**Schedule:**

9:40 – 10:30 Pavao Mardesic (Unversity of Bourgogne) The Geometric Origin of the Tennis Racket Effect

10:40 – 11:30 Bohuan Lin (XJTLU) Covariance Matrices and Essential Parameters for Many-Body Systems

11:40 – 12:30 Andrea Giacobbe (University of Catania) A Geometric Approach to Bifurcations of Linear Systems

**Speaker**: Pavao Mardesic

**Title**: The Geometric Origin of the Tennis Racket Effect

**Abstract**:

The tennis racket effect is a geometric phenomenon which occurs in a free rotation of a three-dimensional rigid body, when a tennis racket thrown in the air does not perform a simple rotation. We explain geometrically that its appearence originates from a pole of a Riemann surface as a result of Picard-Lefschetz formula. We study also related phenomena: a skate board trick called Monster Flip and the Dzhanibekov effect in which a wing nut, spinning around its central axis suddenly makes a half-turn flip. The talk is mainly based on a joint work with Gabriela Gutierrez Guillen, Leo Van Damme and Dominique Sugny

**Speaker**: Bohuan Lin

**Title**: Covariance Matrices and Essential Parameters for Many-Body Systems

**Abstract**:

The description of molecular configurations plays a central role in the study of molecular dynamics. While rotation of a molecule does not change its configuration, molecular vibration does and is known to give rise to rotation due to certain topological structures. In this talk, we discuss the geometry of the configuration spaces for molecular vibrations, and take a look at the information of the molecular configuration and vibration revealed by relevant covariance matrices.

**Speaker**: Andrea Giacobbe

**Title**: A geometric approach to bifurcations of linear systems

**Abstract**:

A parameter-dependent dynamical system is always subject to bifurcations that generally affect its qualitative evolution and the stability of the equilibria in particular. The results of Routh and the so-called Routh-Hurwitz conditions are the most commonly used techniques to investigate the linear stability of a given equilibrium and to define the domain in parameter space in which a given equilibrium is stable. In this presentation, we revisit these results concentrating on marginal hypersurfaces at which bifurcations take place. Marginal hypersurfaces are algebraic and semi-algebraic varieties in invariant space that can be pulled back to the space of parameters of the dynamical system. They possess a stratified structure and labeling due to the type of bifurcations and their codimension.

Due to Grobman-Hartman theorem, generic equilibria can be associated with four natural numbers (whose sum is the dimension of the phase space), a bifurcation generates a change in such numbers, and is connected to one of the following marginal events:

– an eigenvalue vanishes;

– two complex-conjugate eigenvalues touch the purely imaginary axes;

– a real eigenvalue has higher multiplicity.

An analysis of the characteristic polynomial allows us to characterize these marginal events, define hypersurfaces that separate the parameter space in connected domains in which the four natural numbers are constant, and characterize the hypersurfaces depending on the type of change that takes place crossing them. The four natural numbers can be computed analytically, using residues and a sequence of integers called Sturm’s sequence. The analysis can be extended to the Hamiltonian case and Floquet theory.