Matematica discreta e computazionale

Si investigano diversi aspetti della matematica discreta sia da un punto di vista astratto che computazionale. Metodi omologici, combinatori e propri della teoria delle categorie sono combinati per studiare algebre associative che si presentano in svariati contesti, affrontando problemi di classificazione, di natura algebrica e geometrica, che trovano applicazione in fisica teorica. Si studiano e si sviluppano inoltre algoritmi efficaci per la soluzione numerica di problemi matematici sia discreti che continui. Un’enfasi particolare viene posta sulla soluzione numerica di equazioni alle derivate parziali e su problemi di interpolazione e data fitting. Siamo attivi anche nella teoria e negli algoritmi per l’ottimizzazione, tra cui programmazione lineare e non lineare e ottimizzazione combinatoria, particolarmente nel contesto della Ricerca Operativa. Ci si occupa di Fondamenti della Matematica, sia per meglio comprendere e validare i metodi matematici di risoluzione dei problemi che per render più solido lo sviluppo dell’apprendimento degli studenti. A tal fine, la Logica Matematica è studiata per stabilire potenzialità e limiti dei linguaggi formali, e per valorizzare, ove possibile, l’approccio costruttivista e le applicazioni all’informatica. Un’attenzione particolare la si dedica infine a problematiche, prospettive e tecniche della comunicazione utili nella formazione degli insegnanti di Matematica.

Documenti

pdf Brochure di presentazione dell'area  (pdf,  it, 805 KB)
pdf Presentazione Research Day 2017  (pdf,  it, 185 KB)
Lidia Angeleri
Professore ordinario
Massimo Cairo
Dottorando
Carlo Comin
Assegnista
Enrico Gregorio
Professore associato
Francesca Mantese
Ricercatore
Giuseppe Mazzuoccolo
Professore associato
David Pauksztello
Assegnista
Davide Rinaldi
Professore a contratto
Romeo Rizzi
Professore associato
Peter Michael Schuster
Professore associato
Simone Ugolini
Professore a contratto
Competenze
Argomento Persone Descrizione ISI-CRUI
Associative rings and algebras - - aderente allo standard  MSC
Modules, bimodules and ideals Lidia Angeleri
Jorge Nuno Dos Santos Vitoria
Francesca Mantese
Jan Frederik Marks
Indecomposable decompositions. Approximations. Purity. Endoproperties of modules. Mathematics
Representation theory of rings and algebras Lidia Angeleri
Fabiano Bonometti
David Pauksztello
Alessandro Rapa
Infinite dimensional modules over finite dimensional algebras. Classification of tilting objects in module categories and in associated geometric categories. Mathematics
Rings and algebras arising under various constructions Lidia Angeleri
Jorge Nuno Dos Santos Vitoria
Francesca Mantese
Jan Frederik Marks
Localization of rings. Ring epimorphisms. Endomorphism rings of tilting and cotilting modules. Mathematics
Category theory; homological algebra - - aderente allo standard  MSC
Abelian categories Lidia Angeleri
Fabiano Bonometti
Jorge Nuno Dos Santos Vitoria
Francesca Mantese
Jan Frederik Marks
David Pauksztello
Alessandro Rapa
Torsion pairs and cotorsion pairs in abelian categories. Approximations in abelian categories. Heart of t-structures associated to torsion pairs. Mathematics
General theory of categories and functors Enrico Gregorio
Francesca Mantese
Adjoint functors. Equivalence and dualities between module categories. Triangulated and derived functors. Equivalence and dualities between triangulated and derived categories. Mathematics
Homological algebra Lidia Angeleri
Fabiano Bonometti
Jorge Nuno Dos Santos Vitoria
Francesca Mantese
Jan Frederik Marks
David Pauksztello
Alessandro Rapa
Tilting theory. Homological Conjectures. Localization in abelian and triangulated categories. Mathematics
Combinatorics - Graph theory aderente allo standard  MSC
Graph theory Giuseppe Mazzuoccolo
Romeo Rizzi
Graphs are a flexible model for core combinatorial problems as arising in various applications. In particular, graphs are encountered in various fields of mathematics, computer science, science in general, and technology. With this, graph theory is not only fun, but it is also a well established and central area of discrete mathematics of topmost interdisciplinarity. Some topics we are interested in: matching, factoring, edge-coloring, flows, cycle basis, packing, covering and partitioning, graph classes, algorithmic graph theory. Mathematics
