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Mini-courses

Introduction to Pseudodifferential Operators

Abstract: This series of lectures will provide an introduction to pseudodifferential operators. After having defined the concept of oscillatory integral we will focus on the mapping properties of pseudodifferential operators and the corresponding symbolic calculus. In the last four hours we will construct a parametrix for operators with hypoelliptic symbol and we will see the connections between pseudodifferential operator theory and microlocal analysis. Concluding we will give a survey on how this theory can be generalised in the Colombeau framework.

Michael Kunzinger

Global Lorentz Geometry and the Singularity Theorems of General Relativity

Abstract: This series of lectures will provide insights into certain aspects of global Lorentz geometry that have
exerted considerable influence in mathematical physics (general relativity) over the past decades. Based on
global variational methods in Riemannian geometry we present an introduction to the theory of causality in Lorentz manifolds. These in turn will be used to prove the celebrated singularity theorems of Steven Hawking
and Roger Penrose. The idea is to show that mild hypotheses on causality and curvature in Lorentz
manifolds imply the existence of generic singularities. These results have been employed in mathematical
physics to predict singularities of space-time, both on the cosmological scale ("big bang") and in the local context ("black holes").

Michael Oberguggenberger

Stochastic Partial Differential Equations and Generalized Functions

Abstract: This course of lecture is intended to introduce the students to generalized stochastic processes and
applications in stochastic partial differential equations (SPDEs). We start with the construction of processes
with distribution valued paths, suitable for solving linear SPDEs. Next we introduce Colombeau type
stochastic processes, suitable for solving nonlinear SPDEs. Finally, asymptotic and limiting properties
of the solutions will be studied.

Michael Ruzhansky

Pseudo-differential Operators on Lie Groups

Abstract: In this course of lectures we will discuss different quantisations of operators on spaces with
symmetries. The action of a Lie group on the space allows one to introduce a notion of the full symbol
of an operator. We will discuss basic properties of such quantisation and possible applications. In particular,
we will show in detail how this construction works in the case of periodic operators, as well as operators
on the three dimensional sphere. Finally, we will discuss open problems and challenges of this theory.

Roland Steinbauer

Normally Hyperbolic Operators

Abstract: Building on a global approach to differential operators on vector bundles we discuss normally hyperbolic
operators. These generalize the notion of wave operators on Lorentzian manifolds and are second order
operators with their principal symbol given by (minus) the metric. We prove the Weitzenböck formula
which says that up to an order zero term every normally hyperbolic operator is the connection
d'Alembertian for a suitable connection of the vector bundle. Finally we present some recent results
in the low regularity case.

Hans Vernaeve

Nonstandard Principles in Colombeau Theory

Abstract: Already in the 1992 book by Michael Oberguggenberger, it is noted that the Colombeau algebras allow to model microscopic or infinitesimal behaviour of (generalized) functions, and that this is much in the spirit of nonstandard analysis. We introduce the nonstandard-like objects from the above-mentioned book, explain which principles they satisfy, and illustrate how they can be used to solve problems in Colombeau theory. Moreover, those principles provide a conceptual way to deal with Colombeau generalized functions, avoiding to turn to representatives each time one wants to show a statement about them.

Sanja Konjik

Introductory Part: Foundations of Differential Geometry

Abstract: We recall basic notions from differential geometry such as manifolds, tangent and cotangent bundles, vector bundles, vector fields and 1-forms, tensors, semi-Riemannian manifolds, the Levi-Civita connection, parallel transport, geodesics, exponential map, curvature.

