Affine W-algebras form a family of vertex algebras indexed by the nilpotent orbits of a simple finite dimensional complex Lie algebra. Each of them is built as a noncommutative Hamiltonian reduction of the corresponding affine Kac-Moody algebra. In this talk, I will present a joint work with Naoki Genra about the problem of reduction by stages for these affine W-algebras: given a suitable pair of nilpotent orbits in the simple Lie algebra, it is possible to reconstruct one of the two affine W-algebras associated to these orbits as the Hamiltonian reduction of the other one. I will insist on how this problem relates to our previous work about reduction by stages between Slodowy slices, which are Poisson varieties associated with affine W-algebras. I will also mention some applications and motivations coming from Kraft-Procesi rule for nilpotent Slodowy slices, and isomorphisms between simple affine admissible W-algebras.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006).

Let X be a complete measure space of finite measure. The Lebesgue transform of an integrable function f on X encodes the collection of all the mean-values of f on all measurable subsets of X of (finite and) positive measure. In the problem of the differentiation of integrals, one seeks to recapture f from its Lebesgue transform. In previous work we showed that, in all known results, f may be recaptured from its Lebesgue transform by means of a limiting process associated to an appropriate family of filters defined on the collection A of all measurable subsets of X of (finite and) positive measure.
The first result of the present joint work with Steven G. Krantz, is a precise proof of a result announced in a previous work: the existence of such a limiting process is equivalent to the existence of a Von Neumann-Maharam lifting of X.
In the second result of this work, we provide an independent argument that shows that the recourse to filters is a necessary consequence of the requirement that the process of recapturing from its mean-values may be extended to a natural transformation, in the sense of category theory. This result essentially follows from the Yoneda lemma. As far as we know, this is the first instance of a significant interaction between category theory and the problem of the differentiation of integrals.
In a third result, we have proved, in a precise sense, that natural transformations fall within the general concept of homomorphism. As far as we know, this is a novel conclusion: Although it is often said that natural transformations are homomorphisms of functors, this statement appears to be presented as a mere analogy, not in a precise technical sense. In order to achieve this result, we had to bring to the foreground a notion that is implicit in the subject but has remained hidden in the background, i.e., that of partial magma.

La classificazione delle varietà di Fano di dimensione 3 e indice 1 è uno dei risultati fondamentali in geometria algebrica, completata da Iskovskikh e Mukai più di trent'anni fa. In questo seminario, basato su un progetto in collaborazione con Arend Bayer e Alexander Kuznetsov, presenterò una nuova dimostrazione, basata sempre sulle idee di Mukai, che si estende anche al caso singolare e in dimensione superiore. I will present an approach to the problem using logarithmic geometry which allows us to extend the theory of linear series to arbitrary nodal curves. A prominent role is played by vector bundles on Olsson fans, which I will introduce. This is joint work in progress with Luca Battistella and Jonathan Wise.

Knots form an infinite and complex data set, with topological invariants that are often intertwined in ways not yet fully understood. Many foundational challenges in knot theory and low-dimensional topology can be recast as problems in reinforcement learning and generative machine learning. A key decision in approaching knot theory through an ML lens is determining how to represent knots in a machine-readable format, which can be thought of as selecting a suitable prior distribution over the space of all knots. In this talk, I will explore the challenges of representing knots for ML applications and showcase recent examples where machine learning has been successfully applied to problems in low-dimensional topology.

Equations defining projective varieties and their syzygies have been classically studied. In this talk, starting from the case of curves, I will recall several results about syzygies of projective varieties, especially focusing on some recent ones about abelian and Kummer varieties.

  We will review some theory of algebraic groups over Q_p and the construction of the Bruhat-Titts building for a split group G over Q_p . At the end, we will see some applications and we will mention some results about disconnected groups.

  Hopf algebras (and variations of them) are the algebraic counterpart of (strict, rigid) tensor categories. As such, they appear as symmetries of different categorial, geometrical and physical objects. In particular, several applications may be found in diverse areas of mathematics, physics and theoretical computer sciences.
  Hopf algebras were already studied in the 60's and had a big impulse in the 80's after the work of Drinfeld on quantum groups. Despite more than 60 years of study, not much is known about them: general results are sparse and the classification is only known for (quite) small dimensions or for families with different properties.
  This talk will be about a joint project with N. Andruskiewitsch and G. Carnovale in our attempt to determine finite-dimensional pointed Hopf algebras over finite-simple groups of Lie type. The main idea we exploit is the reduction of the problem to group-theoretical criteria to determine the finite-dimensionality of our objects, which boils down to the use of different properties of conjugacy classes, root systems and computational tools.

In operator algebra theory central sequences have long played a significant role in addressing problems in and around amenability, having been used both as a mechanism for producing various examples beyond the amenable horizon and as a point of leverage for teasing out the finer structure of amenable operator algebras themselves. One of the key themes on the von Neumann algebra side has been the McDuff property for II_1 factors, which asks for the existence of noncommuting central sequences and is equivalent, by a theorem of McDuff, to tensorial absorption of the unique hyperfinite II_1 factor. We will show that, for a topologically free minimal action of a countable amenable group on the Cantor set, the von Neumann algebra of the associated dynamical alternating group is McDuff. This yields the first examples of simple finitely generated nonamenable groups for which the von Neumann algebra is McDuff. This is joint work with Spyros Petrakos.

  A celebrated result in graph theory links the chromatic polynomial of a graph to the Tutte polynomial of the associated graphic matroid. In 2005, Helme-Guizon and Rong proved that the chromatic polynomial is categorified by a cohomological theory called chromatic cohomology.
  In this talk, I will describe how to associate a matroid to a directed graph G, called the multipath matroid of G, which encodes relevant combinatorial information about edge orientation. We also show that a specialization of the Tutte polynomial of the multipath matroid of G provides the number of certain "good" digraph colorings.
  Finally, analogously to the relationship between the chromatic polynomial and chromatic cohomology, I will show how the polynomial expressing the number of "good" digraph colorings is linked to multipath cohomology, introduced in a work with Caputi and Collari in 2021.

  The small quantum groups u_q(g) are finite-dimensional quotients of quantum universal enveloping algebras U_q(g) at a root of unity q for g a semisimple complex Lie algebra. After the work of Lusztig, the representation theory of these quantum objects was intensively studied because of its relation with the representation theory of semisimple algebraic groups in positive characteristic. In this talk, I will present some results on the representation theory of what we call "generalized" small quantum groups. A particular feature of these objects is that the role of the corresponding Cartan subalgebra is played by a finite non-abelian group. Nevertheless, they still admit a triangular decomposition and share similar properties with the standard quantum groups, like the existence of weights (that are no longer one-dimensional) and Verma modules.
  This talk is based on a joint work with Cristian Vay [Simple modules of small quantum groups at dihedral groups, Doc. Math. 29 (2024), 1-38].