Upcoming Talks

Two examples of fast-slow dynamics in mechanical engineering ▼

Speaker: Baptiste Bergeot (INSA-CVL / LaMé, Blois, France)

Date: May 07, 2025 Room: BB 5161.0293

Time: 3-5pm

This presentation will explore the complex dynamics of two types of fast-slow mechanical systems. The first system is a nonlinear passive vibration absorber, referred to as a Nonlinear Energy Sink (NES), which is coupled to a self-sustained oscillator requiring vibration attenuation. The second system is a self-sustained wind musical instrument (such as the clarinet) in which one of the bifurcation parameters - specifically, the air pressure within the musician's mouth - varies over time accounting for attack transients performed by the musician. Although these two systems differ greatly in term of applications, they are both modeled by systems of differential equations featuring a small parameter highlighting their singularly perturbed, fast-slow nature. As a result, their dynamics (i) cannot be fully explained by the concepts traditionally used in engineering such as bifurcation diagrams and basins of attraction and (ii) can be significantly influenced by the presence of noise. For each of these systems, the deterministic dynamics will be examined within the framework of the geometric singular perturbation theory but also using more recent concepts developed for analyzing rate-induced tipping phenomena. In selected cases, the impact of noise will also be explored using both numerical simulations and analytical techniques.

Past Talks

An atomistic K-test framework for general grain boundaries and triclinic single crystals ▼

Speaker: Florian Brunner (University of Groningen)

Date: April 09, 2025 Room: BB 5161.0293

Time: 3-5pm

Fracture toughness, i.e. the resistance of a material to fracture, often plays a key role in the design process of materials. Premature failure of engineering components due to material property degradation can cause damage in the order of millions of Euros or people to lose their lives. Prime examples of such degradation mechanisms are hydrogen embrittlement, liquid metal embrittlement and stress corrosion cracking. Those mechanisms influence material properties at the atomic scale and often cause a reduction of grain boundary cohesion in polycrystalline materials, resulting in premature, intergranular fracture.

K-tests are a widely-used approach for atomistic simulations of fracture. In a nutshell, these K-tests are numerical fracture toughness tests in which only the region close to a stressed crack tip is simulated. The remaining, not explicitly modelled part of the material is replaced by boundary conditions based on the theory of linear elastic fracture mechanics (LEFM). K-tests are frequently employed for studying the fracture behaviour of monoclinic single crystals and special tilt grain boundaries. Also for the treatment of more general (up to triclinic) grain boundaries and single crystals mathematical frameworks like e.g. the $6^{\text{th}}$-order Stroh or Lekhnitskii formalism are available. These theories are elegant due to their analytical formulation but also predict spatial oscillations in the relevant field quantities that can lead to unphysical self-interpenetration of cracks for grain boundaries that do not possess at least monoclinic material symmetry. For this reason, typically simplified ($4^{\text{th}}$-order) versions of the mentioned approaches are employed. Such $4^{\text{th}}$-order theories are by default restricted to certain material symmetry classes, are therefore simpler and do not exhibit the mentioned oscillatory field components.

For the investigation of the aforementioned degradation mechanisms, such $4^{\text{th}}$-order approaches and their symmetry requirements are, however, too restrictive. Taking liquid metal embrittlement as an example, it is well-established that grain boundaries with a wide range of orientations, that are not necessarily aligned with the symmetry planes of the underlying crystal structure, are of major importance for a full understanding of this complex phenomenon. Grain boundaries with such general orientations require an up to triclinic description and therefore a $6^{\text{th}}$-order formalism.

Therefore, the present work aims to contribute to the development of an accurate, yet simple K-test framework for the simulation of generally oriented grain boundaries and single crystals. To this end, we first re-visit the $6^{\text{th}}$-order Stroh formalism and examine its applicability. We investigate in which cases the oscillations occur and quantify their influence on the relevant field quantities for a broad range of cubic crystals. Next, we examine the impact of the choice of LEFM approach ($6^{\text{th}}$- or $4^{\text{th}}$-order) on the K-test results. For this, Fe and Cu are considered as representative examples of body-centered cubic and face-centered cubic metals, respectively. Finally, we propose a K-test simulation strategy that includes crack tip tracing and is unambiguous regarding the determination of critical stress intensity factors of generally oriented grain boundaries and triclinic single crystals.

