### Research team

### Expertise

1) Integrable Hamiltonian systems and their singularities (elliptic, hyperbolic, focus-focus) and bifurcation behavior; interaction with Hamiltonian S^1-actions. 2) Symplectic geometry, Floer theory and its applications to symplectic and contact dynamics (homoclinic points, growth behaviour). 3) Hyperkähler Floer theory and associated Hamiltonian PDEs on Hilbert spaces; Bubbling-off analysis; non-squeezing etc. 4) Morse theory and its application to n-categories and opetopes. 5) Optimal transport and its application to integrable systems and integer partitions.

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Francqui research professor.

#### Abstract

Dynamical systems are a very versatile and interdisciplinary field since it describes and models phenomena and processes occurring in mathematics, physics, biology, chemistry etc. by means of differential equations. Moreover, pure and applied aspects often go hand in hand and complement each other naturally. Of particular interest to us are integrable and nonintegrable Hamiltonian dynamical systems and their singularities as well as Hamiltonian PDEs and Floer and Morse theory in symplectic and hyperkähler geometry.#### Researcher(s)

- Promoter: Hohloch Sonja

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Optimal transport and the almost rigidity of the positive mass theorem.

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The aim of this project is to apply the power and versatility of optimal transport to the study of the almost rigidity of the positive mass theorem. The positive mass theorem is motivated by general relativity, and it states that asymptotically flat Riemannian manifolds with non-negative scalar curvature must have non-negative mass. Furthermore, the only asymptotically flat Riemannian manifold with non-negative scalar curvature and zero mass is Euclidean space. It is natural to ask if the positive mass theorem satisfies almost rigidity: if an asymptotically flat Riemannian manifold has small mass, is it true that it must be close to Euclidean space? The appropriate setting for this question is non-smooth Riemannian geometry, and metrics on the space of Riemannian manifolds. Optimal transport has been extremely successful in these areas, and so it is natural to attempt to apply its techniques to the almost rigidity of the positive mass theorem. However, this has not been done yet. In this project, we propose to do the following: 1) Study the almost rigidity of the Riemannian positive mass theorem using metrics arising from optimal transport. 2) Relate metrics currently being used to study the almost rigidity of the positive mass theorem with metrics arising from optimal transport.#### Researcher(s)

- Promoter: Hohloch Sonja
- Fellow: Bryden Edward

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Symmetry reduction and unreduction in mechanics and geometry.

#### Abstract

`Geometric mechanics' usually stands for the application of differential geometric methods in the study of dynamical systems appearing in mathematical physics. The objectives in this project are all centered around such geometric techniques for reduction and unreduction of a Lagrangian system that is invariant under the action of a symmetry Lie group. One encounters such systems in the context of the calculus of variations and in Finsler geometry. On a principal fibre bundle, the terminology symmetry reduction refers to the fact that an invariant Lagrangian system on the full manifold can be reduced to a system of differential equations on the quotient manifold (the so-called Lagrange-Poincare equations). Unreduction, on the other hand, has the opposite goal: to relate a Lagrangian system on the quotient manifold to a system of differential equations on the full manifold. In this proposal we will investigate the conditions under which the unreduced system can be brought back in the form of a set of Euler-Lagrange equations, for some (yet unknown) Lagrangian on the full manifold. The main tool will be the so-called Inverse Problem of the Calculus of Variations. Besides, we will both extend and specify the method of unreduction in such a way that it fits the needs of the research on isometric submersions between two Finsler manifolds.#### Researcher(s)

- Promoter: Hohloch Sonja
- Co-promoter: Mestdag Tom
- Fellow: Yasaka Kenzo

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Euler Disk versus Spinning Top.

#### Abstract

Tops are well-known to the general public as toys that spin on a tip around an axis of rotation before they start to wobble and finally topple over. Some types of tops may display `funny & unexpected' behavior like e.g. the Tippe top that may invert itself and spin on its stem or the rattleback that suddenly may start shaking and then change directions. Moreover, the Euler disk is a mathematical toy that is inspired by a rolling and spinning coin on a table. During its final stage of motion it issues a whirring sound of rapidly increasing frequency. From a mathematical point of view, studying the motion of tops, disks or, more general, rigid bodies in 3-space is a classical topic with a long and rich history. The present project will study a spinning top with truncated tip: Playing with a self-made top with small truncated tip shows the point of contact of the top with the plane first moving on a circle causing a certain wobbling of the top before it becomes rather quickly the whole circle, i.e., the spinning axis becomes vertical. Changes in the size of the truncation cause drastic changes in behavior. Intuitively this looks like a competition between the behavior of an Euler disk and of a spinning top. The aim is to model this behavior mathematically and to determine under which conditions (size/thickness of the circle, friction etc.) which behavior dominates and how the transition looks like.#### Researcher(s)

- Promoter: Hohloch Sonja

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Beyond Symplectic Geometry.

