Many well-known examples of "noncommutative spaces", like Woronowicz' quantum SU(2) or Vaksman-Soibelman odd-dimensional quantum spheres, can be described by C*-algebras associated to directed graphs. More generally, many compact quantum groups and quantum homogeneous spaces, can be described by convolution C*-algebras of "nice" groupoids. C*-algebras associated to combinatorial data (graphs, diagrams, groupoids) allow efficient models to attack key open problems in noncommutative geometry.
  The aim of this talk is to present some basic ideas of noncommutative geometry, using graph and groupoid C*-algebras as examples of "noncommutative spaces that one can see".

In the first part of this talk I will present the Solvability Complexity Index — a theory developed with several coauthors over the last decade to help us classify the complexity of everyday computations. This theory serves as a bridge between classifications of degrees of complexity in theoretical computer science, and applications in numerical analysis. In the second part of the talk, I will discuss recent results where these ideas were applied to the computation of resonances, providing the first algorithms that work in any dimension.

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

The construction of feedback-like control fields for kinetic models in phase space are discussed. The purpose of these controls is to drive an initial density of particles in the phase space to reach a desired cyclic trajectory on the phase space and follow it in a stable way. For this purpose, an ensemble optimal control problem governed by the kinetic model is formulated in a way that is amenable to a Monte Carlo approach. The proposed formulation allows to define a one-shot solution procedure consisting in a backward solve of an augmented adjoint kinetic model. Results of numerical experiments demonstrate the effectiveness of the proposed control strategy.

We determine the largest rate of exponential decay at infinity for non-trivial solutions to the stationary Dirac equation in presence of a (possibly non-Hermitian) matrix-valued perturbation V such that |V(x)| goes as |x|^(-e) at infinity, for -infty < e < 1. Also, we show that our results are sharp for n=2,3, providing explicit examples of solutions that have the prescribed decay, in presence of a potential with the related behaviour at infinity. In this sense, our work is a result of unique continuation from infinity for the Dirac operator.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006

Blackadar and Kirchberg introduced the notion of a Matricial Field (MF) C*-algebra; an algebra is MF if it can be embedded into a corona of matrix algebras. We will discuss this property in crossed-products arising from C*-dynamical systems and more generally in cross-sectional C*-algebras constructed from Fell bundles. The C*-analogue of Connes' embedding conjecture is the Blackadar-KIrchberg Question (BKQ) which asks whether every stably-finite algebra admits the MF property. Using K-theoretic techniques we can answer this question in the affirmative for certain crossed products of classificable algebras by free groups.

The classical mean value property characterizes harmonic functions. It can be extended to characterize solutions of many linear equations. We will focus in an asymptotic form of the mean value property that characterizes solutions of nonlinear equations. This question has been partially motivated by the connection between Random Tug-of-War games and the normalized p-Laplacian equation discovered some years ago, where a nonlinear asymptotic mean value property for solutions of a PDE is related to a dynamic programming principle for an appropriate stochastic game. Our goal is to show that an asymptotic nonlinear mean value formula holds for several types of non-linear elliptic equations. Our approach is flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both infimum and supremum, of linear operators. We study both when the set of coefficients is bounded and unbounded (each case requires different techniques). Examples include Pucci, Issacs, Monge-Ampere, and k-Hessian operators and some of their parabolic versions. This talk is based in joint work with Pablo Blanc (Buenos Aires), Fernando Charro (Detroit), and Julio Rossi (Buenos Aires).

The Einstein equations are the governing equations of general relativity and admit an initial value formulation as a system of nonlinear wave equations. The talk will present some stability and instability results for stationary (black hole) solutions to such evolution equations, enlightening interesting connections between the dynamics of solutions and the number of dimensions considered. No prior exposure to general relativity will be assumed.

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

There are various notions of singular characteristics in the theory of propagation of singularities to viscosity solutions. In this talk, we will discuss a variational construction of generalized characteristics and strict singular characteristics. We proved the solution of generalized characteristics can be constructed in an intrinsic way using the idea of minimizing movements and certain process of homogenization. Moreover, the relation between the strict singular characteristics and the EDI-EVI frame of gradient flow was also discussed. These results bridge various different topics such as Hamiltonian dynamical systems, weak KAM theory, cut locus in geometry, homogenization and gradient flow theory.
This talk is based on our latest work with Piermarco Cannarsa, Jiahui Hong and Kaizhi Wang.
Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023-2027)

See the attached pdf file

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

Non-autonomous or random dynamical systems (RDS) provide useful and flexible models to investigate systems whose evolution depends on external factors, such as noise and seasonal forcing. Modern developments on multiplicative ergodic theory and transfer operators allow us to get useful insights into the long-term behavior of these systems. In this talk, we will present results in this direction, including (quenched) limit theorems and thermodynamic formalism for a class of RDS. Our results will be illustrated with examples, including random open and closed intermittent maps and non-transitive systems. This talk is based on collaborations with J. Atnip, D. Dragicevic, G. Froyland and S. Vaienti.