On a new approach to scattering for dispersive equations
Resumo: The concentration-compactness-rigidity method, pioneered by Kenig and Merle, has become standard in the study of global well-posedness and scattering in the context of dispersive and wave equations. Albeit powerful, it requires building some heavy machinery in order to obtain the desired space-time bounds.
In this talk, we present a simpler method, based on Tao's scattering criterion and on Dodson-Murphy's Virial/Morawetz inequalities, first proved for the 3d cubic nonlinear Schrödinger (NLS) equation. Tao's criterion is, in some sense, universal, and it is expected to work in similar ways for dispersive problems. On the other hand, the Virial/Morawetz inequalities need to be established individually for each problem, as they rely on monotonicity formulae. This approach is versatile, as it was shown to work in the energy-subcritical setting for different nonlinearities, as well as for higher-order equations.
Derived brackets and generalized complex structures
Resumo: Let $M$ be a smooth manifold. A generalized complex structure is a complex structure on $V= TM \oplus T^*M$ which is closed under a peculiar bracket called the Courant bracket. I will motivate the definition of the Courant bracket using the language of derived brackets developed by J.-L. Loday and Y. Kosmann-Schwarzbach. The talk is supposed to be elementary and accessible to all students with some knowledge of differential geometry (manifolds, tensors, de Rham differential).
Renormalization of Hénon maps
Resumo: For surface diffeomorphisms, a strictly positive entropy is associated with the existence of 'horseshoes'. I will focus on surface diffeomorphisms with zero entropy: can the dynamics of these 'simple' systems be described? how does it bifurcate to positive entropy systems? These questions will be discussed for a class of volume-contracting surface diffeomorphisms whose dynamics is intermediate between one-dimensional dynamics and general surface dynamics. It includes the dynamics of any Hénon diffeomorphism with Jacobian smaller than 1/4. De Carvalho, Lyubich and Martens have built a renormalization for the (real) Hénon maps with Jacobian close to 0 and described the boundary of the parameters with zero entropy. With E. Pujals and C. Tresser we extend some of these results up to Jacobian 1/4: any Hénon map with zero entropy can be renormalized. As a consequence, we obtained a two-dimensional version of Sharkovsky’s theorem about the set of periods of interval maps.
Cohomogeneity one curvature homogeneous manifolds in dimension 4
Resumo: A Riemannian manifold M is curvature homogenous if, given two points in M there exists a linear isometry f between the corresponding tangent spaces such that the curvature tensor R of M is preserved, i.e. f^* R=R. This is trivially true if the manifold is homogeneous but there are inhomogeneous examples. This condition can be formulated as a system of (highly overdetermined) pde's. We will talk about a joint paper with W. Ziller where we classify the curvature homogeneous manifolds in dimension four that carry an isometric action with principal orbits of codimension one, i.e. the highest possible degree of symmetry of a non homogenous manifold.
Recent projects at the intersections of art, science and technology
Resumo: The talk will present recent projects at the intersections of art, science and technology such as “Mechanical Unconscious”, a sound installation consisting of a real-time dialogue between old-fashioned telegraph machines, synthetic voices and telephone sounds, sometimes blurring, sometimes imposing the limitsbetween natural and artificial language.
Otavio Schipper holds a degree in Physics from Universidade Federal do Rio de Janeiro. He has received the KLAS Award from the Max Planck Society in 2017 and the Berlin Fellowship from the Akademie der Künste in 2015. Schipper has been invited as Guest Artist by the Arts at CERN program in 2019.
Through the presentation of ready-made objects such as antique telegraph machines, tuning forks, eyeglasses, elevator cabins and electric poles, Otavio Schipper’s work connects past physical worlds with our present mental landscapes. he realms of imagination become present in installations that deal with the perception of time and the cultural memory of objects, leading to the questioning of boundaries between fiction and reality. In Schipper’s installations, the spectator often experiences a spectrum of sensations, from enlightenment to nostalgia. Elements of science and technology lead the viewer into territories associated with dream states