In Classical Probability, a sequence of random variables is said to be exchangeable if its joint distributions are invariant under all finite permutations. Ryll-Nardzeski’s Theorem establishes that exchangeability is the same as spreadability, the a priori weaker symmetry where all subsequences of the given sequence have the same joint distributions. In the non-commutative setting, it is known that the two symmetries no longer coincide for general quantum stochastic processes. We show that under very natural hypothesis there is an extension of the Ryll-Nardzewski Theorem in the noncommutative setting which covers a wide variety of models. Furthermore we obtain an extended De Finetti’s Theorem for various models including processes based on the CAR algebra and on the infinite noncommutative torus. This talk is based on joint work in progress with Valeriano Aiello and Stefano Rossi.

We consider a scalar Euclidean QFT with interaction given by a bounded, measurable function V such that V± := lim w → ±∞ V(w) exist. We find a field renormalization such that all the n-point connected Schwinger functions for n ≠ 2 exist non-perturbatively in the UV limit. They coincide with the tree-level one-particle irreducible Schwinger functions of the erf(ϕ/ √ 2) interaction with a coupling constant (V+ - V-)/2. By a slight modification of our construction we can change this coupling constant to (V+ - V-)/2., where V± := lim w → 0± V(w). Thereby, non-Gaussianity of these latter theories is governed by a discontinuity of V at zero.

Boundary integral methods represent a powerful class of numerical techniques for the solution of partial differential equations, particularly in problems involving infinite or semi-infinite domains, such as those arising in potential theory, acoustics, elasticity, and fluid dynamics. By reducing the dimensionality of the problem and focusing computations on the boundary, these methods can offer significant advantages in terms of accuracy, efficiency, and mesh simplicity. This talk will provide an overview of the numerical challenges that arise when using boundary integral methods, discuss recent advances in efficient algorithms, and present some applications in microfluidics. The talk will highlight both the theoretical elegance and the practical utility of boundary integral methods, and will provide insight into when and why these methods are especially effective.

Tomita-Takesaki modular theory has become a powerful tool in the analysis of quantum field theories. Although generally the modular objects are difficult to calculate explicitly, in the setting of Half-sided Modular Inclusions we have more control over them. The representation theory of a single Half-sided Modular Inclusion is closely related to the canonical commutation relations and is therefore well understood, but it is not so clear what is possible when multiple different half-sided modular inclusions arise within the same standard subspace/von Neumann algebra. After introducing Half-sided Modular Inclusions and their relation to so-called Standard Pairs, I will discuss a recent result which relates inclusions of standard subspaces, both included as half-sided modular inclusions in a surrounding standard subspace, to inclusions of associated complex subspaces. This allows one to relate back to the representation theory to construct concrete examples of non-trivial phenomena, which we also discuss.

In this talk, I will introduce a new statistic on Weyl groups called the atomic length, and clarify this terminology by drawing parallels with the usual Coxeter length. It turns out that the atomic length has a natural Lie-theoretic interpretation, based on crystal combinatorics, that I will present. Last but not least, I will explain how this can be used as a tool for tackling a broad range of enumeration problems arising from modular representation theory (and related to the study of core partitions).
Part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006).

Let V be a representation of a connected complex reductive group G. The group acts on the ring of regular functions on V: the asymptotic support of this representation is a closed convex polyhedral cone, called moment cone. We will present an algorithm that determines the minimal list of linear inequalities for this cone. Some aspect are relevant from algorithm and convex geometry and others from algebraic geometry.
Part of the activity of the MIUR Excellence Department Project MatMod@TOV (CUP E83C23000330006).

NB:This colloquium is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

We consider linear stable operators L of order 2s whose spectral measure is positive only in a relative open subset of the unit sphere, the aim being to present Liouville type results, in a half space, for the inequality -Lu ≥ u^p. In particular we will show that u≡0 is the only nonnegative solution for 1 ≤ p ≤ (N+s)/(N-s). The optimality of the exponent (N+s)/(N-s) will also be discussed. Based on a joint work with I. Birindelli and L. Du (Sapienza Università di Roma)
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006

We consider a class of periodic solutions of second order difference equations with symplectic structure. We obtain an explicit condition for their stability in terms of the 4-jet of the generating function. This result can be seen as a Lagrangian counterpart of the problem of Lyapunov stability of fixed points of area-preserving diffeomorphisms. An application is given to the model of a bouncing ball. Joint work with Rafael Ortega.

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