Commutative algebra - - aderente allo standard  MSC
Arithmetic dynamical systems Simone Ugolini
Dynamics of polynomial and rational maps. Arithmetic dynamics on algebraic varieties. Mathematics
Computer science - - aderente allo standard  MSC
Algorithms for numerical algorithms; for combinatorics and graph theory Massimo Cairo
Carlo Comin
Romeo Rizzi
When we say that our approach to graph theory and combinatorics is algorithmic we not only want to underline the fact that we are most often interested in the obtaining effective algorithms for the problems investigated but also that we indulge unraveling the mathematical problems down till the bottom most level to achieve a most elementary comprehension. Also, we rest on computational complexity as the methodological lighthouse of our research approaches and investigations. This depth and awareness characterizes the strength of the research by our department in Verona. Mathematics
Discrete mathematics in relation to computer science Romeo Rizzi
Discrete mathematics has a privileged link to computer science, and the converse is also true. As algorithmists, we tangle discrete mathematics in order to give our contribution to computer science. Discrete mathematics in relation to computer science is a huge factory all over the world, and our computer science department here in Verona is well present in all this. Mathematics
Theory of computing Romeo Rizzi
The theory of computation is the branch of mathematics and computer science that deals with whether and how efficiently problems can be solved on a model of computation, using an algorithm. In more than one way, this fascinating field has affected our perception of the world and of mathematics itself. In mathematics, it is an eye opener and a source of methodology and philosophical inspiration. This is particularly true for its two main branches of computability theory and computational complexity. Mathematics
Convex and discrete geometry - Polytopes and polyhedra aderente allo standard  MSC
Polytopes and polyhedra Giuseppe Mazzuoccolo
Romeo Rizzi
Polytopes and polyhedra are objects of study in topology, computational geometry, mathematical programming, and combinatorial optimization. The last two perspectives offer tools of operations research which find employment in some of the applied mathematics research lines in Verona. Mathematics
Mathematical logic and foundations - - aderente allo standard  MSC
General logic Gianluigi Bellin
Mathematics
Hilbert's Programme for Abstract Mathematics Davide Rinaldi
Peter Michael Schuster
Daniel Wessel
Extracting the computational content of classical proofs in conceptual mathematics. Particular attention is paid to invocations of logical completeness in mathematical form, typically as variants of Zorn's Lemma. Mathematics
Philosophical aspects of logic and foundations Ruggero Ferro
Several results in mathematical logic point out and explain the limitations, possibilities and advantages of formalization (the use of formal languages). An increasing precision in determining of the role of formal languages is basic to a critical attitude in philosophy of mathematics, spotting untenable positions and supporting others. An empiricist point of view is being developed that overcomes the vagueness and difficulties of know presentations. This type of research has developed, and will continue to support, competences on the following themes: Mathematical logic; Understanding, acquiring, and constructing basic mathematical notion, in particular the primitive ones; The role of logic in the construction and acquisition of mathematical notions; The role of the language in mathematics; The role of formalism in mathematics; Mathematics teacher’s initial and life long education; Mathematical motivations for the teaching of mathematics. Mathematics
Proof theory and constructive mathematics Gianluigi Bellin
Davide Rinaldi
Peter Michael Schuster
Daniel Wessel
Proof theory at large studies mathematical proofs, which thus become themselves objects of mathematics. In a nutshell, the goal is to understand "what can be proved with what" and to gain computational information from proofs. Constructive mathematics aims at direct proofs from which one can read off algorithms; any such algorithm comes with a certificate of correctness for free, which just is the original proof. Mathematics
