We describe the recent construction of Birkhoff cones which are contracted by the action of transfer operators corresponding to dispersing billiard maps. The explicit contraction provided by this construction permits the study of statistical properties of a variety of sequential and open billiards. We will discuss some applications of this technique to chaotic scattering and the random Lorentz gas. This is joint work with C. Liverani.

Combinatorial graphs can serve as a nice tool for description of dynamical systems on Cantor set. A classical example of this type are Bratelli- Vershik diagrams. Recently, Shimomura, motivated by works of Akin, Glasner and Weiss, developed an alternative approach, which helps to describe dynamical systems on Cantor set by employing inverse limit of graphs. This approach provides a useful tool for description of dynamical systems on Cantor set. As a particular application of the above approach we will present a method of construction of Cantor set $C$ with prescribed dynamics and its extension to interval maps with derivative zero on $C$. Starting motivation for this study is an old question whether invariant subset $Csubset [0,1]$ on which derivative of interval map $f$ vanishes must contain a periodic point.

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Note: This event is part of the activity of the MIUR Department of Excellence Project MatMod@TOV.

We recall and discuss the result of Baladi & Tsujii which tells that, under mild conditions, a linear operator considered acting on two different Banach spaces will have spectra which coincide outside of the essential spectrum. * [Lemma A.3 of "Dynamical zeta functions and dynamical determinants for hyperbolic maps" by Viviane Baladi].

We will be interested in the analysis of a system of PDEs modeling continuously stratified fluids under the influence of gravity. The system is obtained by a linearization of the equations of incompressible non-homogeneous fluids (non-homogeneous Euler equations) around a background density profile that increases with depth (spectrally stable density profile). I will present some mathematical problems related to (asymptotic) stability and long-time dynamics.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

  On a pseudo Riemannian manifold consider a 2-form taking value in the adjoint representation of some (semisimple) Lie algebra. It is well known that the corresponding Yang-Mills functional is conformally invariant just in four dimensions. A natural question is whether there are natural replacements of the Yang-Mills functional that are conformally invariant.
  In the first part of the talk we will describe the main tools needed to answer this question, namely conformal defining densities for conformally compact manifolds and the (adjoint) tractor bundle; then we will show how to set up and to formally solve the Yang-Mills boundary problem on conformally compact manifolds. In general, smooth solutions are obstructed by an invariant of boundary connections. Specializing to Poincaré-Einstein manifolds with even boundary dimension parity, this obstruction is a conformal invariant of boundary Yang-Mills connections. This yields conformally invariant, higher order generalizations of the Yang-Mills equations and their corresponding energy functionals.

  While in classical differential geometry one is given a unique differential structure, the de Rham calculus, such a canonical choice does not exist in noncommutative geometry. Moreover, while the de Rham differential is equivariant with respect to a given Lie group action, a noncommutative calculus might not be compatible with a corresponding Hopf algebra symmetry.
  We give a gentle introduction to noncommutative differential geometry, reviewing seminal work of Woronowicz (covariant calculi on Hopf algebras) and Hermisson (covariant calculi on quantum homogeneous spaces). The latter invokes the notion of faithful flatness and Takeuchi/Schneider equivalence. Afterwards we discuss an original construction of a canonical equivariant calculus for algebras in symmetric monoidal categories, with main examples including algebras with (co)triangular Hopf algebra symmetry, particularly Drinfel’d twisted (star product) algebras. The approach relies on and is essentially dual to the concept of ‘braided derivations’ and we show that the corresponding braided Gerstenhaber algebra of multi-vector fields combines with the noncommutative calculus, forming a braided Cartan calculus. If time permits we illustrate how to formulate Riemannian geometry in this framework, proving that for every equivariant braided metric there is a unique quantum Levi-Civita connection. The second half of the talk is based on the thesis of the speaker.

Denoting with H^n the n-dimensional hyperbolic space, we show that constant mean curvature hypersurfaces in H^n x R with small boundary contained in a horizontal slice P are topological disks, provided they are contained in one of the two half-spaces determined by P. This is the analogous in H^n x R of a result in R^3 by A. Ros and H. Rosenberg. The proof is based on geometric and analytic methods : from one side the constant mean curvature equation is a quasilinear elliptic PDE on manifolds, to the other the specific geometry of the ambient space produces some peculiar phenomena. This talk is based on a joint work with Barbara Nelli.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C18000100006

Link to the abstract

Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

  In the 80s Bryan Hartley conjectured that if the unit group a torsion group algebra FG satisfies a group identity, then FG satisfies a polynomial identity. In this talk we aim to review the most relevant results that arose from its solution and to discuss some recent developments concerning group identities for the set of symmetric units of FG.