# Prosjekt- og masteroppgaver for Alexander Schmeding

Prosjektforslagene nedover beskrives på engelsk siden faglitteraturen er på engelsk. (english see below)

Det er ingen krav å skrive oppgaven på engelsk eller å diskutere på engelsk (jeg snakker norsk).

Prosjektene som beskrives nedover er vanligvis tenkt for masterstudenter og er for vanskelig for Bachelor prosjekter (men det kan gå ann å gjøre noe enklere som bachelor prosjekt i samme retning). Selvfølgelig skal vi tilpasse prosjektene for dere og det dere kan.

### Generell informasjon

My work concerns aspects of infinite-dimensional differential geometry and Lie theory. This topic might seem to be abstract and of little practical relevance. However, nothing could be further from the truth! There are many, often surprising, connections between finite-dimensional phenomena and infinite-dimensional geometry.

As an example, consider the problem to forecast the weather. You might have experienced that the weather forecast (especially in Trondheim) is often unreliable. In fact, it is practically impossible to accurately forecast the weather for longer than just a few days. The reason is that that the equations governing weather models are sensitive to errors. They amplify any initial error from the observation data. The mathematical proof for this employs infinite-dimensional geometry!

For more examples and some basic objects from differential geometry look at my book on infinite-dimensional geometry (not a prerequisite for the projects). In case you prefer a video lecture on why you should care about infinite-dimensional geometry:

Here are some keywords of structures and objects I am interested in (most projects will involve only one or two items from the list and we will adapt them to your background and knowledge once we agree on a project).

- infinite-dimensional Lie groups (e.g. diffeomorphism groups)
- weak and strong Riemannian metrics
- manifolds of mappings
- shape spaces (geometric properties of these spaces, check joint article with E. Celledoni on shape spaces and computer animation )
- differential equations on infinite-dimensional manifolds
- geometric hydrodynamics
- geometry of rough paths

#### Recommended courses/knowledge

Almost all of my projects involve manifolds. Thus I recommend taking

The course MA3402 Differential Forms on Manifolds is useful, but not necessary for many of the projects.

For students interested in working with Lie groups I recommend MA3407 Introduction to Lie theory

Courses which are potentially useful (depending on master thesis topic), but not a must:

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#### Bachelor projects

Could consist of getting to know the basic ideas and some tools for an aspect of the above topics. Example: Learn what a Riemannian metric is. Then work out some examples for infinite-dimensional Riemannian manifolds.

#### Master projects

For the master projects, you find some concrete project proposals below. They are non-exclusiv and I am happy to discuss variations and proposals for projects. Get in touch if you want to discuss these or other projects connected to my research interests.

### Constructions on manifolds with boundary

Often in analysis we care about spaces which are subsets of euclidean space with additional structure. These special subsets are often manifolds (as are introduced for example in the "Differential topology" course. Manifolds allow us to talk about differentiability of functions defined on them. However, one often wants to relax the definition of a manifold. One possibility is to consider points on the manifold which are boundary points. The mental picture one should have is to consider a half space in 2D-space. Then the points on the boundary of the half-space are exactly the boundary points. Especially for applications in numerical analysis and to PDE in hydrodynamics it is important to study manifolds of mappings between manifolds *with boundary*. While there are some classical works, especially for smooth and differentiable functions, the boundary case is often not dealt with in the literature. The master projects described below therefore will be concerned with filling in the gaps in the literature. Note that they are independent of each other,

##### 1. Constructions for manifolds of mappings

Manifolds of mappings are infinite-dimensional manifolds constructed from sets of regular (=continuous, often differentiable) mappings between manifolds. For example, we can look at all smooth mappings from one manifold to another. The resulting space can be turned into an infinite-dimensional manifold. These objects appear in a variety of relevant applications such as hydrodynamics and shape analysis. In applications one wants to study these spaces on manifolds with boundary and needs several geometric constructions on the manifold to obtain the properties one wants. In particular, we shall

- construct explicitely charts and geometric objects at the boundary
- exploit these results to construct Sobolev functions between manifolds with boundary and establish some basic properties for these spaces.

The second point will depend on the progress with the first one. It generalises well known results in the theory of Sobolev functions on manifolds without boundary.

*Recommended reading (parts of the following)*:

- Lee, J. Introduction to smooth manifolds 2012
- Inci, H. ; Kappeler, T. ; Topalov, P. On the regularity of the composition of diffeomorphisms Memoirs of the AMS, 2013, Vol.226 (1062), p.1-72

##### 2. Manifolds with polyhedral boundary

In many problems, one would like to consider manifolds whose boundary is "rough". As an example, consider the unit cube in euclidean space (see some pictures below). Since the boundary has corners and edges, this set can not be modelled as a manifold with smooth boundary. However, it is well known that the cube is a "manifold with corners". Still, manifolds with corners are often too restricted and one would like to be able to treat even more general boundary terms. Recently Glöckner introduced the notion of a locally polyhedral manifold, which is modelled on (open) subsets of polyhedra. We will investigate this notion and its generalisation and develop basic tools from differential geometry in this setting.

*Recommended reading (parts of the following)*:

- Knapp A.W.: Stokes’s theory and Whitney manifolds, 2021
- Glöckner, H.: Diffeomorphism groups of convex polytopes, 2022

### Geometry on manifolds of mappings with infinite-dimensional target

The set of smooth mappings between two manifolds appears naturally in many constructions and applications. For example, smooth maps from the unit circle with values in a manifold are called loops and the resulting space of mappings is called a "loop space. These spaces appear naturally in physics applications or in shape analysis. One can endow the space of smooth mappings with a suitable topology and even an (infinite-dimensional) manifold structure which turns natural operations between these manifolds of mappings into smooth mappings. These constructions are well known for finite dimensional manifolds, but one may wonder what happens if one considers a manifold of mappings whose mappings take values in an infinite dimensional manifold. As long as the manifold we are mapping out of stays finite dimensional, the function space topologies still make sense (this was the main result of an earlier project with one of my students) and it is well known that the construction of the manifold structure in special cases (such as a compact source manifold).

There are several important results which one would like to generalise to the more general setting. For example, the so called **Stacey-Roberts Lemma** asserts that a submersion between *finite dimensional* manifolds gives rise to a submersion of the manifolds of mappings.
I have a few ideas as to how to generalise this statement to the case of a submersion between infinite dimensional manifolds (which the project will explore).

Other possibilities would be to construct the manifold structure for the smooth mappings on a **non-compact** manifold taking values in an infinite-dimensional manifold.

##### Recommended reading (parts of)

- Hjelle, E.O. and Schmeding, A.: Strong topologies for spaces of smooth maps with infinite-dimensional target, Expo. Math. 35 no. 1 (2017), pp. 13-53
- Michor, P.W.: Manifolds of differentiable mappings, Shiva press 1980