Upcoming events

02 October 2023 15:45 till 16:45

[STAT/AP] Carla Groenland: Counting graphic sequences via integrated random walks

Via a new probabilistic result, we provide (1+o(1))-asymptotics for the number of integer sequences n-1>= d_1 >= ... >= d_n >= 0 that form the degree sequence of an n-vertex graph (improving both the upper and lower bound by a multiplicative n^{1/4}-factor). In particular, we determine the asymptotic probability that the integral of a (lazy) simple symmetric random walk bridge remains non-negative. This talk will explain how this problem arose, what the connection is with the problem about random walks (including what all the words in this abstract mean) and then provide a short sketch of the proof. This is based on joint work with Paul Balister, Serte Donderwinkel, Tom Johnston and Alex Scott.

05 October 2023 16:00 till 17:00

[PDE & Applications seminar] Kerstin Lux-Gottschalk: The effect of parameter uncertainty on climate tipping points

Several subsystems of the Earth might undergo critical transitions under sustained anthropogenic forcing, i.e. these systems are at risk of passing a so-called tipping point (TP). That induces a drastic sudden change in the system’s phase portrait that is often irreversible. We approach these tipping phenomena through the lense of nonlinear dynamical systems theory. In particular, we use tools from bifurcation theory, which provides the mathematical framework for the creation and loss of equilibria as well as changes in their stability properties under variation of a deterministic control parameter. In this talk, I will address the question how uncertainty in model parameters affects the tipping behavior of the dynamical system in terms of the location of TPs and how they are approached. I will present a workflow for climate tipping points from a Bayesian inference on the uncertain model input parameter to the forward propagation of the obtained input probability distribution through the nonlinear dynamics turning the bifurcation curve into a random object [1]. I will show numerical results on a conceptual model of the Atlantic Meridional Overturning Circulation, which is one of the identified tipping elements. This talk covers joint work with Peter Ashwin (University of Exeter, UK), Richard Wood (Met Office, UK), and Christian K ̈uhn (Technical University of Munich, Germany).

19 October 2023 16:00 till 17:00

[PDE & Applications seminar] Zachary Adams: A quasi-stationary approach to metastability for weakly interacting particle systems

We consider systems of N particles moving as Brownian motions interacting via an attractive potential. For instance, the Glauber dynamics associated with the classical mean-field O(2) spin system of statistical mechanics. In the large particle limit, the empirical measure of such systems is known to converge to a nonlocal parabolic PDE of McKean-Vlasov type. While the McKean-Vlasov system is known to possess no non-trivial stationary solutions, numerical experiments demonstrate the existence of an almost-synchronized state that persists over a long time scale. In this talk, we characterize this almost-synchronized state and the time scale on which it persists using methods involving sub-Markov semigroups, quasi-stationary distributions, and the spectral theory of Schrödinger operators. Control on the time scale in terms of the noise amplitude and particle number are obtained. This is joint work with Maximilian Engel (FU Berlin) and Rishabh Gvalani (Max Planck Institute for Mathematics in the Sciences).

30 October 2023 15:45 till 16:45

[STAT/AP] Reka Szabo: Stability results via Toom contours

In this talk I will review Toom's classical result about stability of trajectories of cellular automata. Informally, we say that a cellular automaton is stable if it does not completely lose memory of its initial state when subjected to noise. Using a contour argument Toom gave necessary and sufficient conditions for the cellular automaton to be stable. I will introduce an alternative definition of Toom contours that allows us to extend his method to more general models. I will show how this method can be used to obtain bounds for the critical parameters for certain models, as well as discuss possible applications and limitations of this extension. (Based on joint work with Jan Swart and Cristina Toninelli.)

