastic network Sem 1 We will show that, in the 2D static case , any positive semidefinte and balanced matrix$W$is the response matrix of a purely elastic planar network. Moreover we will present a constructive proof for the fact that the network we design for a particular response matrix$W$can fit within an arbitrarily small neighborhood of the convex hull of the terminal nodes, provided the springs and masses occupy an arbitrarily small volume. 13:00-14:00 Sandwich Lunch at INI 19:30-22:00 Conference Dinner at Emmanuel College (supported by Meiji Institute for Advanced Study of Mathematical Sciences-MIMS)  Thursday 29 July 09:00-10:00 Grieser, D (Oldenburg) Fat graphs: Variations on a theme Sem 1 A fat graph is, generally speaking, a family of spaces depending on a parameter$\varepsilon$which converge metrically to a metric graph as$\varepsilon\to 0$. The problem of studying the behavior of the spectrum of the Laplacian or other geometric operators, under this limit, arises in various contexts and can be approached by a variety of techniques. In this survey talk I will explain some of these techniques and mention some recent results. 10:00-10:30 Morning Coffee 10:30-11:15 Zuk, A (Paris) On a problem of Atiyah Sem 1 We present constructions of closed manifolds with irrational L2 Betti numbers 11:15-12:00 Law, C-K (National Sun Yat-sen) A Solution to an Ambarzumyan Problem on Trees Sem 1 The classical Ambarzumyan problem states that when the eigenvalues$\lambda_n$of a Neumann Sturm-Liouville operator defined on$[0,\pi]$are exactly$n^2$, then the potential function$q=0$. In 2007, Carlson and Pivovarchik showed the Ambarzumyan problem for the Neumann Sturm-Liouville operator defined on trees where the edges are in rational ratio. We shall extend their result to show that for a general tree, if the spectrum$\sigma(q)=\sigma(0)$, then$q=0\$. In our proof, we develop a recursive formula for characteristic functions, together with a pigeon hole argument. This is a joint work with Eiji Yanagida of Tokyo Institute of Technology. 12:30-13:30 Sandwich Lunch at INI 14:00-14:45 Zuazua, E (Basque) Hardy inequalities and asymptotics for heat kernels Sem 1 We will discuss some Hardy inequalities and its consequences on the large time behavior of diffusion processes. Roughly speaking, the Hardy inequality ensures a further and faster decay rate. Two differente situations will be addressed. First we shall consider the heat equation with a singular square potential located both in the interior of the domain and on the boundary, following a joint work with J. L. Vázquez and a more recent one with C. Cazacu. We shall also present the main results of a recent work in collaboration with D. Krejciírk in which we consider the case of twisted domains. In this case the proof of the extra decay rate requires of important analytical developments based on the theory of self-similar scales. As we shall see, asymptotically, the twisting ends up breaking the tube and adds a further Dirichlet condition, wich eventually produces the increase of the decay rate. Some consequences in which concerns the control of these models will also be presented. References: J. L. Vázquez and E. Zuazua The Hardy inequality and the asymptotic behavior of the heat equation with an inverse square potential. J. Functional Analysis, 173 (2000), 103--153.} J. Vancostenoble and E. Zuazua. Hardy inequalities, Observability and Control for the wave and Schr\"odinger equations with singular potentials, SIAM J. Math. Anal., Volume 41, Issue 4, pp. 1508-1532 (2009) D. Krejcirik and E. Zuazua The heat equation in twisted domains, J. Math pures et appl., to appear. C. Cazacu and E. Z. Hardy inequalities with boundary singular potentials, in preparation. 14:45-15:30 Carlson, R (Colorado) Dirichlet to Neumann Maps for Infinite Metric Graphs Sem 1 Motivated by problems of modeling the human circulatory system, boundary value problems for differential operators -D2 + q are considered on the metric completions of infinite graphs with finite volume, finite diameter, or other smallness conditions. For a large family of graphs, the existence of an ample family of simple test functions permits a generalized definition of the Dirichlet to Neumann map taking boundary functions to their normal derivatives. Properties of this map, problems exhibiting more regular derivatives, and approximation by finite subgraphs will be discussed. 15:30-16:00 Afternoon Tea 16:00-16:45 Eastham, M (Cardiff) The continuous and discrete spectrum of an asymptotically straight leaky wire Sem 1 The approach to quantum graphs developed by Exner and his co-workers is based on a Hamiltonian which contains a singular potential term with a delta-function support on (in two dimensions) a curve C. Here we give conditions on the potential and on the geometry of C under which the associated spectrum is either a semi-infinite interval or the whole real line. The geometry is expressed in terms of a new and simpler concept of asymptotic straightness which does not rely on an asymptotic estimate for the curvature, and which is only imposed on disjoint long sections of C. We also discuss the case where C is a star graph with N rays and the lower spectrum is discrete. We obtain an estimate for the lowest eigenvalue and we contribute to the conjecture that this eigenvalue is maximised for a given N when the star graph is symmetric. A number of open spectral problems related to this work are mentioned. (Joint work with Malcolm Brown and Ian Wood.) 16:45-17:30 Veselic, I (Chemnitz) Discrete alloy type models: averaging and spectral properties Sem 1 We discuss recent results on discrete alloy type models, in particular those with non-monotone parameter dependence. Among others Wegner estimates, averaging techniques, appropriate transformations on the probability space and decoupling properties are dicussed. This is related to the exponential decay of the Green's function and localisation properties. 