Programme & Slides

Schedule

Titles & Abstracts 

Valeriano Aiello

The Motzkin subproduct system

Subproduct systems, defined by Shalit and Solel in 2009, provide a framework for constructing interesting C*-algebras. More recently, Habbestad and Neshveyev introduced Temperley-Lieb subproduct systems. We extend their construction using Motzkin algebras and their Jones-Wenzl idempotents. We show that the resulting Toeplitz C*-algebra is universal, that is, it can be defined in terms of generators and relations, and we investigate aspects of its representation theory. 

This is joint work with Simone Del Vecchio and Stefano Rossi.

Jacopo Bassi

On isometries for spectral triples on AF-algebras

In this talk I will discuss some results on the structure of natural non-commutative analogues of groups of isometries for the spectral triples on AF-algebras introduced by Christensen and Ivan. This is a joint work with R. Conti. If time permits I will briefly explain the connection of this work with a non-commutative analogue of Gromov-Hausdorff convergence, which was studied in a joint work with R. Conti, C. Farsi and F. Latrémolière.

References:

J. Bassi and R. Conti, On isometries of spectral triples associated to AF-algebras and crossed products, J. Noncommut. Geom. 18:2 (2024), 547–566.

J. Bassi, R. Conti, C. Farsi and F. Latrémolière, Isometry groups of inductive limits of metric spectral triples and Gromov-Hausdorff convergence, J. Lond. Math. Soc. 108:4 (2023), 1488–1530.

Erik Bédos

Actions of Fell bundles

In a recent joint work with Roberto C., we introduce the concept of (left) action of a Fell bundle (over a discrete group) on a (right) Hilbert bundle (over another Fell bundle). After presenting some examples, we will discuss the connection with the notion of positive definiteness for bundle maps between Fell bundles, and explain how such actions can be used to induce completely positive maps and construct C*-correspondences.

Francesco Brenti

Combinatorics of Cuntz algebra automorphisms and their subgroups

I will survey the main connections between combinatorics and the automorphisms of the Cuntz algebras and then use them to investigate subgroups of the automorphism group and outer automrphism group of the Cuntz algebra O_n, with particular emphasis on the case n=4. 

This is joint work with Roberto Conti and Gleb Nenashev.

Arnaud Brothier 

Moduli of representations of the Cuntz algebra

Vaughan Jones technology permits to lift functors to group actions of fraction groups. This provides new ways to build explicit representations of the Richard Thompson groups F,T,V. I will explain how this approach permits to promote representations of certain basic algebras to more complicated ones like the Cuntz algebra. Thanks to some rigidity phenomena this will provide moduli spaces of irreducible classes of representations of the Cuntz algebra. 

These are joint works with Vaughan Jones and with Dilshan Wijesena.

Sebastiano Carpi

Holomorphic vertex operator algebras and Teichmüller modular forms

Classical modular forms are  holomorphic functions on the upper half plane H satisfying certain functional equations related to the action of the modular group SL(2,Z) on H. They are deeply related to the geometry of the moduli space of genus one compact complex curves. Vertex operator algebras (VOAs) gives a mathematical description of chiral two-dimensional conformal field theories (chiral CFTs). VOAs with trivial representation theory are called holomorphic and their genus 1 partition function gives rise to classical modular forms. Teichmüller modular forms are higher genus generalizations of classical modular forms.  In this talk I will review some recent results of an ongoing joint work with Giulio Codogni. If V is a holomorphic VOA of central charge c we show that,  for every non-negative integer g,  the genus g partition function of V gives rise in a natural way to a Teichmüller modular form of weight c/2. This gives strong constraints on the partition functions of holomorphic VOAs. Moreover, we clarify the relation between unitary VOAs having the same genus g partition function for all g. Finally, we relate the important open problem of the uniqueness of the moonshine VOA with a weak form of the Harrison-Morrison slope conjecture about the geometry of the moduli spaces of compact Riemann surfaces.

Pierluigi Contucci

Statistical Mechanics and Artificial Intelligence

This talk explores key rigorous results from the statistical mechanics of disordered systems, emphasizing their connection to Boltzmann Machines—a fundamental model in Artificial Intelligence. By drawing on this interplay, we will highlight how concepts from disordered systems can provide deeper insight into learning processes. The presentation will conclude with a discussion of open problems and emerging perspectives at the intersection of statistical mechanics and AI.

Corrado Falcolini

Algoritmi e modelli parametrici da nuvole di punti: il progetto di ricostruzione degli affreschi tardo medievali dell’Abbazia di San Vincenzo al Volturno

Lo studio e l’analisi di elementi, architettonici ed archeologici, sono alla base della conservazione, della ricostruzione e del monitoraggio del nostro patrimonio culturale. Oggi questo studio parte da rilievi, fatti con laser scanner e programmi di fotogrammetria sempre più avanzati, che restituiscono liste di milioni di punti (“nuvole di punti”) sulla superficie degli oggetti rilevati e ne rappresentano quindi la base dati per la loro digitalizzazione. Un esempio è stata la ricostruzione virtuale dell’Arco di Tito al Circo Massimo di Roma che ho contribuito a realizzare con il collega Marco Canciani e in collaborazione con la Sovrintendenza Capitolina ai Beni Culturali. Presenterò parte di un PRIN sulla ricostruzione degli affreschi tardo medievali dell’Abbazia di San Vincenzo al Volturno: alcuni esempi di algoritmi originali per aiutare la ricomposizione di frammenti tridimensionali.

