# Abstracts

Anna Capietto, Università di Torino

Title: Existence and multiplicity of special solutions to a nonlinear eigenvalue problem in the half-line arising from the Dirac equation
Abstract: We provide the existence of a global continuum of solutions to a nonlinear PDE which have a special form. To this aim, we exploit the notion of partial wave subspace in relation with the well-known concept of rotation number.

Alberto Boscaggin, Università di Torino

Title: Periodic solutions of superlinear systems of ODEs: a symplectic approach
Abstract: We prove the existence of infinitely many nodal periodic solutions of  a weakly coupled superlinear system of ODEs. The proof relies on an elementary (higher dimensional) version of the Poincaré-Birkhoff fixed  point theorem, valid for exact symplectic maps with monotone twist. Joint work with Rafael Ortega (University of Granada).

Graham Hugh Cox, University of North Carolina at Chapel Hill

Title: Relating the Morse and Maslov index in an arbitrary domain
Abstract: In a recent paper of Deng and Jones, it was shown that the Morse index of an elliptic boundary value problem can be determined using methods from infinite-dimensional symplectic geometry. This idea extends the classic work of Smale, on higher-dimensional Morse theory, to more general boundary conditions, but has only been carried out for star-shaped domains. In this talk I will explain a generalization these results to domains that are diffeomorphic to a ball, and discuss some open problems for domains with nontrivial topology.

Walter Dambrosio, Università di Torino
Title:  On the boundedness of solutions to a nonlinear singular oscillator
Abstrac: We study a second order scalar equation of the form

$x''+V'(x)=p(t),$

where $p$ is a perodic function and $V$ is a singular potential. We give sufficient conditions on $V$ and $p$ ensuring that all solutions are bounded; we prove the existence of Aubry-Mather sets as well.

Francesca Dalbono, Università di Palermo
Title:  Nodal solutions for supercritical Laplace and p-Laplace equations
Abstract: We study asymptotic behaviour and nodal properties of the radial solutions to a superlinear p-Laplace equation. Our approach is based on the Fowler transformation combined with invariant manifold theory. (This is a joint work with Matteo Franca).

Russell Johnson, Università di Firenze

Title:  The nonautonomous version of the Yakubovich frequency theorem
Abstract: The Frequency Theorem is a basic result in linear-quadratic optimal control theory which has applications to problems concerning absolute stability, dissipative systems, and nonconvex optimization. It was formulated and proved by Yakubovich for control systems with time-periodic coefficients, and later extended to cover the general nonautonomous case when the coefficients  depend aperiodically on time. We will discuss the nonautonomous version of the Frequency Theorem, and indicate in particular how the rotation number for linear Hamiltonian ODEs can be used to verify its hypothesis. The results to be presented are contained in joint work with Roberta Fabbri and Carmen Nunez.

Chris Jones, University of North Carolina at Chapel Hill
Title:  Topological characterizations of the stability index for nonlinear waves
Abstract:TBA

Alessandro Portaluri, Università di Torino
Title:  Linear (in)stability for relative equilibria in singular Lagrangian systems
Abstract: Relative equilibria are maybe the simplest periodic orbit of Newtonian type singualr Lagrangian systems.  The goal of this talk is to  study  the linear and spectral (in)stability of relative equilibria for singular Lagrangian systems. By using symplectic and topological techniques we provide a sufficient condition for  a relative equilibrium to be linearly unstable as well as a necessary condition in order to be spectrally stable. (Joint with Vivina L. Barutello, R. Jadanza)

Nils Waterstraat, Politecnico di Torino
Title:  On the Morse index theorem in semi-Riemannian geometry
Abstrac: In the first part of our talk, we consider the Fredholm Lagrangian Grassmannian of a symplectic Hilbert space, the Maslov index and symplectic reductions.
Afterwards we give a brief recapitulation of the classical Morse index theorem from Riemannian geometry and its generalisation to geodesics in semi-Riemannian manifolds.
Finally, we introduce a new proof of the semi-Riemannian Morse index theorem, which is based on symplectic reductions. (Joint work with A. Portaluri)

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