Programme

The conference will consist in two courses given by renowned researchers, some plenary talks, several contributed short talks, and a poster session.

Courses

■ Narciso Román-Roy (Universitat Politècnica de Catalunya)

Title: An introductory course on geometric field theories: the multisymplectic setting

Abstract:

This course provides an introduction to the multisymplectic formulation of first-order classical field theories. Multisymplectic geometry offers the most general and powerful framework for a covariant description of such theories. The course begins with a concise overview of geometric field theories and introduces the fundamental concepts of multisymplectic manifolds, together with the geometry of first-order jet bundles and bundles of differential forms that form the mathematical setting of the multisymplectic approach.

Within this framework, the Lagrangian formalism for first-order classical field theories is developed, covering both regular and singular cases, and the corresponding Hamiltonian formalism is constructed for regular and almost-regular theories. The course also addresses the basic notions of symmetries and conservation laws, and illustrates the general theory through a selection of relevant examples drawn from modern physics.

■ Alice Barbora Tumpach (Wolfgang Pauli Institut Vienna and Lille University)

Title: Geometry of infinite-dimensional manifolds and applications to integrable systems

Abstract:

This lecture is an introduction to geometric structures on infinite-dimensional manifolds, with applications to the study of integrable systems. We will start with basic notions of manifolds and fiber bundles with model spaces Hilbert, Banach or Fréchet spaces, and then explore geometric structures on infinite-dimensional manifolds, in particular symplectic and Poisson structures. We will show that the Leibniz rule for a Poisson bracket on a Banach manifold does not imply the existence of a Poisson tensor. The existence of a Hamiltonian vector field on a Banach manifold endowed with a Poisson tensor is also not guaranteed. We will review possible definitions of Poisson structures in the infinite-dimensional context that are general enough to include non-trivial examples, avoid possible pitfalls and include all weak symplectic Banach manifolds. Examples of some Banach Poisson-Lie groups and their homogeneous spaces related to the Korteweg-de-Vries hierarchy will be presented. In particular, the construction of a Banach-Poisson-Lie group structure on the unitary group of a Hilbert space will serve as a guide for other unitary groups, like the restricted unitary group. We will show that the restricted Grassmannian inherite a Bruhat-Poisson structure from the restricted unitary group, and that the action of a triangular Banach Lie group on it by "dressing transformations" is a Poisson map which generates the Korteweg-de Vries hierarchy.

Invited talks

■ Luca Vitagliano (Università di Salerno, Italy)

Title: Homogeneous Boundaries of Geometric Structures

Abstract:

Under appropriate homogeneity conditions, a hypersurface in a symplectic manifold inherits from the ambient a contact or a cosymplectic structure. There are similar statements for Poisson manifolds as well as for complex manifolds. Using ideas from the homogeneous symplectic approach to contact geometry, we present a very general theorem putting all these statements under the same umbrella. This also allows generalizations, e.g., to Kahler Geometry, Dirac Geometry, Generalized Complex Geometry, etc. This is joint work with Alfonso Tortorella.

■ Jesús Clemente-Gallardo (University of Zaragoza, Spain)

Title: TBA

Abstract:

TBA

■ Chenchang Zhu (Georg-August-Universitat Gottingen, Germany)

Title: TBA

Abstract:

TBA

■ José Miguel Figueroa-O’Farrill (University of Edinburgh, United Kingdom)

Title: TBA

Abstract:

TBA

■ Melanie Graf (University of Hamburg, Germany)

Title: Singularity theorems from a mathematical perspective

Abstract:

The classical singularity theorems of R. Penrose and S. Hawking from the 1960s are beautiful examples of mathematical results in Lorentzian Geometry with wide physical relevance for General Relativity showing that any spacetime with a smooth Lorentzian metric satisfying certain energy conditions and causality assumptions must be geodesically incomplete. Despite their great success these classical theorems still had and have some drawbacks both in their assumptions and conclusions. Focusing on the assumptions side of the picture I will review the classical theorems and some recent mathematical progress allowing us to relax some of the usual demands placed on the metric regularity as well as the classical pointwise energy conditions, which makes the theorems applicable to a wider class of physically relevant situations.

■ Gonçalo Oliveira (Instituto Superior Tecnico, Lisbon, Portugal)

Title: TBA

Abstract:

TBA

■ Zibo Wang (University of Chinese Academy of Sciences, China)

Title: TBA

Abstract:

TBA

More speakers are to be confirmed soon...

Contributed talks

Poster session