17:30-17:45 Matter, M (Université de Genève) Limits of self-similar graphs and criticality of the Abelian Sandpile Model Sem 1 We consider covering sequences of (Schreier) graphs arising from self-similar actions by automorphisms of rooted trees. The projective limit of such an inverse system corresponds to the action on the boundary of the tree and its connected components are the (infinite) orbital Schreier graphs of the action. They can be approximated by finite rooted graphs using Hausdorff-Gromov convergence. An interesting example is given by the Basilica group acting by automorphisms on the binary rooted tree in a self-similar fashion. We give a topological as well as a measure-theoretical description of the orbital limit Schreier graphs. In particular, it is shown that they are almost all one-ended with respect to the uniform distribution on the boundary of the tree. We study the statistical-physics Abelian Sandpile Model on such sequences of graphs. The main mathematical question about this model is to prove its criticality -- the correlation between sites situated far away each from the other is high -- what is typically done by exhibiting, asymptotically, a power-law decay of various statistics. In spite of many numerical experiments, the criticality of the model was rigorously proven only in the case of the regular tree. We show that the Abelian Sandpile Model on the limit Schreier graphs of the Basilica group is critical almost everywhere with respect to the uniform distribution on the boundary of the tree.
 Friday 30 July 09:00-09:45 Fulling, SA (Texas A&M University) Index Theorems for Quantum Graphs Sem 1 Work in collaboration with P. Kuchment and J. Wilson 09:45-10:30 Morning coffee 10:30-11:15 Zaidenberg, M (Institut Fourier) Variations on discrete Floquet-Bloch Theory in positive characteristic Sem 1 The classical Floquet theory deals with Floquet-Bloch solutions of periodic PDEs. A discrete version of this theory for difference vector equations on lattices, including the Floquet theory on infinite periodic graphs, was developed by Peter Kuchment. Here we propose a variation of this theory for matrix convolution operators acting on vector functions on lattices with values in a field of positive characteristic. 11:15-12:00 Post, O (Humboldt) Convergence results for thick graphs Sem 1 We will give an overview of convergence results for several natural Laplace-like operators on the thick graph to candidates on the underlying metric graph. Of particular interest are the glueing conditions which can be obtained at a vertex by a pure Laplacian. Moreover we show convergence of the Dirichlet-to-Neumann map of a thick graph with boundary to the corresponding operator of the metric graph. 12:30-13:30 Sandwich Lunch at INI 14:00-14:45 Pavlov, B (Auckland) Dirichlet-to-Neumann techniques for periodic problems Sem 1 Spectral analysis in low energy region will be developed for the Schrodinger operator on periodic multi-dimensional lattices and, in particular a method of estimation of mobility of electron/holes in Silicon=Boron sandvich structures will be suggested. I am able to provide a draft of the paper as soon as you need it. 14:45-15:30 Vainberg, B (UNCC) Necklace graphs and slowing down of the light Sem 1 A possible device for slowing down of the light (propagation of wave packets) will be discussed which is based on periodic branching waveguides. Reduction to a quantum graph with a specific boundary conditions at vertices plays a crucial role. 15:30-16:00 Afternoon tea 16:00-16:45 Keller, M (Jena) Absolutely continuous spectrum for trees of finite forward cone type Sem 1 We study a class of rooted trees which are not necessarily regular but exhibit a lot of symmetries. The spectrum of the corresponding graph Laplace operator is purely absolutely continuous and consists of finitely many intervals. Moreover for trees of the class which are not regular the absolutely continuous spectrum is stable under small perturbations by radially symmetric potentials. (This is joint work with Daniel Lenz and Simone Warzel.) 16:45-17:30 Haeseler, S (Jena) The parabolic Harnack inequality for quantum graphs Sem 1 We consider quantum graphs with Kirchhoff boundary conditions. We study the intrinsic metric, volume doubling and a Poincaré inequality. This enables us to prove a parabolic Harnack inequality. The proof involves various techniques from the theory of strongly local Dirichlet forms.