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Symplectic geometry was created as the mathematical foundation of classical mechanics. At the start of the twentieth century it became the foundation also of quantum mechanics. And since the advent of string theory it has played a key role in quantum field theory too. The aim of this project is to take the ideas and techniques of symplectic geometry and apply them to new fields, not in physics, but in geometry itself. We will attack a wide variety of problems, from studying surfaces which minimise area to spaces with negative curvature. These areas are, at first sight, unrelated to symplectic geometry. By bringing symplectic techniques to bear on these questions we will be able to make progress where none was previously possible. To do this, we will need to refine the techniques themselves, which will also lead to progress in symplectic geometry itself.#### Researcher(s)

- Promoter: Hohloch Sonja

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Contact semitoric integrability

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Differential equations are a natural way to model phenomena in mathematics, physics, biology, chemistry and many other areas. When considered under the ect of evolution in time, one speaks of `dynamical systems'. This proposal focuses on systems on three and four dimensional manifolds whose flow induces an S^1 x S^1- or an S^1 x R-action. These cases are called 'toric' and 'semitoric', respectively. We are interested in these systems in the context of symplectic and contact geometry. During the last 3 decades, these topics grew into a very active and influential field with ramifications to many areas in mathematics and physics. Symplectic geometry only exists in even dimensions and is the natural setting for Hamiltonian dynamics. Contact geometry only exists in odd dimension and provides the setting for Reeb dynamics. `Symplectization' and `contactization' admit in certain cases to pass from one to the other. When toric systems on compact symplectic manifolds were classified in the 1980s, the classification of toric systems in contact geometry followed not long after. But toric systems are very rare such that the search for less restrictive classes of systems began. About 10 years ago, semitoric systems were classified in the symplectic Hamiltonian context. The main goal of this proposal is to define and classify semitoric systems in contact geometry and study the interaction of toric and semitoric systems in contact geometry with semitoric systems in symplectic geometry.#### Researcher(s)

- Promoter: Hohloch Sonja
- Fellow: Ignoul Senne

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The topological entropy of Reeb flows and its relations to symplectic topology.

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The objective of this project is to study the topological entropy for Reeb flows and its relations with symplectic and contact topology. Reeb flows are a special class of dynamical systems which lie at the intersection of geometry, topology and mathematical physics. The class of Reeb flows includes the geodesic flows of Riemannian metrics and important examples of Hamiltonian dynamical systems. The dynamical properties of Reeb flows are strongly related to the topological properties of contact and symplectic manifolds. In this project we study the behaviour of the topological entropy of Reeb flows. The topological entropy is an important dynamical invariant which codifies in a single non-negative number the exponential complexity of a dynamical system. If the topological entropy of a dynamical system is positive then the system exhibits some type of chaotic behaviour. In this project we propose to: A) better understand the dynamics of Reeb flows with positive topological entropy using invariants coming from Floer theory; B) to use topological methods to construct new examples of Reeb flows with zero topological entropy; C) to use Floer theory to study how topological entropy varies under perturbations of Reeb flows.#### Researcher(s)

- Promoter: Hohloch Sonja
- Fellow: Ribeiro de Resende Alves Marcelo

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Singularities in semitoric and Poisson geometry.

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Integrable systems are dynamical systems with sufficiently many conserved quantities. Semitoric systems are especially well-behaved, yet still flexible enough to show interesting phenomena. Semitoric geometry strongly interacts with Poisson geometry, which in turn is linked to Hamiltonian dynamics and many other areas in mathematics and physics. This project aims at making use of these connections and at establishing new ones. Consequently, we will prove results on the interface of these fields. We will strengthen the connection to theoretical physics by studying a form of symmetry for singular torus fibrations called T-duality, which we will apply to generalized complex geometry. Within geometry we will establish a new relation between singular and foliated geometric structures. We will compare the description of singular differential forms in algebraic and differential geometry, via Lie algebroids and log structures respectively. Explicitly our objectives are: (1) Study and classify semitoric manifolds with singular symplectic structures. (2) Extend T-duality to singular torus fibrations. (3) Transform singular geometric structures into foliated geometric structures. (4) Establish a dictionary between singular symplectic and algebraic log geometry. Finishing these projects will pave the way for interesting new interactions between these areas of research.#### Researcher(s)

- Promoter: Hohloch Sonja
- Fellow: Witte Aldo

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Geometric structures and applications to control theory and numerical integration.

