# Summer School 2019

**Multiscale Phenomena in Geometry and Dynamics**takes place from

**22 to 26 July 2019**. It is aimed at young researchers studying or researching in the fields of dynamics, analysis, geometry, and related fields.

## Multiscale Phenomena in Geometry and Dynamics

The summer school **Multiscale Phenomena in Geometry and Dynamics** at the Department of Mathematics of the Technical University Munich is aimed at Master's students, PhD students and postdocs whose interests are related to the topics of multiscale methods for ordinary and/or partial differential equations. It covers a broad spectrum of topics of current interest in the area.

We are delighted to welcome four renowned mathematicians in the fields of geometric dynamics and multiscale analysis to the Technical University Munich. Each will present four lectures on a certain area of their expertise, as listed below.

#### Date

The summer school takes place from **22 - 26 July 2019.**

#### Location

Insitute for Advanced Studies (IAS),

TUM - Garching Forschungszentrum

Lichtenbergstraße 2a

85748 Garching bei München

#### Application

Please apply for a place at the Summer School using our webform below before **23rd June 2019 **(extended deadline!).

There is no participation fee thanks to the generous support of the Collaborative Research Center "Discretization in Geometry and Dynamics", the International School of Applied Mathematics (ISAM) and the Elite Program TopMath.

### Contact

### Administrative coordination

### Application

### Scientific organization

## Speakers and topics of the 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.

#### 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.

Slides to the talks of Tere M-Seara

#### 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.

Slides to the talks of Peter Szmolyan

## Summer School 2019: Schedule

This is the schedule of the Summer School "Multiscale Phenomena in Geometry and Dynamics" from 22 to 26 July 2019.

#### Locations

All lectures take place at the Institute for Advanced Study (IAS) on the TUM Campus, Forschungszentrum Garching.

#### Social Events

**Biergarten:** We will walk together to the Mühlenpark Biergarten in Garching (approx. 2,5km, 30 minutes walk) leaving the campus after the last lecture on Monday evening.

**Conference Dinner:** Group travel from Garching with the U-Bahn to the Augustiner Bierkeller, Arnulfstr. 52, 80335 München, where our group has a reservation for 18:30.

## Accomodation support

Accommodation support is available for a limited number of applicants in shared twin rooms at the Hoyacker Hof in Garching from Sunday/Monday to Friday. Should you wish to apply for accommodation support, please indicate this on your online application and upload a letter of motivation accordingly.

## Organization

Christian Kühn & Marco Cicalese are the scientific organizers of the Summer School "Multiscale Phenomena in Geometry and Dynamics". Contact persons for administrative questions and all enquiries are Katja Kröss, TopMath coordinator, and Diane Clayton-Winter, SFB TRR109 coordinator. Please contact via email under gradoffice@ma.tum.de.

We thank the Collaborative Research Center "Discretization in Geometry and Dynamics", the International School of Applied Mathematics (ISAM) and the Elite Program TopMath for the generous support of the Summer School.