Termine

Given a Riemannian manifold, the geodesic X-ray transform of a symmetric tensor field is defined by the line integrals of the latter over geodesics, and it is a central object in geometric inverse problems. Broadly speaking, one is interested in recovering information about the unknown tensor field from its X-ray transform, and the extent to which this is possible depends heavily on the underlying geometry. In this talk, we focus on the geodesic X-ray transform in the geometric setting of two-dimensional asymptotically hyperbolic manifolds, which are non-constant curvature generalizations of hyperbolic space. This setting is interesting in part due to its connections to theoretical physics, specifically the AdS-CFT Correspondence. The geodesic X-ray transform on symmetric tensor fields of positive rank has a natural nullspace, implying that such a tensor field cannot be uniquely recovered from its X-ray transform. On asymptotically hyperbolic surfaces we propose gauge representatives modulo the nullspace of the transform to be reconstructed from the data, by proving a ``transverse traceless-conformal-potential'' decomposition. We then use our tensor decompositions to provide range characterizations of the geodesic X-ray transform in the special case of 2-dimensional hyperbolic space, as well as to develop reconstruction procedures. Based on joint work with François Monard and Yuzhou Zou.

Understanding the melting process of a constructed Electric Arc Furnace is crucial in today´s steel industry. To obtain that understanding, we utilize a controlled ODE system capable of simulating said process. Unknown parameters of the underlying furnace, that serve as input data to the ODE system, are needed to accurately sim- ulate the melting process. Due to the complexity of the system, Inverse Parameter Estimation is not feasible. We therefore train a Neural Operator on simulation data which will then serve as a surrogate of the Forward Operator and thus make Parameter Estimation feasible.

In this talk, I will begin with the classical isoperimetric inequality for closed curves in Euclidean space and its connection to the Plateau problem, as resolved by Douglas and Radó. The notion of non-positive curvature naturally enters the picture via the Gauss equation, which links the intrinsic and extrinsic geometry of surfaces. We then explore the relationship between the isoperimetric inequality and the intrinsic geometry of minimal discs. From there, the discussion extends beyond Euclidean ambient spaces to those of non-positive curvature, and from curves to higher-dimensional cycles. This talk includes results from joint work with Cornelia Druţu, Urs Lang, Alexander Lytchak, Panos Papasoglu, and Stefan Wenger.

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