General Mathematics Colloquium

This colloquium takes place every other Wednesday afternoon, 16:00-17:00 in the MathLab 9th floor, NU building (the seminar room next to the common area). For the moment we are on Zoom. Please ask for the meeting ID and Passwords. For more information, please contact one of the organizers Dennis DoblerIlke Canakci, and Fahimeh Mokhtari.

A database of earlier years' talks can be found here.

Upcoming talks in 2021:

May, 17th: Martijn Kool (UU)

Title: The Geometry of Magnificent Four

Solid partitions are piles of boxes in the corner of a 4-dimensional room. Their enumeration is a mystery since MacMahon proposed an incorrect formula around 1916. Motivated by super-Yang-Mills theory on (complex) 4-dimensional affine space, Nekrasov recently assigned a measure to solid partitions and proposed a conjectural formula for their weighted enumeration.

We give a geometric definition of this measure using Hilbert scheme of points on 4-dimensional affine space. Although this Hilbert scheme is very singular and has "higher obstructions", we can use recent work of Oh-Thomas to localise our invariants and prove Nekrasov's conjecture. This is a non-technical talk based on joint work with J. V. Rennemo.

Previous talks in 2021:

April, 21st: Tere M-Seara (UPC)

Title: Chaos and oscillatory motions in the planar three body problem.

The planar three body problem models the motion of three bodies under the Newtonian gravitational force. In 1922 Chazy classified the possible final motions of the three bodies, that is, the behaviours the bodies may have when time tends to infinity. One of them are what is known as oscillatory motions, that is, solutions of the three body problem such that the positions of the bodies  is unbounded but comes back infinitely many times to a bounded region of the configuration space. At the time of Chazy, all types of final motions were known, except the oscillatory ones. We prove that, if all three masses are not equal, such motions exist. In fact, we prove the existence of chaotic behaviour on the motion of the bodies. The oscillatory orbits are one of the consequences of the existence of this chaotic behaviour.

April, 7th: Jose Mujica (VU Amsterdam) [Slides]

Title: Slow Manifolds, Invariant Manifolds and their interactions: a tale of slow-fast dynamical systems 

Slow-fast dynamical system arise in several applications. They describe phenomena in which system variables evolve in different timescales. Since Fenichel’s seminal work in the late 70’s, a method known as Geometric Singular Perturbation Theory (GSPT) has proven to be successful in the study of slow-fast systems. One of the main ideas of GSPT is to exploit the timescale separation in order to construct trajectories as a concatenation of slow and fast segments, obtained as solutions of subsystems describing the limiting slow and fast motion, respectively. This was one gets understanding of the geometry of so-called slow manifolds, along which the flow behaves considerably slow with respect to the rest of the dynamics; togethter with classical invariant manifolds, slow manifolds organize the phase space globally and locally.

In this talk we discuss some of the ideas of GSPT and describe the dynamics of a family of slow-fast systems with one fast and two slow variables. The focus is on a bifurcation that occurs in the slow dynamics known as a folded saddle-node of type II. This scenario provides a crash between classical dynamical systems and slow-fast systems, in the sense that there is an interaction of a slow manifold with a global invariant manifold. This has consequences for the local and global dynamics of the system. In particular we discuss the organization of recurrent dynamics in the form of oscillatory patterns known as mixed-mode oscillations, and the homoclinic scenarios nearby. The way we approach this is via the numerical approximation of these manifolds in a boundary-value-problem setup with the software AUTO, and track the manifolds when system parameters are varied.

March, 17th: Vanja Nikolic (RU) [Slides]

Title: The mathematics behind nonlinear sound waves

Sound waves with sufficiently large amplitudes are known to exhibit nonlinear behavior. The nonlinearity will be apparent sooner in high-frequency waves because these effects accumulate over the distance measured in wavelengths. This makes ultrasonic waves inherently nonlinear. Their many applications range from non-invasive surgery to non-destructive material testing and motivate the mathematical investigation into nonlinear acoustics. In this talk, we will give an overview of research questions and some recent results in the analysis and numerics of partial differential equations that model the propagation of such nonlinear waves.

March, 3rd: Thomas Rot (VU Amsterdam) [Slides]

Title: Most ropes have an even number of ends.

Compact one-dimensional manifolds can be classified: They are finite unions of circles and intervals. A simple consequence of this classification is that the number of boundary components of a compact one-dimensional manifold is even. This innocent looking observation has far reaching consequences. It allows you to determine quickly if you can escape a maze, can prove the fundamental theorem of algebra, and much more. I will discuss some classical consequences that I particularly enjoy. I will end my talk with discussion of recent work, joint with Federica Pasquotto, on degree theory for orbifolds, In this setting ropes might have an odd number of ends, but we can still say something. No prior knowledge of (differential) topology will be assumed.

Februar, 17th: Chris Bick (VU Amsterdam) [Slides]

Title: Coupled Oscillator Networks: Structure, Interactions, and Dynamics

The collective dynamics of coupled oscillatory processes govern many aspects crucial to our lives, whether it is the synchronous beating of our heart cells, collective activity of neurons in the brain, or power grid networks that operate in a stable frequency regime. In this talk we discuss how the collective network dynamics are shaped by the network structure (what oscillator is coupled to what other oscillator) and the network interactions (how one oscillator is coupled to another). We discuss in particular how "higher-order" interactions, which have attracted tremendous attention in recent years, give rise to heteroclinic and chaotic dynamics.

Wednesday February, 3rd: Svetlana Dubinkina (VU Amsterdam) [Slides]

Title: Shadowing approach to data assimilation

Data assimilation is broadly used in atmosphere and ocean science to correct error in the state estimation by incorporating information from measurements (e.g. satellites) into the mathematical model. The widely-used variational data assimilation method has a drawback of a drastic increase of the number of local minima of the corresponding cost function as the number of measurements increases. The shadowing approach to data assimilation, which was pioneered by K. Judd and L. Smith in Physica D (2001), aims at estimating the whole trajectory at once. It has no drawback of several local minima. However, it is computationally expensive, requires measurements of the whole trajectory, and has an infinite subspace of solutions. 

We propose to decrease the computational cost by projecting the shadowing approach to the unstable subspace that typically has much lower dimension than the phase space. Furthermore, we propose a novel shadowing-based data assimilation method that lifts up the requirement of a fully-observed state. We prove convergence of the method and demonstrate in numerical experiments with Lorenz models that the developed data assimilation method substantially outperforms the variational data assimilation method.