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Timetable (AGAW06)

Analysis on Graphs and its Applications Follow-up

Monday 26th July 2010 to Friday 30th July 2010

Monday 26th July 2010
08:30 to 08:55 Registration
08:55 to 09:00 Welcome from Sir David Wallace (INI Director)
09:00 to 10:00 B Davies ([Kings])
Non-Weyl Asymptotics for Resonances of Quantum Graphs
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 \[ N(r)=\#\{ \mbox{ resonances $\lambda=k^2$ of $H$ such that $|k|<r$}\} \] one should regard eigenvalues of $H$ as special kinds of resonance.

One might hope that $N(r)$ obeys the same leading order asymptotics as $r\to\infty$ as in the case of ${\cal G}_0$, but this is not always the case. A Pushnitski and EBD have proved the following theorem, whose proof will be outlined in the lecture.

Theorem 1 The resonances of $H$ obey the Weyl asymptotic law if and only if the graph ${\cal G}$ does not have any balanced vertex. If there is a balanced vertex then one still has a Weyl law, but the effective volume is smaller than the volume of ${\cal G}_0$.

10:00 to 10:30 Morning Coffee
10:30 to 11:15 R Band ([Weizmann])
Scattering from isospectral graphs.
In 1966 Marc Kac asked 'Can one hear the shape of a drum?'. The answer was given only in 1992, when Gordon et al. found a pair of drums with the same spectrum. The study of isospectrality and inverse problems is obviously not limited to drums and treats various objects such as molecules, quantum dots and graphs. In 2005 Okada et al. conjectured that isospectral drums can be distinguished by their scattering poles (resonances). We prove that this is not the case for isospectral quantum graphs, i.e., isospectral quantum graphs share the same resonance distribution. This is a joint work with Adam Sawicki and Uzy Smilansky.
11:15 to 12:00 G Berkolaiko ([Texam A and M University])
Nodal domains and spectral critical partitions on graphs
The $k$-th eigenfunction of a Schrodinger operator on a bounded regular domain $\Omega$ with Dirichlet boundary conditions defines a partition of $\Omega$ into $n$ nodal subdomains. A famous result by Courant establishes that $n \leq k$; the number $k-n$ will be referred to as the nodal deficiency. The nodal subdomains, when endowed with Dirichlet boundary conditions, have equal first eigenvalue, which coincides with the $k$-th eigenvalue of the original Schrodinger problem. Additionally, the partition is bipartite, i.e. it consists of positive and negative subdomains (corresponding to the sign of the eigenfunction), with two domains of the same sign not sharing a boundary. Conversely, for a given partition, define the energy of the partition to be the largest of the first Dirichlet eigenvalues of its subdomains. An $n$-partition with the minimal energy is called the minimal $n$-partition. It is interesting to relate the extremal properties of the partitions to the eigenstates of the operator on $\Omega$. Recently, Helffer, Hoffmann-Ostenhof and Terracini proved that $n$-th minimal partition is bipartite if and only if it corresponds to a Courant-sharp eigenfunction (an eigenfunction with nodal deficiency zero). We study partitions on quantum graphs and discover a complete characterization of eigenfunctions as critical equipartitions. More precisely, equipartitions are partitions with all first eigenvalues equal. We parameterize the manifold of all equipartitions and consider the energy of an $n$-equipartition as a function on this manifold. For a generic graph and large enough $n$ we establish the following theorem: a critical point of the energy function with $b$ unstable directions is a bipartite equipartition if and only if it corresponds to an eigenfunction with nodal deficiency $b$. Since by constructions it has $n$ nodal domains it is therefore the $n+b$-th eigenfunction in the spectral sequence. Since at a minimum the number of unstable directions is $b=0$, our results include the quantum graph analogue of the results of Helffer et al. They also provide a new formulation of known bounds on the number of nodal domains on generic graphs. This is joint work with Rami Band, Hillel Raz and Uzy Smilansky.
12:30 to 13:30 Sandwich Lunch at INI
14:00 to 14:45 J Keating ([Bristol])
Quantum statistics on Graphs
I will discuss possible quantum exchange statistics in the case of graphs
14:45 to 15:30 J Bolte ([Royal Holloway])
Many-particle systems on quantum graphs with singular interactions
