Luogo: Dipartimento di Matematica, via Sommarive, 14 - Povo (TN) - Aula Seminari "-1"
Seminario periodico del dipartimento di matematica
Prossimo seminario: giovedì 26 aprile
Massimiliano Mella - Dipartimento di Matematica e Informatica Università degli Studi di Ferrara
Lʼubiquità della geometria birazionale
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.
Valeria Simoncini - Alma Mater Studiorum Università di Bologna
Computational methods for large-scale matrix equations and application to PDEs
Riccardo Ghiloni - Dipartimento di Matematica Università di Trento
The geometry of Nash: between algebra, analysis and topology
Gianni Dal Maso - Scuola Internazionale Superiore di Studi Avanzati
Homogenisation and Gamma-convergence of free discontinuity problems
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.
Peter Michael Schuster - Università degli Studi di Verona
On the Computational Content of Krull's Lemma
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.
Andrea Pinamonti - Università degli Studi di Trento
Introduction to sub-Riemannian geometry
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.
Daniela Cadamuro - Technische Universität München
An introduction to algebraic quantum ﬁeld theory
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.
Eva Riccomagno - Università degli Studi di Genova
Algebraic Statistics for data analysis
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.
|26/04||Massimiliano Mella - Università degli Studi di Ferrara|
|10/05||Romeo Brunetti - Università degli Studi di Trento|
|24/05||Leonard Peter Bos - Università degli Studi di Verona|
|07/06||Olivia Caramello - Università degli Studi dell’Insubria|
|21/06||Sonia Mazzucchi - Università degli Studi di Trento|
Referenti: Ana Maria Alonso Rodriguez - Eduardo Luis Sola Conde