maths bites trento

Abstract
Linear matrix equations such as the Lyapunov and Sylvester equations and their generalizations have classically played an important role in the analysis of dynamical systems, in control theory and in eigenvalue computation. More recently, matrix equations have emerged as a atural linear algebra framework for the discretized version of (systems of) deterministic and stochastic partial differential equations ((S)PDEs), and new challenges have arisen. In this talk we will review some of the key methodologies for solving large scale linear matrix equations. We will also discuss recent strategies for the numerical solution of more involved equations, such as multiterm linear matrix equations and bilinear systems of equations, which are currently attracting great interest due to their occurrence in new application models associated with (S)PDEs.

Abstract Linear matrix equations such as the Lyapunov and Sylvester equations and their generalizations have classically played an important role in the analysis of dynamical systems, in control theory and in eigenvalue computation. More recently, matrix equations have emerged as a atural linear algebra framework for the discretized version of (systems of) deterministic and stochastic partial differential equations ((S)PDEs), and new challenges have arisen. In this talk we will review some of the key methodologies for solving large scale linear matrix equations. We will also discuss recent strategies for the numerical solution of more involved equations, such as multiterm linear matrix equations and bilinear systems of equations, which are currently attracting great interest due to their occurrence in new application models associated with (S)PDEs.

The stochastic homogenisation of free-discontinuity functionals is studied assuming stationarity of the random volume and surface energy densities. Combining the deterministic results on Gamma- convergence of free-discontinuity functionals with the Subadditive Ergodic Theorem, we characterise the homogenised volume and surface energy densities in terms of limits of the solutions of auxiliary minimum problems on large cubes.

Zorn's Lemma (ZL) presumably is the most common incarnation of the Axiom of Choice in abstract mathematics. The invocation of ZL, however, allegedly obscures any algorithmic content of proofs, and thus is deemed non-constructive in general. Yet ZL often allows for proofs shorter and more elegant than those in which one sticks to explicit computations, especially when ZL is used together with proof by contradiction. This is of particular interest when the wording of the claim to be proved is completely elementary, as is the case for Joyal's version of Gauss's lemma that the product of two primitive polynomials is primitive as well. A paradigmatic example indeed is Krull's Lemma (KL), one of the basic forms of ZL in commutative algebra: every proper (radical) ideal can be extended to a prime ideal. In fact KL makes possible to reduce certain proofs about reduced rings, alias semiprime rings, to the more convenient special case of integral domains. For any such use of KL, however, one just needs that the characteristic axioms of integral domain be conservative, for definite Horn clauses H, over the axioms of reduced ring. In other words, if any such H can be proved from the former, stronger axioms, then H already can be proved from the latter, weaker ones. Now this proof-theoretic conservation theorem has a perfectly constructive proof, and thus contains a conversion algorithm. While the case of KL and similar single cases were already known before, e.g. in dynamical algebra, with Rinaldi and Wessel we recently have generalised the method to abstract entailment relations, which are known to faithfully represent reasoning in algebraic structures with little logical notation. Building upon work of Scott we give an equivalent mathematical criterion for proof-theoretic conservation. On top of KL and the related Lindenbaum Lemma, instances include the theorems of Hahn-Banach and Artin-Schreier.

The aim of this talk is to introduce some basic ideas in sub-Riemannian geometry. We will start by introducing sub-Riemannian manifolds and we characterize their tangent space using the celebrated Mitchell's theorem. This will allow us to introduce and study Carnot groups. We will conclude by describing some challenging open problems in sub-Riemannian geometry.

Quantum field theory aims at unifying quantum theory (which describes particle physics at microscopic sacales) with the principles of special relativity (which describes objects moving near the speed of light). While being one of the most successful theories in theoretical physics, its precise mathematical description remains a challenging problem. A consistent mathematical framework for quantum field theory (due to Haag and Kastler) can be formulated in the language of C*- or von Neumann algebras, their automorphisms and representations. A net of C*-algebras becomes the fundamental object for the description of physical phenomena, and the lecture will give a brief introduction and motivate the conceptual foundations. A quite different and technically more involved problem is to construct relevant examples that fit into this framework. However, the abstract setting provides us with more flexibility here than a more pedestrian approach. This will be illustrated by recent examples which make use of the concept of “wedge algebras”, an intermediate step to the construction that helps controlling the functional analytic properties of physically relevant operators.

Polynomial representations of probability density functions and of designed experiments are at the core of Algebraic Statistics. Some ideas and techniques used in Algebraic Statistics will be exemplified by three applications. An example is given by a symbolic-numeric approach for the analysis of datasets, represented as sets of limited precision points. A second one refers to a novel statistical model class for the understanding of discrete processes and its causal interpretation. The third one considers independence in Gaussian models and structural meta-analysis.

Una peculiarità della geometria algebrica è lʼuso di mappe che non siano definite ovunque. La loro principale applicazione è la classificazione birazionale delle varietà algebriche, il rinomato Programma dei Modelli Minimali. La loro utilità è però, sempre più spesso, testimoniata da applicazioni in svariati ambiti della matematica. Il seminario vuole essere una galleria di possibili applicazioni di queste tecniche a: decomposizioni tensoriali, sistemi dinamici e crittografia.

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Spiegherò il senso in cui i topoi di Grothendieck possono servire da ʻpontiʼ unificanti utili a collegare tra loro teorie matematiche differenti e a studiarle da una molteplicità di punti di vista diversi. Inizierò col richiamare i preliminari necessari per poi presentare la visione generale dei topoi come ʻpontiʼ e alcune applicazioni selezionate in diversi settori della Matematica.