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Geometric mechanics refers to a variety of topics that lie at the intersection of differential geometry, dynamical systems, and analytical mechanics. The main idea is to identify the geometric structures underlying many classes of physical and engineering systems. These geometric structures can be useful for the qualitative study of the system and can also be used for instance in the design of control laws and geometric integrators. The property that a system can be derived from a variational principle is one of these useful structures. In this project we will use the inverse problem of the calculus of variations to find stabilizing controls for a variety of mechanical systems. One of the advantages of finding a variational structure is that we can then use energy methods to show stability, or find conditions for stability. We will also introduce more flexibility to the classical inverse problem to extend its possible applications. More precisely, for the inverse problem on a Lie algebra we will allow variable structure constants in order to have more freedom in the energy shaping step. The theory of exterior differential systems has been applied successfully to the inverse problem to identify variational cases. We will also adapt these techniques to the inverse problem for constrained systems with an eye towards the problem of Hamiltonization of nonholonomic systems. Finally we will also study geometric integrators for metriplectic and dissipative systems.#### Researcher(s)

- Promoter: Mestdag Tom
- Co-promoter: Hohloch Sonja
- Fellow: Farre Puiggali Marta

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From semitoric systems to Floer theory and integrable dynamics.

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Semitoric systems are a type of dynamical system (such as a spinning top) which satisfy certain symmetries. These systems can be well understood in terms of five invariants which can be recovered from the system. Semitoric systems lie in the field of symplectic geometry, another subfield of symplecitc geometry is Floer theory, which attempts to compute and understand certain invariants of symplectic manifolds and their Lagrangian submanifolds (a type of submanifold which arises naturally in the study of semitoric systems). We propose to (1) initiate research to better understand the invariants of semitoric systems (2) expand results and ideas from semitoric systems to more general systems (including those with so-called hyperbolic points, which are more common in nature) (3) explore the connection between semitoric systems, integrable systems, and Floer theory.#### Researcher(s)

- Promoter: Hohloch Sonja
- Fellow: Palmer Joseph

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From semitoric systems to integrable dynamics and Floer theory (Int Sys Floer).

#### Abstract

Semitoric systems are a type of dynamical system which satisfy certain symmetries. These systems can be well understood in terms of five invariants. Semitoric systems lie in the field of symplectic geometry, another subfield of symplecitc geometry is Floer theory, which attempts to compute and understand certain invariants of symplectic manifolds and their Lagrangian submanifolds (a type of submanifold which arises naturally in the study of semitoric systems). This project will (1) initiate research to better understand the invariants of semitoric systems (2) expand results and ideas from semitoric systems to more general systems (including those with so-called hyperbolic points, which are common in nature) (3) explore the connection between semitoric systems, integrable systems, and Floer theory.#### Researcher(s)

- Promoter: Hohloch Sonja
- Fellow: Palmer Joseph

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Symplectic Techniques in Differential Geometry.

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During the past decades, research of symplectic geometry accelerated providing lots of new tools and applications for very different topics in mathematics. This Excellence of Science (EoS) research project pushes these ideas further to areas not immediately connected to symplectic geometry.#### Researcher(s)

- Promoter: Hohloch Sonja

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Rigidity and conservation laws of Hamiltonian partial differential equations in hyperkähler Floer theory.

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Named after the Irish physicist, astronomer, and mathematician W. R. Hamilton (1805-1875), Hamiltonian systems are an important class of dynamical systems with certain conservation laws and rigidity features. A well-known classical example is the n-body problem (`movement of the planets around the sun'). In fact, Hamiltonian systems appear in many shapes throughout mathematics, physics, chemistry, biology, and engineering. Classical Hamiltonian problems are formulated as systems of ordinary differential equations on finite dimensional spaces. Nevertheless, there are also equations that can be reformulated as Hamiltonian systems, but this time on infinite dimensional spaces. Such systems are called Hamiltonian partial differential equations, in short Hamiltonian PDEs. Examples are the Korteweg-de Vries equation, the Sine-Gordon equation, the nonlinear Schrödinger equation, nonlinear sigma models etc. This project has two main aspects: - On the one hand, we start with a `triholomorphic' Dirac-type equation on a so-called hyperkähler manifold that can be transformed into a Hamiltonian PDE on the infinite dimensional loop space of the manifold, and then studies conservation laws, integrability (`extra symmetries'), and features from modern symplectic geometry (`non-squeezing' properties, symplectic capacities etc.) of this Hamiltonian PDE. - On the other hand, we investigate the occurrence and bifurcation behavior of singularities with hyperbolic components in 4-dimensional integrable Hamiltonian systems and classify the associated fibers.#### Researcher(s)

- Promoter: Hohloch Sonja
- Fellow: Gullentops Yannick

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Modern symplectic geometry in integrable Hamiltonian dynamical systems.

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This research project studies aspects of interactions between integrable Hamiltonian systems and modern symplectic geometry: The symplectic classification in the sense of Pelayo & Vu Ngoc's is completed for two seminal examples of semitoric systems, namely the coupled spin oscillator and coupled angular momenta. In addition, more general families are studied and partially classified.#### Researcher(s)

- Promoter: Hohloch Sonja
- Fellow: Alonso Fernandez Jaume

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