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This talk focuses on a mathematical model of dengue fever transmission, incorporating delay terms to reflect realistic time delays in the disease dynamics. Using the next-generation matrix approach, the basic reproduction number ($R_0$) is derived, providing critical insight into the conditions for disease persistence or eradication. The model is further analyzed through nondimensionalization to simplify the system and identify key parameters. Equilibrium points are determined, and their stability is examined in the presence of delays. Numerical simulations are presented for specific parameter sets to illustrate the dynamics, highlighting the impact of delays on disease progression and stability. These findings provide valuable insights into the role of temporal factors in infectious disease modeling and contribute to a deeper understanding of disease dynamics.

Abstract

We investigate the identification of the time-dependent source term in the diffusion equation using boundary measurements. This facilitates tracing back the origins of environmental pollutants. Employing the concept of dynamic complex geometrical optics (CGO) solutions, a variational formulation of the inverse source problem is analyzed, leading to a proof of uniqueness result. Our proposed two-step reconstruction algorithm first determines the point source locations and subsequently reconstructs the Fourier components of the emission concentration functions. Numerical experiments on simulated data are conducted. The results demonstrate that the proposed two-step reconstruction algorithm can reliably reconstruct multiple point sources and accurately reconstruct the emission concentration functions. Additionally, by partitioning the algorithm into online and offline computations, and concentrating computational demand offline, real-time pollutant traceability becomes feasible. This method, applicable in various fields - especially those related to water pollution, can identify the source of a contaminant in the environment, thus serving as a valuable tool in environmental protection.

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We are pleased to welcome some EDDy alumni for a discussion panel about experiences since EDDy, reflections on the EDDy time, and advice to the current generation of EDDys. Insights into the job market, interviews, etc are welcome.

Following the discussion, we will go for dinner.

17:30-18:00 Arrival, chit-chat

18:00-18:50 Panel discussion: Life after EDDy

19:00- Dinner at the Elisenbrunnen Restaurant

Abstract

We consider the task of characterizing and quantifying the effect of geometric uncertainties on the behavior of a system whose physics is described by partial differential equations (PDEs). We focus in particular on uncertainty in the shape of the physical domain or of an internal interface. We first address how such uncertainties can be modeled, and how to efficiently compute different realizations of the PDE solution. Then, we will address both the forward problem, that is the task of propagating the uncertainty from the geometry to the PDE solution, and the inverse problem in a Bayesian setting, that is using indirect measurement data to characterize the shape uncertainty. For both cases, we will discuss computational methods for efficient uncertainty quantification and their theoretical guarantees.

Abstract

Evolutions by self-similarity constitute an important paradigm in geometric flows. They often convey information about the dynamical properties of a given flow. This principle is particularly evident in the well-known curve-shortening flow (CSF), where homothetic shrinkers and translators naturally arise in singularity analysis. Furthermore, stability analysis shows that closed CSF-shrinkers act as saddle points for the asymptotic behavior of the flow. Such shrinkers are fully classified as either circular or belonging to the family of Abresch-Langer curves.

In this talk, we present an analog classification scheme for a non-local, area-preserving version of CSF. As a corollary, we obtain a rigidity result for closed $\lambda$-curves, confirming a conjecture by J.-E. Chang. Finally, we establish a saddle-point property for this constrained setting.

Abstract

In this talk, we present an algebraic-topological framework designed to elucidate the behavior of diverse dynamical systems. The central idea is to construct a filtered chain complex that captures connections between the system’s invariant sets. The unfolding of the associated spectral sequence reveals a rich algebraic structure, offering insights into the dynamical properties of a one-parameter family of flows. This analysis sheds light on phenomena such as bifurcations resulting from the cancellation of singularities and periodic orbits, as well as the emergence of new periodic orbits. Our approach bridges algebraic topology and dynamical systems, providing a fresh perspective on complex dynamical behaviors.