02 November 2023 16:00 till 17:00

[PDE & Applications seminar] Julie Rowlett

06 November 2023 15:45 till 16:45

[STAT/AP] Aernout van Enter: Dyson models with random boundary conditions

I discuss the low-temperature behaviour of Dyson models (polynomially decaying long-range Ising models in one dimension) in the presence of random boundary conditions. For typical random (i.i.d.) boundary conditions Chaotic Size Dependence occurs, that is, the pointwise thermodynamic limit of the finite-volume Gibbs states for increasing volumes does not exist, but the sequence of states moves between various possible limit points. As a consequence it makes sense to study distributional limits, the so-called "metastates" which are measures on the possible limiting Gibbs measures.

The Dyson model is known to have a phase transition for decay parameters α between 1 and 2. We show that the metastate obtained from random boundary conditions changes character at α =3/2. It is dispersed in both cases, but it changes between being supported on two pure Gibbs measures when α is less than 3/2 to being supported on mixtures thereof when α is larger than 3/2.

Joint work with Eric Endo (NYU Shanghai) and Arnaud Le Ny (Paris-Est)

We also discuss the relation with a recent high-temperature result by Johansson Oberg and Pollicott about regularity of eigenfunctions of Transfer Operators. ( work in progress with Evgeny Verbitskiy and Mirmukshin Makhmudov).

16 November 2023 16:00 till 17:00

[PDE & Applications seminar] Laura Scarabosio

20 November 2023 15:45 till 16:45

[STAT/AP] Collin Drent

30 November 2023 16:00 till 17:00

[PDE & Applications seminar] Marianne Bauer

08 December 2023 15:45 till 16:45

[STAT/AP] Wioletta Ruszel

TBA

14 December 2023 16:00 till 17:00

[PDE & Applications seminar] Clarice Poon

18 December 2023 15:45 till 16:45

[STAT/AP] Bert Zwart

08 January 2024 15:45 till 16:45

[STAT/AP] Alethea Barbaro

11 January 2024 16:00 till 17:00

[PDE & Applications seminar] Oliver Tse: Generalized gradient structures for interacting population dynamics

In this talk, we discuss a class of non-local evolution equations arising from reversible interacting birth-and-death processes that can be given a generalized gradient structure, and further motivate the structure via the large-population limit of measure-valued processes in population dynamics.

22 January 2024 15:45 till 16:45

[STAT/AP] Melvin Drent

25 January 2024 16:00 till 17:00

[PDE & Applications seminar] Daniele Avitabile: Uncertainty Quantification for Neurobiological Networks

This talk presents a framework for forward uncertainty quantification problems in spatially-extended neurobiological networks. We will consider networks in which the cortex is represented as a continuum domain, and local neuronal activity evolves according to an integro-differential equation, collecting inputs nonlocally, from the whole cortex. These models are sometimes referred to as neural field equations.

Large-scale brain simulations of such models are currently performed heuristically, and the numerical analysis of these problems is largely unexplored. In the first part of the talk I will summarise recent developments for the rigorous numerical analysis of projection schemes [1] for deterministic neural fields, which sets the foundation for developing Finite-Element and Spectral schemes for large-scale problems.

The second part of the talk will discuss the case of networks in the presence of uncertainties modelled with random data, in particular: random synaptic connections, external stimuli, neuronal firing rates, and initial conditions. Such problems give rise to random solutions, whose mean, variance, or other quantities of interest have to be estimated using numerical simulations. This so-called forward uncertainty quantification problem is challenging because it couples spatially nonlocal, nonlinear problems to large-dimensional random data.

I will present a family of schemes that couple a spatial projector for the spatial discretisation, to stochastic collocation for the random data. We will analyse the time- dependent problem with random data and the schemes from a functional analytic viewpoint, and show that the proposed methods can achieve spectral accuracy, provided the random data is sufficiently regular. We will showcase the schemes using several examples.

Acknowledgements This talk presents joint work with Francesca Cavallini (VU Amsterdam), Svetlana Dubinkina (VU Amsterdam), and Gabriel Lord (Radboud University).

[1] Avitabile D. (2023). Projection Methods for Neural Field Equations. https://www.danieleavitabile.com/projection-methods-for-neural-field-equations/