J. Gil: A conic manifold perspective of elliptic operators on graphs

We give a simple, explicit, sufficient condition for the existence of a sector of minimal growth for second order regular singular differential operators on graphs. We specifically consider operators with a singular potential of Coulomb type and base our analysis on the theory of elliptic cone operators.

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Y. Kopylov: Lp,q-cohomology of warped cylinders

We extend some results by Gol'dshtein, Kuz'minov, and Shvedov about the Lp-cohomology of warped cylinders to Lp,q-cohomology for p unequal q. As an application, we establish some sufficient conditions for the nontriviality of the Lp,q-torsion of a surface of revolution.

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F. Alouges: A preconditioning technique for the resolution of tridimensional electromagnetic scattering problems

In the numerical resolution of tridimensional electromagnetic scattering problems, a rather new technique has been proposed by D. Levadoux to find intrinsically well conditioned integral equations, which are sought from the discretization of perturbations of the identity operators. A strategy, in the heart of which one finds approximations of the Dirichlet to Neumann map by merging local contributions has been developed which works well for smooth surfaces. After presenting the method and numerical results, we will show its behavior when the surface is not regular and the kind of problems we are aware of for improvements.

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T.G. Ayele: On partial hypoellipticity of differential operators

The spaces of basic functions D and S, the spaces of generalized functions (Distributions) S' and D' and a nonzero linear differential operator P(D) with constant coefficients are considered. The notions of convolution, differentiation and Fourier Transform in distributional sense are used. Under certain assumptions it is proved that the convolution of a finite number of generalized functions exists. Using these, partial hypoellipticity of differential operator P(D) is described in terms of its fundamental solutions.

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Y. Chiba: Boundary value problems for degenerate hyperbolic equations

For weakly hyperbolic operators whose characteristic roots degenerate only on the initial hypersurface, we construct microfunctional solutions whose singularities are only either one of the characteristic roots.

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A. Dasgupta: Ellipticity and Fredholmness on Lp(Rn) of Pseudo-Differential Operators with Exit at Infinity

Based on the works of D.Grieme and B.-W.Schulze, we prove that ellipticity and Fredholmness are equivalent for pseudo-differential operators with exit at infinity on Lp(Rn).

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D. Grieser: Thin tubes in mathematical physics, global analysis and spectral geometry.

A thin tube is a family of compact Riemannian manifolds (possibly with boundary) depending on a parameter ε> 0, which looks like an interval times a fixed compact Riemannian manifold scaled to size ε, plus ends of ’size’ ε attached. More generally, several such tubes can be glued together along the ends to form a graph-like structure. We give a survey of problems and results that concern the asymptotics, as ε → 0, of spectral data (eigenvalues, eigenfunctions) of the Laplace operator on a thin tube. Equivalently, one may rescale a thin tube by the factor ε−1 to obtain a ’long cylinder’ as it is often considered for example in the derivation of glueing theorems for spectral invariants in global analysis.

In the graph situation the following problem has been discussed extensively in the mathematical physics community: Find a differential operator on the graph (considered as one-dimensional simplicial complex) whose spectral data determines the asymptotics of the spectral data on a thin tube around this graph! While the answer has been known in the case of Neumann boundary conditions (at the boundary of the tube) for a long time, the case of Dirichlet (or more general) boundary conditions turns out to be much more subtle. I will give an outline of the recent solution of this problem in the case of general boundary conditions.

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E. Hunsicker

Talk 1: A formal overview:
In this talk, I will recall the formal structure of the calculus of pseudodifferential operators on a compact manifold, and how it permits the construction of parametrices for elliptic differential operators. Then I will introduce the b-pseudodifferential calculus of operators on manifolds with (approximately) cylindrical ends as a formal extension of the standard pseudodifferential operator algebra, and explain how this extended calculus permits the construction of parametrices for elliptic b-differential operators on these manifolds.

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Talk 2: A simple example:
In this talk, I will go through the simplest example possible, the first derivative operator on R, as well as some generalizations of this, as a motivation for the rigorous b-calculus definitions we will see in Talk 3.

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Talk 3: Rigorous definitions:
In this talk, I will define the objects that make up the formal structures of the b-calculus we saw in Talk 1, using motivation from the examples we saw in Talk 2. In addition, I will define the b-trace, which will be used in Pat MacDonald's talks.

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A. Kokotov: Compact polyhedral surfaces and determinant of Laplacian

We’ll discuss the spectral theory of compact polyhedral surfaces (Riemann surfaces with flat conformal metric having conical singularities) of an arbitrary genus. The main goal is to study the determinant of the Laplacian as a functional on the space of pairs (X, m), where X is a Riemann surface and m is a conformal flat conical metric on X. For special case of metrics with trivial holonomy an explicit formula for this functional is found.

