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.

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.

  A polynomial identity of an algebra A is a polynomial (in non-commuting variables) vanishing under all evaluations in A. Algebras satisfying at least one of these non-trivial relations are called PI-algebras. The first goal of this talk is to show how it is possible to characterize PI-algebras by means of numerical invariants related to their identities. Along the way we will highlight the combinatorial and analytical aspects of the theory, its connection with invariant theory, representation theory and growth problems. In the last part, we will look at some of the latest developments in this area, when it comes to algebras with additional structure.

alpha-induction is known to be a construction of a modular invariant full Q-system from a chiral Q-system of a conformal net. In this talk, I would like to introduce the equivariant version of this procedure. Indeed, we need a new construction of tensor categories, the neutral double construction, introduced by the speaker. Moreover, I would like to explain the relationship between the neutral double construction and an equivariant version of the center construction, which is also introduced by the speaker.

The tensor functor called alpha-induction arises from a Q-system in a braided unitary fusion category. In the operator algebraic language, it gives extensions of endomorphism of N to M arising from a subfactor N subset M of finite index and finite depth giving a braided fusion category of endomorpshisms of N. We study this alpha-induction for bi-unitary connections, which give a characterization of finite-dimensional nondegenerate commuting squares. We show that the resulting alpha-induced bi-unitary connections are flat if we have a local Q-system.

  The celebrated Cauchy identity rewrites a certain infinite product as a sum of products of Schur functions. The identity has a vast number of applications and interpretations; in particular, one can understand the infinite product as the character of polynomial functions on the space of square matrices and the products of Schur functions as characters of tensor products of irreducible gl(n) modules.
  The classical Cauchy identity has (at least) three natural generalizations: a nonsymmetric version, a q-version and their mixture. The representation theory of the nonsymmetric Cauchy is governed by the Borel subalgebra, the q-Cauchy identity is controlled by the representations of the current algebras and the nonsymmetric q-Cauchy identity has to do with the modules over the Iwahori algebra. We will discuss all these identities and explain the relevant representation theory.
  Based on joint works with Anton Khoroshkin, Ievgen Makedonskyi and Daniel Orr.

  Quiver Grassmannians are projective varieties parametrizing subrepresentations of quiver representations. Originating in the geometric study of quiver representations and in cluster algebra theory, they have been applied extensively in recent years in a Lie-theoretic context, namely as a fruitful source for degenerations of (affine) flag varieties. This approach allows for an application of homological methods from the representation theory of quivers to the study of such degenerate structures. The resulting varieties being typically singular, a construction of natural desingularizations is very desirable.
  We construct torus equivariant desingularizations of quiver Grassmannians for arbitrary nilpotent representations of an equioriented cycle quiver. This applies to the computation of their torus equivariant cohomology.