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This talk was part of the Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications" held at the ESI January 13 -- February 14, 2025. In this talk, I will survey some aspects relating classes of PDEs with metrics on a 2-dimensional manifold with non zero constant Gaussian curvature. The notion of a differential equation (or system of equations) describing pseudo-spherical surfaces (curvature -1) or spherical surfaces (curvature 1) will be introduced. Such equations have rema... read more
https://phaidra.univie.ac.at/o:2114329
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This talk was part of the Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications" held at the ESI January 13 -- February 14, 2025. In this tutorial we will give a brief presentation of the several equivalent ways to define a covariant derivative in the infinite dimensional Grassmann manifolds, and we will show how to use some operator theoretic methods can be used to study conjugated points along geodesics. read more
https://phaidra.univie.ac.at/o:2114328
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This talk was part of the Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications" held at the ESI January 13 -- February 14, 2025. The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. In this talk we will discuss universality properties of the stationary solutions to the Euler equations. The study of these universality features was suggested by Tao as a novel way to address the problem of global existence ... read more
https://phaidra.univie.ac.at/o:2114327
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This talk was part of the Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications" held at the ESI January 13 -- February 14, 2025. We use local symplectic Lie groupoids to approximate Hamiltonian dynamics for generic Poisson structures. More precisely, recursively obtained solutions of a Hamilton-Jacobi-like equation are interpreted as Lagrangian bisections in a neighborhood of the unit manifold, that, in turn, provide the desired approximation. This approximation provide... read more
https://phaidra.univie.ac.at/o:2114326
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This talk was part of the Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications" held at the ESI January 13 -- February 14, 2025. In 1995, Claude Roger conjectured that the universal central extension of the Lie algebra of exact divergence-free vector fields should have a particularly straightforward description in terms of the de Rham cohomolgy in codimension 2. We give an outline of a recent proof of this result, and use it to construct the universal central extension ... read more
https://phaidra.univie.ac.at/o:2114325
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This talk was part of the Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications" held at the ESI January 13 -- February 14, 2025. This talk explores the geometric and stability aspects of quasi-geostrophic flows on the sphere, with a focus on the curvature of the configuration space. We will analyze the long-term behavior of some exact solutions generated by spherical harmonics, examining the relationship between curvature, conjugate points, and the Coriolis force. Speci... read more
https://phaidra.univie.ac.at/o:2114324
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This talk was part of the Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications" held at the ESI January 13 -- February 14, 2025. We present an overview of several integrable systems that can be described in the context of Banach (Poisson or symplectic) manifolds, including KdV and Toda-lattice. Time permitting results obtained together with A. Odzijewicz and A.B. Tumpach will also be mentioned. read more
https://phaidra.univie.ac.at/o:2114323
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This talk was part of the Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications" held at the ESI January 13 -- February 14, 2025. According to J.L. Lagrange, the variational description of mechanics "reduces all the laws of motion of bodies to their equilibrium and thus brings dynamics back into statics." Since statics is conveniently formulated in the language of differential geometry, we can also view variational calculus as a way to integrate differential geometry int... read more
https://phaidra.univie.ac.at/o:2114322
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This talk was part of the Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications" held at the ESI January 13 -- February 14, 2025. According to J.L. Lagrange, the variational description of mechanics "reduces all the laws of motion of bodies to their equilibrium and thus brings dynamics back into statics." Since statics is conveniently formulated in the language of differential geometry, we can also view variational calculus as a way to integrate differential geometry int... read more
https://phaidra.univie.ac.at/o:2114321
All rights reserved
This talk was part of the Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications" held at the ESI January 13 -- February 14, 2025. This course will provide an introduction to the theory of integrable systems from the perspective of infinite-dimensional geometry. read more
https://phaidra.univie.ac.at/o:2114320
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