Summer School 2019

Die Summer School Multiscale Phenomena in Geometry and Dynamics findet vom 22. bis 26. Juli 2019 statt. Sie richtet sich an junge Forscher, die sich mit den Bereichen Dynamik, Analyse, Geometrie und verwandten Feldern beschäftigen.

Multiscale Phenomena in Geometry and Dynamics

Die Summer School Multiscale Phenomena in Geometry and Dynamics an der Fakultät für Mathematik der Technischen Universität München richtet sich an Master-Studierende, Doktorand*innen und Postdocs, die sich für multiskale Methoden für normale und/oder partiale Differentialgleichungen interessieren. Dabei deckt sie ein breites Spektrum an aktuellen Themen ab.

Wir freuen uns, vier renommierte Mathematiker aus den Bereichen Geometriedynamik und Multiskalenanalyse an der Technischen Universität München begrüßen zu dürfen. Jeder hält vier Vorträge zu einem bestimmten Fachgebiet, wie unten aufgeführt.

Datum

Die Summer School 2019 findet vom 22. bis 26. Juli 2019 statt.

Veranstaltungsort

Insitute for Advanced Study (IAS), 
TUM - Garching Forschungszentrum
Lichtenbergstraße 2a
85748 Garching bei München

Bewerbung 

Bitte bewerben Sie sich mit unserem Anmeldeformular unten bis zum 23. Juni 2019 (extended deadline!) um einen Platz in der Summer School. 

Dank der großzügigen Unterstützung des Sonderforschungsbereichs "Diskretisierung in Geometrie und Dynamik", der International School of Applied Mathematics (ISAM) und des Elite-Programms TopMath gibt es keine Teilnahmegebühr.

Kontakt

Administrative Organisation

Diane Clayton-Winter

Diane Clayton-Winter
SFB TRR109

clayton (at) ma.tum.de

 

Bewerbungen

Katja Kröss

Katja Kröss
TopMath

gradoffice (at) ma.tum.de

 

Wissenschaftliche Organisation

Prof. Christian Kühn

Sprecher und Themen der Summer School 2019

Andrea Braides, Università di Roma "Tor Vergata": Geometric Flows on Lattices

Daniel Grieser, Universität Oldenburg: Scales, blow-up and quasimode constructions

Tere M-Seara, Universitat Politècnica de Catalunya: Exponentially small splitting of separatrices

Peter Szmolyan, Technische Universität Wien: Advances in Geometric Singular Perturbation Theory

Andrea Braides: Geometric Flows on Lattices

This course will focus on a class of problems that are a prototype for variational evolution in heterogeneous media, using tools of Homogenization Theory, Gradient Flows in Metric Spaces, Geometric Measure Theory, and Discrete-to-Continuum Analysis.

I will consider simple energies on spin functions on lattices inspired by problems in Statistical Mechanics and Image Reconstruction. As we consider an increasing number of nodes (or, equivalently, we scale the lattice by a small parameter) the overall behaviour of such energies (continuum approximation) from a `static standpoint' (described by a Gamma-limit) is that of an interface in the continuum between regions of constant ``magnetization'’. This is a very basic example of Homogenization.

Gradient-flow type evolutions of surface energies are geometric flows such as Mean Curvature Flow. The question is whether the geometric flows related to the continuum approximations describe an evolution at the discrete level (in a sense, if the passage discrete-to-continuum `commutes' with the gradient flow).

I will then give a notion of variational evolution for lattice energies on varying lattices and describe some features of the limit flow in some examples. We will see that the continuum static approximation of the energies must be corrected to obtain the right continuum approximation of the discrete evolution.

Summer School 2019

Daniel Grieser: Scales, blow-up and quasimode constructions

I will explain how the language of manifolds with corners and blow-up can be useful, and indeed is very natural, in dealing with certain types of problems involving different scales, for example for quasimode constructions for ’thin’ domains having non-uniform different scaling behavior. The first part of the course will introduce the basic language and machinery, then some applications will be presented.