Carla Farsi

Isometries  of spectral triples associated to Cantor sets

The spectral triples of Pearson-Bellissard are a famous classical example of spectral triples on ultrametric Cantor sets; in [2] we compute their small and large isometry groups. We also compute in [1] isometry groups of some inductive limit spectral triples, and outline continuity of spectral triple isometry groups with respect to the spectral propinquity of Latremoliere.

References:

[1} J. Bassi, R. Conti, C. Farsi, F. Latrémolière, Isometry groups of inductive limits of metric spectral triples and Gromov-Hausdorff convergence .  J. Lond. Math. Soc. (2) 108 (2023),no. 4,1488–1530.

[2]  R. Conti, C.Farsi Isometries of Kellendonk-Savinien spectral triples and Connes metrics.  Internat. J. Math. 33 (2022), no.13, Paper No. 2250084, 26 pp.

Francesco Fidaleo

Spectral Actions for Free Particles and their Asymptotics

For the spectral action consisting of the average number associated to the gas of free q-particles (including Bose, Fermi and classical ones corresponding to q = ±1 and  respectively) in thermal equilibrium, we compute the asymptotic expansion with respect to the natural cut-off given by (a suitable power of) the inverse temperature. The same analysis can be carried out for other relevant extensive quantities like entropy and mean energy, with only more involved computations. We treat both relevant situations relative to massless and non relativistic massive particles, where the natural cut-off is 1/β = kBT and 1/√β, respectively. We show that the massless situation enjoys less regularity properties than the massive one. We also treat in some detail the relativistic massive case for which the natural cut-off is again 1/β. The ”passage to the continuum” describing infinitely extended open systems in thermal equilibrium is considered, by also discussing the appearance of condensation phenomena occurring for Bose-like q-particles, q ∈ (0, 1]. The more singular situation corresponding to the massless case in a finite (d-dimensional) volume is handled by using the theory of distributions associated to a very particular class of test-functions. Such an analysis relative to such classes of regular functions has a deep connection with the investigation of properties of the Riemann zeta-function.

The present talk is based on: F. Ciolli and F. Fidaleo Spectral actions for q-particles and their asymptotics, J. Phys. A (Math. Theor.) 55 (2022), 424001 (19 pp).

Robin Hillier

Dynamical decoupling of open quantum systems

The talk provides an introduction to dynamical decoupling, an error mitigation procedure in quantum control quantum information theory, which aims to stabilise open quantum systems undergoing decoherence. We look at some of the mathematical challenges, provide conditions as to how and when dynamical decoupling works in given quantum systems, what the resulting dynamics looks like and how close the idealised mathematical description  is to physical reality. 

The talk is based on a series of joint papers with C. Arenz, D. Burgarth, and P. Facchi.

Roberto Longo

A Bekenstein-type inequality in Quantum Field Theory

Florin  Radulescu 

von Neumann dimension and V.F.R. Jones view on automorphic forms

Using von Neumann dimension, V.F.R Jones was able to give a new interpretation of automorphic forms as intertwining operators of type II1 representations of the modular group. He obtained a remarkable result concerning sets of zeros of analytic functions on the upper half plane. This approach can also be used to find new information about properties of the associated Hecke operators and the canonical associated  representation of PSL(2, Z[1/p]), p>1, a prime.

Stefano Rossi

Non-commutative skew-products

In this talk, I'll first acquaint the audience with general skew-products as suitable non-commutative dynamical systems (given by a locally compact group G whose action on a crossed product is assigned through a 1-cocycle of G in $\mathbb{T}$), after quickly recalling the classical counterpart they arise from, i.e. the well-known Furstenberg-Anzai skew products. Then I'll move on to present how they can be classified up to conjugacy. I'll also discuss their main ergodic properties, including unique ergodicity, unique ergodicity w.r.t. the fixed-point subalgebra, which, for our systems, is the same as requiring the uniqueness of an invariant conditional expectation onto the fixed-point subalgebra. Finally, I'll spend a word on what the set of all invariant states of our systems may look like: quite remarkably, it is either a singleton or affinely homeomorphic with the Borel probability measures on the one-dimensional torus $\mathbb{T}$.

The talk is based on recent joint work with V. Crismale, S. Del Vecchio, and M.E. Griseta.

Lyudmyla Turowska

Absolutely dilatable module maps

I will discuss the notion of absolute dilation for maps on von Neumann algebras, focusing primarily on maps on B(H) with an additional modularity condition. The notion was recently defined and studied by C. Duquet and C. Le Merdy. They characterized dilatable Schur multipliers. We extend the results by replacing the requirement of being Schur by being modular over arbitrary von Neumann algebra, instead of maximal abelian selfadjoint algebra.   Such maps   are characterized by the existence of a tracial von Neumann algebra (N, τ), called an ancilla, and a certain unitary operator. Different types of ancillas (abelian, finite-dimensional, etc.) lead to the definition of local, quantum, approximate quantum, and quantum commuting dilatable maps, and I will discuss the relationships between these types. The motivation to study different types of dilations comes from Quantum Information Theory. The interrelation between QIT and dilatable maps will be explained.

The talk is based on an ongoing project with A. Chatzinikolaou and I. G. Todorov.

Ezio Vasselli

Hilbert bimodules and models in Quantum Field Theory

Algebraic relations describing the interplay between a Dirac field and a (gauge) bosonic field are common in quantum field theory, yet their realization on Hilbert space is usually technically problematic. In this talk I describe a simple method, based on Hilbert bimodules, to obtain quantum fields carrying the desired commutation relations. As an application, we exhibit a model fulfilling the commutation relations of the Coulomb gauge at a fixed time.