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

We study opportunistic traders that try to detect and exploit the order flow of dealers hedging their net exposure to the FX fix. We also discuss how dealers can take this into account to balance not only risk and trading costs but also information leakage in an appropriate manner. It turns out that information leakage significantly expands the set of scenarios where both dealers and the clients whose orders they execute benefit from hedging part of the exposure before the fixing window itself. (Joint work in progress with Roel Oomen (Deutsche Bank) and Mateo Rodriguez Polo (ETH Zurich))

TBA

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Invited by Prof. Christian Hainzl

After motivating the Stefan problem from the random growth model perspective, I will discuss its discontinuities in time. These turn out to be characterized by the cascade equation, a second-order hyperbolic PDE. Questions of existence and regularity for the latter can be answered by expressing its solution as the value function of a player in an equilibrium of a suitable mean field game. Based on joint work with Yucheng Guo and Sergey Nadtochiy.

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