Operations research, mathematical programming - - aderente allo standard  MSC
Mathematical programming Romeo Rizzi
In mathematics, statistics, empirical sciences, computer science, or management science, mathematical optimization (alternatively, mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives. Here, optimization includes finding "best available" values of some objective function given a defined domain, including a variety of different types of objective functions and different types of domains. Optimization theory, techniques, and algorithms, comprises a large area of applied mathematics. Among the many sectors of mathematical programming, some of those represented in Verona are the following: linear programming, integer linear programming, combinatorial optimization, multiobjective optimization. Mathematics
Operations research and management science Romeo Rizzi
Operations research is a discipline that deals with the application of advanced analytical methods to help make better decisions. The terms management science and decision science are sometimes used as more modern-sounding synonyms. Employing techniques from other mathematical sciences, such as mathematical modeling, statistical analysis, and mathematical optimization, operations research arrives at optimal or near-optimal solutions to complex decision-making problems. Operations Research is often concerned with determining the maximum (of profit, performance, or yield) or minimum (of loss, risk, or cost) of some real-world objective. Originating in military efforts before World War II, its techniques have grown to concern problems in a variety of industries. Besides its applications in industry and in management, Operations Research is at the very junction of mathematics and economics. Operations research embodies lots of deep results and theory but, at the same time, it is the archetype of applied mathematics. Mathematics
Gruppi di ricerca
Nome Descrizione URL
Algebra Il gruppo lavora in teoria delle rappresentazioni di algebre e teoria dei moduli. http://profs.sci.univr.it/~angeleri/RT%20Verona.html
Logica Logica in matematica ed informatica. https://logicseminarverona.wordpress.com/
Progetti
Titolo Responsabili Fonte finanziamento Data inizio Durata (mesi) 
Álgebra no conmutativa: Anillos, Módulos y C*- álgebras Lidia Angeleri Ministerio de Ciencia e Innovación 01/01/06 36
Algebras and cluster categories Enrico Gregorio, Francesca Mantese, Lidia Angeleri Università degli studi di Padova 01/03/08 24
Decomposition and tilting theory in module, derived and cluster categories Enrico Gregorio, Francesca Mantese, Lidia Angeleri Università degli studi di Padova 01/03/05 24
Differential graded categories Francesca Mantese, Lidia Angeleri Università degli studi di Padova 01/03/11 24
Estructura de anillos, C*-álgebras y categorías de módulos Lidia Angeleri Ministerio de Ciencia e Innovación 01/01/12 36
Estructura y Clasificación de Anillos, Módulos y C*-álgebras Lidia Angeleri Ministerio de Ciencia e Innovación 01/01/09 36
Grup de Recerca en Teoria de Anells 2005-2008 Lidia Angeleri Generalitat de Catalunya 01/01/05 36
Grup de Recerca en Teoria de Anells 2009-2013 Lidia Angeleri Generalitat de Catalunya 01/01/09 60
Strutture algebriche e loro applicazioni: categorie abeliane e derivate, entropia algebrica e rappresentazioni di algebre Francesca Mantese, Lidia Angeleri Fondazione CARIPARO 01/10/12 36
Teoria tilting e cotilting e generalizzazioni; applicazioni alle categorie derivate, alle categorie cluster, alla localizzazione, alle congetture omologiche e ad altri problemi aperti (PRIN 2007) Enrico Gregorio Ministero dell'Istruzione dell'Università e della Ricerca 22/09/08 24
Teoria tilting e cotilting per algebre di artin, anelli astratti e topologici. Confronto fra moduli di lunghezza finita e infinita. (PRIN 2005) Enrico Gregorio Ministero dell'Istruzione dell'Università e della Ricerca 30/01/06 24
Teoria tilting, localizazzione e purità in categorie di moduli e categorie derivate (PRIN 2009) Lidia Angeleri PRIN VALUTATO POSITIVAMENTE 15/07/11 12
TTinDMod (FP7-PEOPLE-2012-IEF) Lidia Angeleri, Jorge Nuno Dos Santos Vitoria Unione Europea 02/09/13 24

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