Skein theory and quantum groups ▼

Speaker: Zhihao Wang (University of Groningen)

Date: April 02, 2025 Room: BB 5161.0293

Time: 3-5pm

In this talk, we will review the representation theory of the quantum group Oq(SL2). We will present how to use knot diagrams to represent representations of Oq(SL2). We will then look at the relations in these knot diagrams obtained from the representation theory of Oq(SL2). At the end of this talk, it will be clear how we derive Kauffman bracket relations via Oq(SL2).

On the affine invariant of hypersemitoric systems ▼

Speaker: Sonja Hohloch (University of Antwerp)

Date: March 10, 2025 Room: EA 5159.0010

Time: 3-5pm

Many naturally occurring dynamical systems have symmetries or preserved quantities (just think of systems with preserved angles, invariance under rotation etc.). Roughly, integrable systems are Hamiltonian dynamical systems that admit a maximal number of independent symmetries/ preserved quantities.

In 1988, Delzant symplectically classified toric integrable systems by means of their momentum map image which is a very nice and special convex polytope, often referred to as `Delzant polytope' of the toric system.

Semitoric systems are integrable systems of the form $F=(J,H): (M, \omega) \to \mathbb{R}$ where $(M, \omega)$ is a 4-dimensional connected symplectic manifold and $J$ is proper and induces an effective Hamiltonian torus action and $F$ admits only nondegenerate singularities and no hyperbolic components. Intuitively, semitoric systems generalize toric systems in dimension four by admitting in addition to elliptic-elliptic and elliptic-regular singularities also focus-focus singularities. In 2009-2011, Pelayo $\&$ Vu Ngoc symplecticaly classified semitoric integrable systems in terms of 5 invariants, among which a `generalized semitoric polytope' deduced from the momentum map image, i.e. generalizing the Delzant polytope.

When admitting also hyperbolic components for the nondegenerate singularities and mildly degenerate (so-called parabolic) points, then one generalizes semitoric systems to so-called hypersemitoric systems. The long term goal is to obtain a classification of hypersemitoric integrable systems on compact connected 4-dimensional symplectic manifolds.

This talk presents one of the expected invariants, the so-called `affine invariant' which is the generalization of the semitoric polytope invariant. This talk is based on ongoing work with N. Flamand (Antwerp) and a joint preprint (arXiv:2411.17509) with K. Efstathiou (Duke Kunshan University) and P. Santos (Antwerp).

TBA ▼

Speaker: Luis Venegas (University of Groningen)

Date: February 27, 2025 Room: TBA

Time: 3-5pm

TBA

Weyl's tube formula in sub-Riemannian geometry ▼

Speaker: Tommaso Rossi (Laboratoire Jacques-Louis Lions - Sorbonne Université)

Date: February 26, 2025 Room: BB 5161.0293

Time: 3-5pm

The tube of radius r around a submanifold is the set of all points within distance r of the submanifold. In this talk, we discuss the volume of a tube around a submanifold in sub-Riemannian geometry. Firstly, we show that the volume of the tube around a non-characteristic submanifold of class $C^2$ is either smooth or real-analytic for small radii, depending on the regularity of the underlying manifold, and we establish a Weyl's tube formula. Secondly, we investigate Weyl's invariance theorem in sub-Riemannian geometry: we show that two curves in the Heisenberg group with the same Reeb angle have the same Weyl's tube formula. This is a joint work with T. Bossio and L. Rizzi.

Cosmic Anisotropy & Bianchi Characterization ▼

Speaker: Robbert Scholtens (University of Groningen)

Date: December 18, 2024 Room: BB 5161.0293

Time: 3-5 pm

The cosmological principle states that at the largest scales, the universe is spatially homogeneous and isotropic, directly leading to the spacetime metric at those scales to be one of the Robertson-Walker types. However, recent observations of e.g. bulk flows and supernovae call into question particularly the assumption of isotropy, the relaxing of which would yield new models of the universe at a fundamental level. One of such potential models is the Bianchi class, which consists of metrics with 3 non-vanishing Killing vector fields that form a spatial frame everywhere (this encodes only homogeneity). These have been studied at length before, e.g. by Jantzen, Ellis, McCallum, and Hawking. Our presentation shows in some sense the reverse: when given a suitable 3D Lie algebra of vector fields, that we can find a spatial metric basis on which these are guaranteed to be Killing. (Then an extension to a spacetime metric can be made.) We do this in a pedagogical way, by gently introducing the problem and using advanced mathematics only to a necessary degree. In so doing we create a table which, when fed the suitable 3D Lie algebra, directly yields the frame for the metric. Finally, we illustrate the use case for this research by linking it to finding wave operators in homogeneous spacetimes. This presentation is based on 2408.04938, in collaboration with Marcello Seri, Holger Waalkens, and Rien van de Weygaeart.

