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.

It is well known that uniformly expanding circle maps whose derivative is Holder continuous have a unique ergodic invariant probability measure absolutely continuous with respect to Lebesgue. This result was extended by Fan and Jiang in 2001 to maps whose derivative is Dini-integrable. However, there exist counterexamples, both to the existence and to the uniqueness, for C^1 maps for which the derivative is less regular. We show that nevertheless, for any given modulus of continuity, there is a C^1 uniformly expanding map of the circle whose derivative has that modulus of continuity and has an invariant probability measure equivalent to Lebesgue.

We introduce a class of one-dimensional full branch maps which may admit up to two neutral fixed points as well as critical points and/or singularities with unbounded derivative. We give a complete classification of the possible physical measures which may appear (or not), study some other statistical properties, and show that different behaviour can be quite intermingled in parameter space. This is joint work with Douglas Coates and Muhammad Mubarak.

  The quantum affine algebras U_q are Hopf algebras with the coproduct Δ defined by Drinfeld and Jimbo; but they have also a "coproduct" Δ_v with values in a completion of U_q ⊗ U_q , introduced by Drinfeld for quantm affinizations. While the relation between Δ and the action of the braid group (and also of the weight lattice, which is a subgroup of the braid group) is complicated and involves the R-matrix, Δ_v is by construction equivariant with respect to the action of the weight lattice.
  In this talk I will show that Δ_v can be obtained as "equivariant limit" of Δ .

  Given a family of smooth projective curves and an arbitrary connected linear algebraic group G, we investigate the Picard group of the stack of relative G-bundles on the family.
  This is a joint work with Roberto Fringuelli.