Short contributions

Stevan Pilipović

Jasson Vindas

Dimitris Scarpalezos

Dora Seleši

Donal Connolly

Nenad Teofanov

Endre Süli

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Mini-courses Claudia Garetto Introduction to Pseudodifferential Operators Abstract: This series of lectures will provide an introduction to pseudodifferential operators. After having defined the concept of oscillatory integral we will focus on the mapping properties of pseudodifferential operators and the corresponding symbolic calculus. In the last four hours we will construct a parametrix for operators with hypoelliptic symbol and we will see the connections between pseudodifferential operator theory and microlocal analysis. Concluding we will give a survey on how this theory can be generalised in the Colombeau framework. Michael Kunzinger Global Lorentz Geometry and the Singularity Theorems of General Relativity Abstract: This series of lectures will provide insights into certain aspects of global Lorentz geometry that have exerted considerable influence in mathematical physics (general relativity) over the past decades. Based on global variational methods in Riemannian geometry we present an introduction to the theory of causality in Lorentz manifolds. These in turn will be used to prove the celebrated singularity theorems of Steven Hawking and Roger Penrose. The idea is to show that mild hypotheses on causality and curvature in Lorentz manifolds imply the existence of generic singularities. These results have been employed in mathematical physics to predict singularities of space-time, both on the cosmological scale ("big bang") and in the local context ("black holes"). Michael Oberguggenberger Stochastic Partial Differential Equations and Generalized Functions Abstract: This course of lecture is intended to introduce the students to generalized stochastic processes and applications in stochastic partial differential equations (SPDEs). We start with the construction of processes with distribution valued paths, suitable for solving linear SPDEs. Next we introduce Colombeau type stochastic processes, suitable for solving nonlinear SPDEs. Finally, asymptotic and limiting properties of the solutions will be studied. Michael Ruzhansky Pseudo-differential Operators on Lie Groups Abstract: In this course of lectures we will discuss different quantisations of operators on spaces with symmetries. The action of a Lie group on the space allows one to introduce a notion of the full symbol of an operator. We will discuss basic properties of such quantisation and possible applications. In particular, we will show in detail how this construction works in the case of periodic operators, as well as operators on the three dimensional sphere. Finally, we will discuss open problems and challenges of this theory. Roland Steinbauer Normally Hyperbolic Operators Abstract: Building on a global approach to differential operators on vector bundles we discuss normally hyperbolic operators. These generalize the notion of wave operators on Lorentzian manifolds and are second order operators with their principal symbol given by (minus) the metric. We prove the Weitzenböck formula which says that up to an order zero term every normally hyperbolic operator is the connection d'Alembertian for a suitable connection of the vector bundle. Finally we present some recent results in the low regularity case. Hans Vernaeve Nonstandard Principles in Colombeau Theory Abstract: Already in the 1992 book by Michael Oberguggenberger, it is noted that the Colombeau algebras allow to model microscopic or infinitesimal behaviour of (generalized) functions, and that this is much in the spirit of nonstandard analysis. We introduce the nonstandard-like objects from the above-mentioned book, explain which principles they satisfy, and illustrate how they can be used to solve problems in Colombeau theory. Moreover, those principles provide a conceptual way to deal with Colombeau generalized functions, avoiding to turn to representatives each time one wants to show a statement about them. Sanja Konjik Introductory Part: Foundations of Differential Geometry Abstract: We recall basic notions from differential geometry such as manifolds, tangent and cotangent bundles, vector bundles, vector fields and 1-forms, tensors, semi-Riemannian manifolds, the Levi-Civita connection, parallel transport, geodesics, exponential map, curvature. Short contributions Stevan Pilipović Jasson Vindas Dimitris Scarpalezos Dora Seleši Donal Connolly Nenad Teofanov Endre Süli

	Mini-courses Introduction to Pseudidifferential Operators, Global Lorentz Geometry and the Singularity Theorems of General Relativity, Stochastic Partial Differential Equations and Generalized Functions, Pseudo-differential Operators on Lie Groups, Normally Hyperbolic Operators, Nonstandard Principles in Colombeau Theory	Organizers WUS Austria Department of Mathematics and Informatics, Novi Sad Serbian Academy of Sciences and Arts Ministry of Science and Technological Development Provincial Secretariat for Science and Technological Development	Contact pilipovic@dmi.uns.ac.rs sanja.konjik@dmi.uns.ac.rs

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