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C. Martin: Corner operators and applications to elliptic complexes

We consider the algebra of corner pseudodifferential operators as introduced by Schulze. In the corner algebra we construct parametrices for elliptic operators, where the ellipticity is defined as the invertibility of a chain of symbols associated to the singular strata of the manifold with corners. Then we construct parametrices for elliptic complexes of such pseudodifferential operators. We then draw conclusions on the cohomology of complexes, especially finite dimensionality, here realized in corner Sobolev spaces with double weights, and we see that the ellipticity entails a finite index (Euler characteristic) of such a complex. Moreover, we obtain a characterization of harmonic forms as elements of those spaces, of infinite smoothness and also an analogue of the Hodge decomposition

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P. McDonald: Relative determinants of elliptic operators on conic spaces

In these lectures we give an introduction to the theory and application of zeta regularized determinants on conic manifolds. In the first lecture we review the construction of zeta regularized determinants and associated invariants for compact manifolds. We discuss the basic properties of such objects and briefly review applications which have driven the de- velopment of the theory, providing a framework for carrying out the required constructions in the context of singular spaces. In the second lecture we discuss extensions of the basic theory to noncompact and singular objects. In particular, we develop the notion and basic theory of relative zeta functions and relative determinants. In the third and final lecture we develop the theory of relative zeta determinants of Laplace type operators on conic manifolds in more detail, establishing a number of formulae for specific examples.

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T. Østergaard Sørensen: Analytic structure of Coulombic wave functions

We prove that near two-particle coalescence points - say, x1=x2 - any Columbic wavefunction ψ (of an atom or a molecule) can be written ψ=ψ1 + |x1 - x2| ψ2, with ψ1 and ψ2 real analytic. This is joint work with M. and T. Hoffmann-Ostenhof (Vienna) and S. Fournais (Aarhus)

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B.-W. Schulze

We give an introduction into algebras of pseudo-differential operators on manifolds with conical singularities and edges, with the typical degenerate differential operators and parametrices of elliptic elements. A simple example of a manifold with edge is a manifold with boundary, where the closed half axis (the inner normal) is the model cone and the boundary the edge. For edges in general the model cone of local wedges has a non-trivial cross section; then the structures are based on a suitably prepared calculus on manifolds with conical singularities, including conical exits to infinity. Our approach contains as a special case the calculus of boundary value problems with symbols without the transmission property at the boundary. There is a principal symbolic hierarchy, with the "standard" interior symbol and the operator-valued edge symbol, responsible for ellipticity, the construction of parametrices, and the Fredholm property in weighted spaces. The cone and edge algebras contain many interesting substructures, especially ideals of smoothing and non-compact operators affecting the index, and symbolic information (meromorphic operator functions) reflecting asymptotics of solutions to elliptic problems.

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M.W. Wong: Fourier-Wigner Transforms and Liouville\'s Theorems for the Sub-Laplacian on the Heisenberg Group

The sub-Laplacian on the Heisenberg group is first decomposed into twisted Laplacians parametrized by Planck's constant. Then using Fourier-Wigner transforms so parametrized, we prove that the twisted Laplacians are globally hypoelliptic in the setting of tempered distributions. An an application, this result on global hypoellipticity is used to obtain Liouville's theorems for harmonic functions for the sub-Laplacian on the Heisenberg group. (This is joint work with Aparajita Dasgupta.)

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J. Toft: Fourier integral operators with non-smooth symbols

We consider a class of Fourier integral operators with symbols in modulation spaces, or more general, in certain types of coorbit spaces, and with phase functions satisfying certain types of non-degeneracy conditions on their second order derivatives. We give some proof sketches on continuity for such operators, when acting on modulation spaces. Furthermore we establish Schatten properties for such operators.

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J. Wirth: Dispersive estimates for wave equations with time-periodic dissipation

Aim of the short presentation is to announce some new results on dissipative wave equations and the influence of oscillations in coefficients. To be precise, we will show that solutions to damped wave equations with a (a.e. positive) periodic dissipation (of locally bounded variation) satisfy the same Matsumura- type decay estimates like equations with constant dissipation. The result is based on a precise investigation of the corresponding monodromy operator in combination with tools from the theory of special functions / Hill's equation.

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I. Witt: Minicourse "The Cauchy problem for hyperbolic differential operators on edge manifolds"

We shall discuss the Cauchy problem for hyperbolic differential operators on edge manifolds. To establish well-posedness, one needs to work in an adapted scale of function spaces and additionally impose conditions along the edges. These conditions take the place of the familiar boundary conditions. They are formulated in terms of the asymptotics near the edges of the solutions one is seeking. Accordingly, function spaces incorporate a priori knowledge of those asymptotics. We will explain the functional-analytic set-up in detail, in particular at places where ideas from singular analysis enter, and also show that it constitutes the correct framework for establishing energy inequalities.

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