Scale groups are closed subgroups of the group of isometries of a regular tree that fixes an end of the tree and are vertex-transitive. They play an important role in the study of locally compact totally di-sconnected groups as was recently observed by P-E. Caprace and G. Willis. In the 80’s they were studied by A. Figa-Talamanca and C. Nebbia in the context of abstract harmonic analysis and amenability. It is a miracle that they are closely related to fractal groups, a special subclass of self-similar groups.
In my talk I will discuss two ways of building scale groups. One is based on the use of scale-invariant groups studied by V. Nekrashevych and G. Pete, and a second is based on the use of liftable fractal groups. The examples based on both approaches will be demonstrated using such groups as Basilica, Hanoi Tower Group, and a group of intermediate growth (between polynomial and exponential). Additionally, the group of isometries of the ring of p-adics and the group of dilations of the field of p-adics will be mentioned in relation with the discussed topics.

The role of automorphic forms as intertwiners between various representations of free group factors was discovered a long time ago by Vaughan Jones, starting with a remarkable formula relating Peterson scalar product with the intrinsical trace. The intertwiner associated to an automorphic form is an eclectic object, not much can be computed, but the Muray von Neuman dimension can be used to get hints on its image. Vaughan Jones used that to settle the problem of finding analytic functions vanishing on the orbit under the modular group of a point in the upper half plane. In past work of the speaker, it was put in evidence that this is related to equivariant Berezin quantization. This leads to a different representation of free group factors and to the existence of a quantum dynamics whose associated unbounded Hochschild 2- cocycle is related to the isomorphism problem. I will explain some concrete formulae and some new interpretation of the associated quantum dynamics

Dynamical systems subject to perturbations that decay over time are relevant in the description of many physical models, e.g. when considering the effect of a laser pulse on a molecule, in epidemiological studies, as well as in celestial mechanics. For this reason, in the present talk, we consider a time-dependent perturbation of a Hamiltonian dynamical system having an invariant torus supporting quasiperiodic solutions. Assuming the perturbation decays polynomially fast as time tends to infinity, we prove the existence of orbits converging in time to the quasiperiodic solutions associated with the unperturbed system. This result generalizes the work of Canadell and de la Llave, where exponential decay in time was considered, and the one of Fortunati and Wiggins, where arithmetic, non-degeneracy conditions, and exponential decay in time are assumed. We apply this result to the example of the planar three-body problem perturbed by a given comet coming from and going back to infinity asymptotically along a hyperbolic Keplerian orbit (modeled as a time-dependent perturbation).
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

Alpha-induction is a tensor functor producing a new fusion category from a modular tensor category and a Q-system. This can be formulated in terms of quantum 6j-symbols and braiding and gives alpha-induced bi-unitary connections. Last year, we showed that locality of the Q-system implies flatness of the alpha-induced connections. We now prove that the converse also holds.
The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0

Given the moduli space of hyperplanes in projective space, V. Alexeev constructed a family of compactifications parametrizing stable hyperplane arrangements with respect to given weights. In particular, there is a toric compactification that generalizes the Losev–Manin compactification for the moduli of points on the line. We study the first natural wall crossing that modifies this compactification into a non-toric one by varying the weights. In particular, we prove that in dimensions two the wall crossing corresponds to blowing up at the identity of the generalized Losev–Manin space. As an application, we show that any Q-factorialization of this blow-up is not a Mori dream space for a sufficiently high number of lines. This is joint work in progress with Patricio Gallardo.

Paul de Faget de Casteljau (1930-2022) was a highly gifted mathematician who worked in industry and made fundamental mathematical contributions. In this talk, I will focus on one central contribution that de Casteljau developed soon after he started working for Citroen in 1958. It is the algorithm of de Casteljau, a simple construction of polynomial curves from control points by iterated linear interpolation. This algorithm is not only very simple, very useful, and very well known, but it also has a great number of properties and generalizations that make it a fundamental and unifying theoretical tool for Geometric Design. As a tribute to an outstanding pioneer in CAGD, I will recall widely and little known generalizations and properties of this algorithm to remind us of its beauty, versatility and importance as THE algorithm and backbone of Computer Aided Geometric Design (CAGD).

Motivated by the problem of determining whether biexactness, the (AO)-property and von Neumann solidity are equivalent properties for a discrete countable group, I will discuss few recent results regarding analytic properties of SL(3,Z), related to hyperbolicity. I will focus on the role of measurable dynamics and proximality arguments in this context. Partly based on joint works with F. Radulescu and T. Amrutam.
Some references:
https://arxiv.org/abs/2305.16277
https://arxiv.org/abs/2111.13885
https://arxiv.org/abs/2403.05948

The log etale topology is a natural analogue of the etale topology for log schemes. Unfortunately, very few things satisfy log etale descent -- not even vector bundles or the structure sheaf. We introduce a new rhizomic topology that sits in between the usual and log etale topologies and show most things do satisfy rhizomic descent! As a case study, we look at tropical abelian varieties and give some exotic examples.

  Folded galleries, as introduced by Peter Littelmann in the 1990s, are combinatorial objects related to certain (subsets of) elements of Coxeter groups. They have shown to have versatile applications in algebra and geometry, making them an object of interest for current research. In this talk we will retrace the roots of their invention 192 years back in history, contemplate colorful illustrations of examples, and discover open questions for future applications.
  N.B.: the talk will be colloquium-style and aimed at a wide audience: no prerequisite of deep algebraic nor group theoretic knowledge is required.

  Given an arbitrary Coxeter system (W,S) and a nonnegative integer m, the m-Shi arrangement of (W,S) is a subarrangement of the Coxeter hyperplane arrangement of (W,S). The classical Shi arrangement (m=0) was introduced in the case of affine Weyl groups by Shi to study Kazhdan-Lusztig cells for W. As two key results, Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in W and that the union of their inverses form a convex subset of the Coxeter complex. The set of m-low elements in W was introduced to study the word problem of the corresponding Artin-Tits (braid) group and they turn out to produce automata to study the combinatorics of reduced words in W.
  In this talk, I will discuss how to Shi's results extend to any Coxeter system and show that the minimal elements in each Shi region are in fact the m-low elements. This talk is based on joint work with Matthew Dyer, Susanna Fishel and Alice Mark.