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Abstract

Nonlocal regularization of conservation laws consists in conservation laws in which the flux function depends on the solution through the convolution with an exponential kernel. We study two singular limits associated to the problem: 1) the convergence of the solutions as the nonlocality shrinks to a local evaluation, i.e., when the kernel tends to a Dirac delta distribution; 2) the long-time behavior as the time tends to infinity.

This talk is based on works in collaboration with G. M. Coclite, M. Colombo, J.-M. Coron, G. Crippa, N. De Nitti, K. H. Karlsen, A. Keimer, E. Marconi, L. Pflug, N. H. Risebro, L. Spinolo, and E. Zuazua.

Abstract

Consider a strictly hyperbolic system of conservation laws, where each characteristic field is either genuinely nonlinear or linearly degenerate. In this standard setting, it is well known that there exists a Lipschitz semigroup of weak solutions, defined on a domain of functions with small total variation. If the system admits a strictly convex entropy, we give a short proof that every entropy weak solution taking values within the domain of the semigroup coincides with a semigroup trajectory. The result shows that the assumptions of “Tame Variation” or “Tame Oscillation”, previously used to achieve uniqueness, can be removed in the presence of a strictly convex entropy.

Combined with a compactness argument, the result yields a uniform convergence rate for a very wide class of approximation algorithms. Some partial estimates on the convergence rate are given.

Abstract

Since the beginning of the 1900s, linearized Cosserat elasticity is well known and often used in the engineering community for modeling micropolar elastic solids. But from a mathematician’s perspective a geometrically nonlinear version of the model is interesting. Existence of solutions for the latter has been known for about 20 years, while regularity questions were investigated only during the last couple of years.

In the first part of the talk, the Cosserat bulk model will be introduced and an overview of recently developed different regularity results will be given, for Cosserat energy minimizers as well as for critical points. Classical regularity theory for harmonic maps into manifolds is an essential tool in deriving those results. At the same time, the geometric nature of the model’s nonlinearities in some cases allows not only regular, but also quite singular solutions to exist, which will be the focus of the second part of the talk.

Finally, those singular solutions do not exist, when we go away from the (3d-) Cosserat bulk model towards a Cosserat model for shells, i.e. very thin materials, whose elastic behaviour is governed by their (2d-) midsurface.

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In this talk, we will discuss a shape space framework for collision-aware geometric modeling, where basic geometric operations automatically avoid interpenetration. Shape spaces are a powerful tool for surface modeling, shape analysis, nonrigid motion planning, and animation, but past formulations permit nonphysical intersections. Our framework augments an existing shape space using a repulsive energy such that collision avoidance becomes a first-class property, encoded in the Riemannian metric itself. In turn, tasks like intersection-free shape interpolation or motion extrapolation amount to simply computing geodesic paths via standard numerical algorithms. To make optimization practical, we develop an adaptive collision penalty that prevents mesh self-intersection. The final algorithms apply to any category of shape, and do not require a dataset of examples, training, rigging, nor any other prior information. For instance, to interpolate between two shapes we need only a single pair of meshes with the same connectivity. We evaluate our method on a variety of challenging examples from modeling and animation.

Abstract

I will focus on the consensus-based optimization model as an example of a McKean-Vlasov dynamics arising from interacting particles.

I will discuss topics around: the mean-field limit, the long-time behavior, and the low-temperature regime. But before doing so, I will try to provide some intuition from the stability of differential equations that shall be useful for the presentation.

This is based on joint works with Hui Huang (Uni. Graz) and Lukang Sun (TUM).

Abstract

Composite materials are important to modern technology as the mixing of two different material properties at fine scales can give rise to unexpected emergent behavior. Therefore, understanding the process of phase separation on such materials is crucial to leveraging these processes for technological applications.

A variational model for the interaction between homogenization and phase separation in a double-periodic material is considered. The focus of the talk is on the regime where the separation happens at a bigger scale than both micro-scales, and a phase separation for the homogenized functional is expected.

A mathematical introduction to the history of the phase separation functional will be given, and with it the tools needed to tackle the double-periodic case, such as Γ-convergence and the periodic unfolding operator.

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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.

Abstract

Differential geometry meets infinite dimensional calculus: A Riemannian metric on the space of embeddings $\mathbb{S}^1 \rightarrow\mathbb{R}^m$

Arnold introduced a new perspective in the field of geometric analysis and geometric mechanics: infinite-dimensional differential geometry. Ebin and Marsden expanded on their ideas and developed a more general framework. This gave rise to a new branch in mathematics. In my talk, I motivate their new perspective through a (finite-dimensional) example. After that, I am going to introduce the geometry of mapping spaces. This will lead to the setting of embeddings and the occurring difficulties. Since the field of infinite-dimensional geometry is quite broad and still under heavy research, I will restrict myself to “strong Riemannian metrics”. My colleagues and I designed a Riemannian metric on the space of embeddings of the circle. Inspired by the tangent-point energies, we were able to incorporate self-repulsion into a metric on the mentioned manifold. I will elaborate on how we were able to overcome the topological difficulties of the manifold and some basic concepts of the theory. In the second half of the talk, I will explain the result we were able to achieve and give short proofs for some theorems. This will clarify the analytical difficulties of this theory, for example, the role of regularity. In addition, I am going to explain some differences between finite- and infinite-dimensional geometry. Finally, we will finish by elaborating on how our geometrical statements affect the dynamics of the deformations and give an outlook onto future work and other directions open for exploration.

Abstract

The main goal of this basic notions seminar is to introduce the audience to the core idea(s) of (co-)homology: To demystify the concept, we will begin with a motivational section, foreshadowing the homological origins of the well-known Poincaré Lemma and the (finite-dim.) Mountain Pass Lemma. Then, we will introduce the (co-)homology theory via general (co-)chain complexes. Next, we will take a look at how to obtain different (co-)chain complexes from various (co-)homology theories yield, and we will try to generate some intuitive understanding of these different theories. At last, we will revisit the examples from the motivational section.

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This course aims to briefly introduce Conley Index Theory, focusing on the fundamental ideas of Conley’s approach to studying dynamical systems. We begin with the basic definitions and key properties of the Conley index, then we explore the construction of Morse decompositions and discuss Morse-Conley inequalities.

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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.