We introduce higher order Verma modules over a Kac-Moody algebra <strong>g</strong> (one may assume this to be <strong>sl</strong>_<em>n</em> throughout the talk, without sacrificing novelty). Using these, we present positive formulas - without cancellations - for the weights of arbitrary highest weight <strong>g</strong>-modules <em>V</em>. The key ingredient is that of "higher order holes" in the weights, which we introduce and explain.

In some classical functional inequalities, optimal functions and optimal constants are known. The next question is to understand which distance to the set of the optimal functions is controlled by the deficit, that is, the difference of the two sides of the inequality written with the optimal constant. In 1991, an answer was given by Bianchi and Egnell in the case of a Sobolev inequality on the Euclidean space, using compactness methods. A major issue with the method is that the new constant is so far unknown. The purpose of this lecture is to review some examples of related functional inequalities in which one can at least give an estimate of the stability constant. <br> <br> Note: </strong> This talk is part of the activity of the MIUR Department of Excellence Project MATH@TOV CUP E83C18000100006

pAQFT defines nets of local algebras by a limiting construction with relative S-matrices. The latter can be constructed perturbatively from an interaction Lagrangian. In many instances, the construction can be improved by adding a total derivative to the interaction Lagrangian (which would have no effect in classical field theory). The LV formalism controls whether and how this modification affects the (relative) S-matrices and provides a tool to identify the local observables of the model.

In 1936, R. de Possel observed that, in the general setting of a measure space with no metric structure, certain phenomena, relative to the differentiation of integrals, which are familiar in the Euclidean setting precisely because of the presence of a metric, are devoid of actual meaning. In this work, in collaboration with Steven G. Krantz, we show that, in order to clarify these difficulties,it is useful to adopt the language of filters, which has been introduced by H. Cartan just a year after De Possel's contribution.

The arc space of a variety is an infinite dimensional scheme whose geometric structure captures, in a way that is not yet fully understood, certain features of the singularities of the variety. Focusing on its local rings and invariants of these rings such as embedding dimension and codimension, we explore the local structure of arc spaces. Our main tools rely on a formula for the sheaf of differentials on arc spaces and some recent finiteness results on the fibers of the map induced at the level of arc spaces from an arbitrary morphism of schemes over a field. The talk is based on joint work with Christopher Chiu and Roi Docampo.

The operator-algebraic construction of 1+1-dimensional integrable quantum field theories has received substantial attention over the past decade. These models are characterized by their asymptotic particle spectrum and their two-particle scattering matrix; so far, those particles have been bosonic. By contrast, we consider the case of asymptotic fermions. Abstractly, they arise from a grading of the underlying operator algebraic structures (Borchers triples); more concretely, one replaces the generating quantum fields fulfilling wedge-local commutation relations with a variant fulfilling anticommutation relations. Many of the technical methods required can be carried over from the bosonic case; most importantly, existing results on the technically hard part of the construction (i.e., establishing the modular nuclearity condition) do not require modification. Thus we are lead to a new family of rigorously constructed quantum field theories which are physically distinct from the bosonic case (with a different net of local algebras). Haag-Ruelle scattering theory confirms that they indeed describe fermions. Also, their local operators fulfill a modified version of the form factor axioms, consistent with the physics literature.

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We are concerned with sequences of graphs with a uniform local structure. The underlying sequence of adjacency matrices has a canonical eigenvalue distribution, in the Weyl sense, and it has been shown that we can associate to it a symbol <i>f</i>. The knowledge of the symbol and of its basic analytical features provides key information on the eigenvalue structure in terms of localization, spectral gap, clustering, and global distribution. We discuss different applications and provide numerical examples in order to underline the practical use of the developed theory. In particular, we show how the knowledge of the symbol <i>f</i> can benefit iterative methods to solve Poisson equations on large graphs and provides insight on the recurrence/transience property of random walks on graphs. This talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006.

We review some recent results on the stabilizability of a wide class of control systems with unbounded inputs, including those with a polynomial dependence on the controls.   <br> We present an extension to these unbounded control systems of the well-known relations between the global asymptotic controllability, the sample stabilizability, and the existence of a control Lyapunov function. The results are based on a reparameterization technique commonly adopted in optimal impulsive control, and in particular, on showing that the unbounded setting can be equivalently recasted in terms of a related, extended control problem with bounded controls.   <br> Finally, we briefly discuss an integral cost associated to the control system and we establish sufficient conditions for the sample stabilizability of the system with regulated cost.   <br> This talk is based on a joint work with Monica Motta, Università di Padova.  <br> <br> <a href="https://teams.microsoft.com/l/meetup-join/19%3a7f273a2aff1145dfae91372d186b1cd9%40thread.tacv2/1653403302031?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22d37d6fea-2e4d-4c35-88e4-99bf4cf68fe9%22%7d"> <strong> MS Teams Link for the streaming </strong> <em> </strong> </a> <br> <strong> Note: </strong> This talk is part of the activity of the MIUR Department of Excellence Project MATH@TOV CUP E83C18000100006

  Stable permutations are a class of permutations that arises in the study of the automorphism group of the Cuntz algebra. In this talk, after introducing the Cuntz algebra and surveying the main known results about stable permutations, I will present a characterization of stable permutations in terms of certain associated graphs. As a consequence of this characterization we prove a conjecture in [Advances in Math. <strong>381</strong> (2021) 107590], namely that almost all permutations are not stable, and we characterize explicitly stable 4 and 5-cycles. <br>   This is a joint work with Roberto Conti and Gleb Nenashev. <br>   <strong>N.B.:</strong> please <a href="https://teams.microsoft.com/l/meetup-join/19%3a342c23eced2540e4bcbb6f938e999db6%40thread.tacv2/1614163430980?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%22cf31dbee-7758-4272-af72-503d7694f2ea%22%7d" target="Teams">click <strong>HERE</strong> to attend the talk</a> in streaming.