Upcoming Talks

Weyl's tube formula in sub-Riemannian geometry ▼

Speaker: Tommasso 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.

TBA ▼

Speaker: Luis Venegas (University of Groningen)

Date: February 27, 2025 Room: TBA

Time: 3-5pm

TBA

Past Talks

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.

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.

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.

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.