It is known by the works of Adamović and Perše that the affine simple vertex algebras associated with G₂ and B₃ at level -2 can be conformally embedded into L_-2(D₄).
In this talk, I will present a join work with Tomoyuki Arakawa, Xuanzhong Dai, Justine Fasquel, Bohan Li on the classification to the irreducible highest weight modules of these vertex algebras.
I will also describe their associated varieties: the associated variety of that corresponding to G₂ is the orbifold of the associated variety of that corresponding to D₄ by the symmetric group of degree 3 which is the Dynkin diagram automorphism group of D₄. This provides new interesting examples in the context of orbifold vertex algebras. These vertex algebra also appear as the vertex operator algebras corresponding to rank one Argyres-Douglas theories in four dimension with flavour symmetry G₂ and B₃.
  N.B.: this talk is part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006)

K-stability (or existence of Kähler-Einstein metrics) of explicit Fano varieties has been studied for a long time. Delta invariants (stability thresholds) detect the K-stability of Fano varieties. Moreover, Abban--Zhuang developed a powerful method to compute the delta invariants by adjunctions. In this talk, I will explain our recent results on the K-stability of some Fano weighted hypersurfaces via the Abban--Zhuang method. This is based on joint work with Luca Tasin.

Magnetic systems are the natural toy model for the motion of a charged particle moving on a Riemannian manifold under the influence of a (static) magnetic force. In this talk we introduce a curvature operator called magnetic curvature which encodes the information of the classical Riemannian curvature together with terms of perturbation due to the magnetic interaction. In a variational setting, we use this new notion of curvature to approach the problem of finding closed trajectories for small energy levels.

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

Understanding how to characterize quantum chaotic dynamics is a longstanding question. The universality of chaotic many-body dynamics has long been identified by random matrix theory, which led to the well-established framework of the Eigenstate Thermalization Hypothesis. In this talk, I will discuss recent developments that identify Free Probability -- a generalization of probability theory to non-commuting objects -- as a unifying mathematical framework to describe correlations of chaotic many-body systems. I will show how the full version of the Eigenstate Thermalization Hypothesis, which encompasses all the correlations, can be rationalized and simplified using the language of Free Probability. This approach uncovers unexpected connections between quantum chaos and concepts in quantum information theory, such as unitary designs.

  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.