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.

"We consider a Kuramoto-Shivashinsky like equation close to the threshold of instability with additive white noise and spatially periodic boundary conditions which simultaneously exhibit Turing bifurcations with a spatial 1:3 resonance of the critical wave numbers. For the description of the bifurcating solutions we derive a system of coupled stochastic Landau equations. It is the goal of this paper to prove error estimates between the associated approximation obtained through this amplitude system and true solutions of the original system. The Kuramoto-Shivashinsky like equation serves as a prototype model for so-called super-pattern forming systems with quadratic nonlinearity and additive white noise."

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))

The exact treatment of Markovian models on complex networks requires knowledge of probability distributions expo- nentially large in the number of nodes n. Mean-field approximations provide an effective reduction in complexity of the models, requiring only a number of phase space variables polynomial in system size. However, this comes at the cost of losing accuracy close to critical points in the systems dynamics and an inability to capture correlations in the system. In this talk, we introduce a tunable approximation scheme for Markovian spreading models on networks based on matrix product states (MPSs). By controlling the bond dimensions of the MPS, we can investigate the effective dimensional- ity needed to accurately represent the exact 2n dimensional steady-state distribution. We introduce the entanglement entropy as a measure of the compressibility of the system and find that it peaks just after the phase transition on the disordered side, in line with the intuition that more complex states are at the ’edge of chaos’. The MPS provides a systematic way to tune the accuracy of the approximation by reducing the dimensionality of the systems state vector, leading to an improvement over second-order mean-field approximations for sufficiently large bond dimensions.

TBA

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Invited by Prof. Christian Hainzl

Quantum mechanics, now about a century old, is a very successful physical theory of matter on a small scale. From its first description until today, it has surprised scientists and laypersons alike by the strange behaviour it attributes to particles, atoms, and molecules. This behaviour can be characterized by the keywords Uncertainty, Superposition, and Entanglement. It took about sixty years before it was realized that these three characteristics do not just express a certain vagueness and strangeness of matter on a small scale but can actually be USEd to our advantage. In 1994 Peter Shor made this idea concrete by devising an algorithm that would enable large arrays of quantum systems to perform specific calculations (factoring large integers), which are impossible to do in practice on any classical device. With this algorithm, present-day cryptographic schemes can be broken, provided such "quantum computers” can be made to work. Starting from a discussion of the "two-slit experiment”, we sketch the working of Shor's algorithm and discuss the possibilities of future quantum computers.

About the speaker: Hans Maassen is a dutch mathematical physicist and emeritus professor specializing in quantum probability and quantum information theory. Standing out among his discoveries is the entropic uncertainty relation, named after himself and Jos Uffink, a fundamental  inequality in quantum mechanics.

This talk is open to the general public and all interested persons, and is presented by the SFB TRR352 "Mathematics of Many-Body Quantum systems and their collective phenomena" in cooperation with the TUM-IAS Workshop "Beyond IID in Information Theory".

Interconnected real-world systems oftentimes contain non-pairwise interactions between agents. These groupwise interactions are referred to as higher order interactions and can be encoded by means of hypergraphs or hypernetworks. Countless works in recent years have pointed out how this structural feature crucially shapes the collective behavior. This talk will, in particular, focus on dynamics of systems with higher order interactions. We observe that the restriction to undirected higher order interactions obstructs the emergence of certain heteroclinic structures in phase space. The directed counterparts, on the other hand, do not. Motivated by this, we define a general class of directed hypernetworks and corresponding maps that respect a given interaction structure, so-called admissible maps. For this class, all robust patterns of (cluster) synchrony that a given hypernetwork supports can be classified. Interestingly, these are only determined by higher degree polynomial admissible maps. In particular, unlike in classical networks, cluster synchronization is a higher order, that is, nonlinear effect. This feature induces a novel type of “reluctant” synchrony breaking bifurcation when a high order tangency of the solution branch to a non-robust synchrony space causes formerly synchronous nodes to separate unusually slowly.

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.

Clinical data often include a mix of continuous measurements and covariates that have been discretized, typically to protect privacy, meet reporting obligations, or simplify clinical interpretation. This combination, along with the nonlinear and tail-asymmetric dependence frequently observed in clinical data, affects the behavior of regression and variable-selection methods. Copula models, which separate marginal behavior from the dependence structure, provide a principled approach to studying these effects. In this talk, we analyze how discretizing a continuous covariate into equiprobable categories impacts conditional quantiles and likelihoods in bivariate copula models. For the Clayton and Frank families, we derive closed-form anchor points: for a given category, we identify the continuous covariate value at which the conditional quantile under the continuous model matches that of the discretized one. These anchors provide an exact measure of discretization bias, which is small near the center but can be substantial in the tails. Simulations across five copula families show that likelihood-based variable selection may over- or under-weight discretized covariates, depending on the dependence structure. Through simulations, we conclude by comparing polyserial and Pearson correlations, as well as Kendall’s tau (-b), in the same settings. Our results have practical implications for copula-based modeling of mixed-type data.