Single quantum particles on graphs have proven to provide interesting models of complex quantum systems; their spectral properties have been studied in great detail. In this talk we discuss extensions to quantum many-particle systems on graphs with singular interactions. We focus on two-particle interactions that are either localised at the vertices, or are of Dirac-delta type on the edges. In both cases the interactions are realised in terms of self-adjoint extensions of suitable Laplacians in two variables. These extensions can be characterised in terms of boundary conditions, and given particular boundary conditions the type of interactions can be identified. (This talk is based on joint work with Joachim Kerner.)
15:30 to 16:00 Afternoon Tea
16:00 to 16:45 S Avdonin ([Alaska])
Recursive Algorithms Solving Inverse Problems on Quantum Graphs
In this talk we describe a new approach to solving boundary inverse problems on quantum graphs. This approach is based on the Boundary Control method and combines the spectral and dynamical approaches to inverse problems on graphs. It was proposed in [1] for the Schr\"odinger equation with standard matching conditions and was extended in [2] to the two-velocity wave equation. Since the number of edges of graphs arising in applications is typically very big, we propose a recursive procedure which may serve as a base for developing effective numerical algorithms. For trees, this procedure allows recalculating efficiently the inverse data from the original tree to the smaller trees, `removing' leaves step by step up to the rooted edge. Numerical tests for inverse problems are impossible without producing accurate inverse data. This means that we have to have reliable numerical algorithms for solving the direct Problems --- given the coefficients of equations and the graph topology find its spectral (and dynamical) data. Even for the simplest graph --- a finite interval or the semi-axis --- this is a rather difficult problem from the numerical point of view. The surprising fact is that to solve numerically, say, the Gelfand--Levitan equation and find the potential from the given spectral function is much easier than to find the spectral function from the given potential. For graphs with many edges these difficulties increase dramatically. Therefore, at the moment there are no efficient algorithms for, or numerical experiments in, solving inverse problems on graphs. Based on the results of [3], we propose a way to reduces the `direct' problem to solving second kind Volterra integral equations. 1. S. Avdonin and P. Kurasov, Inverse problems for quantum trees,} Inverse Problems and Imaging, {2} (2008), 1--21. 2. S. Avdonin, G. Leugering and V. Mikhaylov, On an inverse problem for tree-like networks of elastic strings}, Zeit. Angew. Math. Mech., {90} (2010), 136--150. 3. S. Avdonin, V. Mikhaylov and A.Rybkin, The boundary control approach to the Titchmarsh-Weyl $m-$function,} Comm. Math. Phys., {275} (2007), 791--803.
16:45 to 17:30 K M Schmidt (Cardiff University)
The HELP Inequality on Trees
This is joint work with BM Brown and M Langer. We establish analogues of Hardy and Littlewood's integro-differential equation for Schroedinger-type operators on metric and discrete trees, based on a generalised strong limit-point property of the graph Laplacian.
17:30 to 17:45 R Rueckriemen (Dartmouth College)
The Floquet spectrum of a quantum graph
We define the Floquet spectrum of a quantum graph as the collection of all spectra of operators of the form $D=(-i\frac{\partial}{\partial x}+\alpha(\frac{\partial}{\partial x}))^2$ where $\alpha$ is a closed $1$-form. We show that the Floquet spectrum completely determines planar 3-connected graphs (without any genericity assumptions on the graph). It determines whether or not a graph is planar. Given the combinatorial graph, the Floquet spectrum uniquely determines all edge lengths of a quantum graph.
17:45 to 18:00 J Lipovsky ([Charles University in Prague])
The Absence of Absolutely Continuous Spectra for Radial Tree Graphs
We will introduce a family of Schrödinger operators on tree graphs with coupling conditions given by (b_n-1)^2+4 real parameters where b_n is the branching number. We will show the unitary equivalence of the Hamiltonian on the tree graph and the orthogonal sum of the Hamiltonians on the halflines. We will use this unitary equivalence to prove that for a large family of coupling conditions there is no absolutely continuous spectrum of the Hamiltonian on the sparse tree. On the other hand, we will show nontrivial examples of trees with the spectrum which is purely absolutely continuous.
18:00 to 19:00 Welcome Wine Reception (supported by Meiji Institute for Advanced Study of Mathematical Sciences-MIMS)
Tuesday 27th July 2010
09:00 to 10:00 U Smilansky (Weizmann Institute of Science)
Periodic walks on random graphs and random matrix theory
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 to 10:30 Morning Coffee
10:30 to 11:15 J Harrisson ([Baylor])
