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 .

The shadowing property is a landmark of the dynamical systems theory which is deeply related to stability phenomena. Hyperbolicity is a famous source of systems with the shadowing property. Nevertheless, the shadowing property does not hold for systems beyond the hyperbolic ones. Indeed, the Lorenz attractor is a paradigmatic example of non-hyperbolic flow which reassembles several properties of the hyperbolic ones, although it does not satisfy the shadowing property, as it was showed by M. Komuro. Several years later L. Wen and X. Wen extended Komuro's results and proved that a sectional hyperbolic set does not satisfy the shadowing property, unless it is hyperbolic. In this talk, we will push further this discussion and ask whether the non-shadowableness of singular flows is due to the sectional hyperbolicity or it is, in fact, a consequence of existence of attached singulaties. This is a joint work with A. Arbieto, A Lopez and Y. Sanchez.

In this talk I will discuss relations between chaotic and hyperbolic systems. More specifically, how we can obtain known results from hyperbolic dynamics using stronger notions of sensitivity to initial conditions. I will briefly explain expansiveness, topological hyperbolicity, cw-expansiveness, cw-hyperbolicity and first-time sensitivity.

Starting from Namah, Roquejoffre (1999) and Fathi (1998), the large time asymptotic behavior of solutions to Hamilton-Jacobi equations has been extensively investigated by many authors, mostly on smooth compact manifolds or the N-dimensional torus. Following recent development due to Pozza, Siconolfi (to appear), we extended this asymptotic analysis to time dependent problems on networks. The main difference between this and more traditional settings is that, for the well posedness of the evolutive problem on networks, the equation must be coupled with a ”flux limiter”, that is the choice of appropriate constants on each vertex of the network. These constants, among other things, bond from above the time derivatives of any subsolution on the vertices. In this talk we will show how this new condition impact the asymptotic behavior of the solutions to the Hamilton-Jacobi problem on networks.

  Fundamental superalgebras are special finite-dimensional superalgebras over an alge-braically closed field of characteristic zero defined in terms of certain multialternating graded polynomials. They play a key role in Kemer's Representability Theorem. In the present talk we provide new examples of fundamental superalgebras. Finally, if time allows, we shall give a characterization of fundamental superalgebras in terms of the representation theory of the hyperoctahedral group.
  This is based on a joint work with Antonio Giambruno and Ernesto Spinelli.

  Quiver Grassmannians are projective varieties parametrising subrepresentations of quiver representations. Their geometry is an interesting object of study, due to the fact that many geometric properties can be studied via the representation theory of quivers. For instance, this method was used to study linear degenerations of flag varieties, obtaining characterizations of flatness, irreducibility and normality via rank tuples.
  We give a construction for smooth quiver Grassmannians of a specific wild quiver, realising a class of Schubert varieties inside flag varieties. This allows for a definition of linear degenerations of Schubert varieties.

In the theory of local observables in Algebraic Quantum Field Theory (AQFT) modular theory creates a one-parameter group of modular automorphisms from a single state and this group often has a geometric implementation. If modular groups are contained in finite-dimensional Lie groups, they naturally lead to gradings of the Lie algebra and further to causal symmetric spaces. Conversely, we explain how nets of local observables (resp. of standard subspaces) on causal symmetric spaces can be constructed for all irreducible unitary representations of simple Lie groups which are either linear or locally isomorphic to SL(2,R). This is joint work with Jan Frahm (Aarhus), Gestur Olafsson (Baton Rouge) and Vincenzo Morinelli (Rome)

In this talk, we describe some monotonicity formulas holding along the regular level sets of suitable p-harmonic functions in asymptotically flat Riemannian 3-manifolds with a single end, with or without boundary, having nonnegative scalar curvature. Using such formulas, we obtain some geometric inequalities, including the positive mass inequality and the Riemannian Penrose inequality.

It is well known that a family of tent-like maps with bounded derivatives has no linear response for typical deterministic perturbations changing the value of the turning point. In this note we prove the following result: if we consider a tent-like family with a cusp at the turning point, we recover the linear response. This is a joint work with Stefano Galatolo.

  Trusses are like affine rings: as a ring is an abelian group with a compatible multiplica-tion, a truss is a torsor (over an abelian group) with a compatible multiplication. Introduced by Brzeziński in 2017 to unify the classical theory of rings with the modern theory of braces, trusses unexpectedly revealed the existence of a previously uncharted territory, whose exploration is not just leading to fascinating discoveries, but it is also shedding new light on groups and rings themselves. This talk would like to be a brief walk in the world of ternary operations to meet heaps, trusses, their modules and their novel heaps of modules.