Research team

Fundamental Mathematics

Expertise

1) Symplectic geometry, Floer theory and its applications to symplectic and contact dynamics (homoclinic points, growth behaviour). 2) Hyperkähler Floer theory and associated Hamiltonian PDEs on Hilbert spaces; Bubbling-off analysis. 3) Integrable Hamiltonian systems, in particular semitoric systems, focus-focus singularities, and Hamiltonian S^1-actions. 4) Morse theory and its application to n-categories and opetopes. 5) Optimal transport and its application to integer partitions. 6) Symplectic numerics.

The topological entropy of Reeb flows and its relations to symplectic topology. 01/11/2020 - 31/10/2023

Abstract

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.

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Geometric structures and applications to control theory and numerical integration. 01/10/2020 - 30/09/2023

Abstract

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.

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From semitoric systems to Floer theory and integrable dynamics. 01/10/2019 - 31/03/2024

Abstract

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.

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Symplectic Techniques in Differential Geometry. 01/01/2018 - 31/12/2021

Abstract

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.

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Rigidity and conservation laws of Hamiltonian partial differential equations in hyperkähler Floer theory. 01/10/2017 - 30/09/2021

Abstract

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 starts out from a `triholomorphic' Dirac-type equation on a so-called hyperkähler manifold. It can be transformed it into a Hamiltonian PDE on the infinite dimensional loop space of the manifold. For this new equation, we expect to show conservation laws, integrability (`extra symmetries'), features from modern symplectic geometry (`non-squeezing' properties, symplectic capacities etc.), and estimates on the number of periodic solutions (`infinite dimensional Arnold conjecture'). So far, only three people studied Hamiltonian PDEs with modern symplectic methods. Hamiltonian PDEs with hyperkähler background having symplectic features, as in this project, have never before been investigated. Due to its link to the Cauchy-Riemann-Fueter equation in the supersymmetric sigma model, our results may also be of interest to physicists.

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From semitoric systems to integrable dynamics and Floer theory (Int Sys Floer). 01/05/2019 - 30/04/2020

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.

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Modern symplectic geometry in integrable Hamiltonian dynamical systems. 01/10/2015 - 30/09/2019

Abstract

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.

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