Slides to the talks of Daniel Grieser

Tere M-Seara: Exponentially small splitting of separatrices: examples and techniques

In this course we will review a classical problem in perturbation theory. We will consider a system having some hyperbolic manifold (point, periodic orbit, torus etc) with stable and unstable manifolds which coincide and study how they split when perturbing the system. In the regular case, there is a well established  method, known as Melnikov-Poincaré method, which provides an asymptotic formula which measures the difference of the manifolds in a suitable section in terms of the perturbing parameter.

We will study the so-called singular case, where the system is degenerate and this formula is not always valid. We will focus in analytic systems where the splitting is exponentially small in the perturbing parameter. We will show some famous examples where this phenomenon occurs like the Hopf zero-singularity, the restricted three body problem or the problem of Arnold diffusion in the so-called a priori-stable systems. Then we will give some ideas of how can prove a formula which gives the asymptotic measure of the splitting of separatrices in these cases.

Peter Szmolyan: Advances in Geometric Singular Perturbation Theory

Modelling of problems from natural sciences, engineering and life sciences by ordinary differential equations often leads to singular perturbation problems with solutions varying on several widely separated time-scales. The analysis of such systems by methods from dynamical systems theory  - most notably from invariant manifold theory - has become known as geometric singular perturbation theory (GSPT). By the efforts of many people GSPT has been applied successfully in the analysis of an impressive collection of diverse problems. Fenichel theory for normally hyperbolic critical manifolds combined with the blow-up method at non-hyperbolic points is often able to provide remarkably detailed insight into complicated dynamical phenomena. Much of this work has been carried out in the framework of slow-fast systems in standard form, i.e. for systems with an apriori splitting into slow  and fast variables.

More recently GSPT turned out to be useful for systems for which the slow-fast structures and the resulting applicability of GSPT are somewhat hidden. Problems of this type include singularly perturbed systems in non-standard form, problems depending singularly on more than one parameter, and smooth systems limiting on non-smooth systems as a parameter tends to zero. 

In this course I will survey these developments and explain key features in the context of selected applications.

Summer School 2019: Programm

Das ist der Ablaufplan der Summer School "Multiscale Phenomena in Geometry and Dynamics" vom 22. bis 26. Juli 2019.

Veranstaltungsort

Alle Vorlesungen finden im Institute for Advanced Study (IAS) auf dem TUM Campus des Forschungszentrums Garching statt.

 

Social Events

Biergarten: Wir gehen nach dem letzten Vortrag am Montagabend gemeinsam vom Campus aus zum Mühlenpark Biergarten in Garching (ca. 2,5 km, 30 Minuten Fußweg).

Conference Dinner: In der Gruppe fahren wir von Garching mit der U-Bahn zum Augustiner Bierkeller, Arnulfstr. 52, 80335 München, wo um 18:30 Uhr für uns reserviert ist.

Unterstützung bei der Unterkunft

Interessierten Bewerbern stellen wir kostenfrei eine begrenzte Anzahl von Doppelzimmern für je zwei Personen von Sonntag/Montag bis Freitag im Hoyacker Hof in Garching zur Verfügung. Wenn Sie dort eine Unterkunft beantragen möchten, geben Sie dies bitte bei Ihrer Bewerbung an und laden Sie ein Motivationsschreiben hoch.

Organisation

Christian Kühn und Marco Cicalese sind die wissenschaftlichen Organisatoren der Summer School "Multiscale Phenomena in Geometry and Dynamics". Ansprechpartnerinnen für verwaltungstechnische Fragen und alle anderen Belange sind Katja Kröss, TopMath Koordinatorin, oder Diane Clayton-Winter, Koordinatorin des DFG SFB TRR109. Kontaktieren Sie uns unter gradoffice (at) ma.tum.de.

Wir danken dem Sonderforschungsbereich "Diskretisierung in Geometrie und Dynamik", der International School of Applied Mathematics (ISAM) und dem Elite-Programms TopMath für die großzügige Unterstützung der Summer School 2019.