Properties of zeta functions of quantum graphs
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 to 12:00 A Terras ([San Diego])
Explicit Formulas for Zeta Functions of Graphs
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 to 13:30 Sandwich Lunch at INI
14:00 to 14:45 A Teplyaev ([Connecticut])
Uniqueness of Laplacian and Brownian motion on Sierpinski carpets
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 to 15:30 T Isola ([tbc])
Line integrals of one-forms on the Sierpinski gasket
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 to 16:00 Afternoon Coffee
16:00 to 16:45 D Guido ([Roma])
Some (noncommutative) geometrical aspects of the Sierpisnki gasket
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 to 17:30 B Winn ([Loughborough])
Relationship between scattering matrix and spectrum of quantum graphs
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 to 17:45 V Chernyshev ([BMSTU])
Statistical properties of semiclasssical solutions of the non-stationary Schrödinger equation on metric graphs
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 to 18:00 T Petrillo ([UCSD])
On the Splitting of Primes in Coverings
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 28th July 2010
09:00 to 10:00 G Milton ([Utah])
Complete characterization and synthesis of the response function of elastodynamic networks
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 to 10:30 Morning Coffee
10:30 to 11:15 D Lenz ([Jena])
Unbounded Laplacians on graphs: Basic spectral properties and the heat equation
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 to 12:00 H Kovarik ([Torino])
Heat kernel estimates for Laplace operators on metric trees
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 to 12:45 A Laptev (Imperial College London)
On some sharp spectral inequalities for Schrödinger operators on graphs
12:45 to 13:00 D Onofrei ([Utah])
Characterization of the static response of a two-dimensional elastic network
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 to 14:00 Sandwich Lunch at INI
19:30 to 22:00 Conference Dinner at Emmanuel College (supported by Meiji Institute for Advanced Study of Mathematical Sciences-MIMS)
Thursday 29th July 2010
09:00 to 10:00 D Grieser ([Oldenburg])
Fat graphs: Variations on a theme
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 to 10:30 Morning Coffee
10:30 to 11:15 A Zuk ([Paris])
On a problem of Atiyah
We present constructions of closed manifolds with irrational L2 Betti numbers
11:15 to 12:00 C-K Law ([National Sun Yat-sen])
A Solution to an Ambarzumyan Problem on Trees
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 to 13:30 Sandwich Lunch at INI
14:00 to 14:45 E Zuazua ([Basque])
Hardy inequalities and asymptotics for heat kernels
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 to 15:30 R Carlson ([Colorado])
Dirichlet to Neumann Maps for Infinite Metric Graphs
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 to 16:00 Afternoon Tea
16:00 to 16:45 M Eastham ([Cardiff])
The continuous and discrete spectrum of an asymptotically straight leaky wire
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 to 17:30 I Veselic ([Chemnitz])
Discrete alloy type models: averaging and spectral properties
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 to 17:45 M Matter (Université de Genève)
Limits of self-similar graphs and criticality of the Abelian Sandpile Model
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 30th July 2010
09:00 to 09:45 SA Fulling (Texas A&M University)
Index Theorems for Quantum Graphs
Work in collaboration with P. Kuchment and J. Wilson
09:45 to 10:30 Morning coffee
10:30 to 11:15 M Zaidenberg ([Institut Fourier])
Variations on discrete Floquet-Bloch Theory in positive characteristic
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 to 12:00 O Post ([Humboldt])
Convergence results for thick graphs
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 to 13:30 Sandwich Lunch at INI
14:00 to 14:45 B Pavlov ([Auckland])
Dirichlet-to-Neumann techniques for periodic problems
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 to 15:30 B Vainberg ([UNCC])
Necklace graphs and slowing down of the light
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 to 16:00 Afternoon tea
16:00 to 16:45 M Keller ([Jena])
Absolutely continuous spectrum for trees of finite forward cone type
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 to 17:30 S Haeseler ([Jena])
The parabolic Harnack inequality for quantum graphs
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.
University of Cambridge Research Councils UK
    Clay Mathematics Institute The Leverhulme Trust London Mathematical Society Microsoft Research NM Rothschild and Sons