Colloquium and Seminars

The transition from laminar (smooth) to turbulent (disordered) flow in a pipe remains one of the oldest outstanding problems in fluid mechanics. In a vertically-aligned pipe, under the influence of heating, the opposite process has been observed, i.e. turbulence may be partially or fully suppressed by buoyancy forces. As the suppression of turbulence leads to severe heat transfer deterioration, this phenomenon is undesirable in both heating and cooling applications, such as nuclear reactor cooling systems and geothermal energy capture. In this talk, I will present our recent efforts in modelling and understanding the transitions to and from turbulence in a vertical heated pipe, focusing on the competition between shear- and buoyancy-driven instabilities and the resulting suppression or enhancement of coherent structures. Using tools from dynamical systems theory, stability analysis and optimisation, I will show how we can capture and optimise the transitions between different flow states under varying heating conditions. Connections to turbulence control strategies in ‘ordinary’ (isothermal) pipe flow will also be discussed, where, in contrast, turbulence suppression is typically highly desirable, as it leads to drag reduction and lower energy consumption.

In this talk, we study the emergence of spatially localised coherent structures induced by a compact region of spatial heterogeneity that is motivated by numerical studies into the formation of tornados. While one-dimensional localised patterns induced by spatial heterogeneities have been well studied, proving the existence of fully localised patterns in higher dimensions remains an open problem in pattern formation. We present a general approach to prove the existence of fully localised two-dimensional patterns in partial differential equations containing a compact spatial heterogeneity. This includes patterns with radial and dihedral symmetries, but also extends to patterns beyond these standard rotational symmetry groups. In order to demonstrate the approach, we consider the planar Swift--Hohenberg equation whose linear bifurcation parameter is modified with a radially-symmetric step function. In this case the trivial state is unstable in a compact neighbourhood of the origin and linearly stable outside. The introduction of a spatial heterogeneity results in an infinite family of bifurcation points with finite dimensional kernels, allowing one to prove local and global bifurcation theorems. We prove the existence of local bifurcation branches of fully localised patterns, characterise their stability and bifurcation structure, and then rigorously continue to large amplitude via analytic global bifurcation theory. Notably, the primary (possibly stable) bifurcating branch in the Swift--Hohenberg equation alternates between an axisymmetric spot and a non-axisymmetric `dipole' pattern, depending on the width of the spatial heterogeneity. We also discuss how one can use geometric singular perturbation theory to prove the persistence of the patterns to smooth spatial heterogeneities. This work is in collaboration with Daniel Hill and Matthew Turner.

Multispecies asymmetric exclusion processes (ASEPs) are interacting particle systems characterised by simple, local dynamics, where particles occupy lattice sites and interact only with their adjacent neighbors, following asymmetric exchange rules based on their species labels. I will present recent results on two-point correlation functions in multispecies ASEPs, including models on finite rings and their continuous-space limit as the number of sites tends to infinity. Using combinatorial tools such as Ferrari–Martin multiline queues, projection techniques, and bijective arguments, we derive exact formulas for adjacent particle correlations and resolve a conjecture in the continuous multispecies TASEP (Aas and Linusson, AIHPD 2018). We also extend finite-ring results of Ayyer and Linusson (Trans AMS, 2017) to the partially asymmetric case (PASEP), formulating new correlation functions that depend on the asymmetry parameter. I will briefly outline ongoing work on boundary-driven multispecies B-TASEP and long-time limiting states in periodic ASEPs, suggesting connections between pairwise correlations and stationary-state structure.

Das Beweisen ist für die Mathematik als Disziplin von zentraler Bedeutung und spielt daher auch in der mathematischen Ausbildung eine wichtige Rolle. Lernende sollen die Mathematik als deduktives System begreifen, die Art der Absicherung mathematischer Ergebnisse verstehen, argumentative Herausforderungen erfolgreich bewältigen können und so ein adäquates Verständnis von mathematischen Beweisen aufbauen. Ausgehend von einem theoretischen Rahmenmodell zum mathematischen Beweisverständnis werden Ergebnisse empirischer Studien vorgestellt, die das Beweisverständnis von Lernenden in unterschiedlichen Phasen der mathematischen Ausbildung untersuchen und Möglichkeiten der Förderung aufzeigen. ______________________ Invited by Prof. Stefan Ufer