Linear quantum systems: poles, zeros, invertibility and sensitivity ▼

Speaker: Guofeng Zhang (The Hong Kong Polytechnic University)

Date: December 11, 2024 Room: BB 5161.0293

Time: 3-5pm

The non-commutative nature of quantum mechanics imposes fundamental constraints on system dynamics, which in the linear realm, are manifested through the physical realizability conditions on system matrices. These restrictions give system matrices a unique structure. In this talk I discuss this structure by investigating the zeros and poles of linear quantum systems. Firstly, I show that -s_0 is a transmission zero if and only if s_0 is a pole of the transfer function, and -s_0 is an invariant zero if and only if s_0 is an eigenvalue of the A-matrix, of a linear quantum system. Moreover, s_0 is an output-decoupling zero if and only if -s_0 is an input-decoupling zero. Secondly, based on these zero-pole relations, we prove that a linear quantum system must be Hurwitz unstable if it is strongly asymptotically left invertible. Stable input observers are constructed for unstable linear quantum systems. Finally, the sensitivity of a coherent feedback network is investigated. We found that the well-known complementarity constraint between sensitivity and complementary sensitivity functions no longer holds in the quantum regime; instead, much richer fundamental performance limitations exist. The fundamental tradeoff between ideal input squeezing and system robustness is studied on the basis of system sensitivity analysis.

Slow-fast processes in sleep-wake modelling ▼

Speaker: Gianne Derks (Leiden University)

Date: November 27, 2024 Room: BB 5161.0293

Time: 3-5pm

We are estimated to spend one third of our life asleep, and it is increasingly apparent that good sleep is essential for overall health. Yet many people suffer from insufficient sleep or sleep disorders. Even though there is only a partial understanding about the why and how of sleep, mathematical models do exist that capture the broad features of sleep-wake regulation and are widely used in safety-critical industries to model fatigue risk.

In this talk we will discuss some mathematical models of sleep-wake regulation and their multi-time scale features. Most mathematical models consider two states: a sleep and a wake state. Biologically, sleep-wake regulation can be understood as the result of the interaction of two oscillatory processes: the circadian oscillation of our body clock, and a relaxation oscillator known as the `sleep homeostat' that results in a sleep pressure that increases during wake and decreases during sleep. The resulting two-process model has been immensely successful, providing the very language which frames most sleep research. An analysis of the two-timescale features of some more physiological based models of sleep-wake regulation can show that such models can be reduced to the so-called two-process model. This allows for a more physiological interpretation of some the parameters in the two-process model.

However, the two state models do not account for the fact that during the night we cycle between two main sleep states (rapid eye movement (REM) and non-rapid eye movement (NREM) sleep). We will also discuss a model that considers three states: these two sleep states and a wake state. We will show that this model can be considered as a three-timescale problem. This three-timescale decomposition reveals additional geometric structure which acts to organise oscillations between REM and NREM states. This deeper geometric understanding of the generation of REM-NREM cycles brings insight into the relationship between model predicted and observed patterns of REM-NREM cycles and suggests ways in which models could be modified to more accurately reflect patterns of human sleep.

This is joint research with Anne Skeldon, Derk Jan Dijk, Rachel Bernasconi, Matthew Bailey, Panos Kaklamanos, and Paul Glendinning.

How scale symmetries lead to attractors and an arrow of time in the universe ▼

Speaker: Sean Gryb (University of Groningen)

Date: November 13, 2024 Room: BB 5161.0293

Time: 3-5pm

A significant and persistent arrow of time is expressed in our universe by means of two important facts: the rapid and monotonic decrease of the red-shift and the relative smoothness of the early matter distribution. In this talk, I will show that an explanation can be given for these phenomena when the theory is expressed invariantly under a particular kind of scale symmetry. This explanation is given in terms of dynamically privileged states along generic solutions that lie on attractors and so-called "Janus points." A preferred temporal direction can be defined using such structures for observers in states close to an attractor. I will show how such a scenario can be realised in two models of the universe in a way that explains the properties of the red-shift and early matter distribution of the universe. Surprisingly, removing the relevant scale symmetry leads to flows on state space that do not preserve the measure. For N-body gravitational systems, this time dependence leads to the remarkable conclusion that 'early' states near a Janus point are typically smooth while most clumpy states occur near 'late'-time attractors.