Discrete and computational mathematics

Different aspects of discrete mathematics are investigated both from an abstract and a computational point of view. Categorical, homological and combinatorial methods are combined to study associative algebras arising in different contexts, to deal with classification problems, and to investigate categories of algebraic or geometric nature that find application also in theoretical physics. Theoretical background and algorithms for optimization, including mathematical programming and combinatorial optimization, are also studied, particularly in the context of operations research. Mathematical logic, especially proof theory and constructive mathematics, is pursued to uncover the computational content of mathematical proofs, e.g. algorithms with certificates of correctness. This partial realization of the revised Hilbert Programme is extended to conceptual areas, such as the theories of commutative rings and Banach algebras, where the finite underpinning of transfinite methods is a particular challenge.

Documents

pdf Brochure di presentazione dell'area  (pdf,  it, 805 KB)
pdf Presentazione Research Day 2017  (pdf,  it, 185 KB)
Lidia Angeleri
Full Professor
Fabiano Bonometti
PhD student
Massimo Cairo
PhD student
Enrico Gregorio
Associate Professor
Francesca Mantese
Assistant Professor
Giuseppe Mazzuoccolo
Associate Professor
Alessandro Rapa
PhD student
Romeo Rizzi
Associate Professor
Peter Michael Schuster
Associate Professor
Simone Ugolini
Temporary Professor
Skills
Topic People Description ISI-CRUI
Associative rings and algebras - - standard compliant  MSC
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
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
Category theory; homological algebra - - standard compliant  MSC
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
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
Adjoint functors. Equivalence and dualities between module categories. Triangulated and derived functors. Equivalence and dualities between triangulated and derived categories. Mathematics
Combinatorics - Graph theory standard compliant  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 - - standard compliant  MSC
Arithmetic dynamical systems Simone Ugolini
Dynamics of polynomial and rational maps. Arithmetic dynamics on algebraic varieties. Mathematics
Computer science - - standard compliant  MSC
Combinatorial Algorithms and algorithmic 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 standard compliant  MSC
Polytopes and Polihedra 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 - - standard compliant  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 - - standard compliant  MSC
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
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
Gruppi di ricerca
Name Description 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/
Projects
Title Managers Sponsors Starting date Duration (months)
Álgebra no conmutativa: Anillos, Módulos y C*- álgebras Lidia Angeleri Ministerio de Ciencia e Innovación 1/1/06 36
Algebras and cluster categories Enrico Gregorio, Francesca Mantese, Lidia Angeleri Università degli studi di Padova 3/1/08 24
Decomposition and tilting theory in module, derived and cluster categories Enrico Gregorio, Francesca Mantese, Lidia Angeleri Università degli studi di Padova 3/1/05 24
Differential graded categories Francesca Mantese, Lidia Angeleri Università degli studi di Padova 3/1/11 24
Estructura de anillos, C*-álgebras y categorías de módulos Lidia Angeleri Ministerio de Ciencia e Innovación 1/1/12 36
Estructura y Clasificación de Anillos, Módulos y C*-álgebras Lidia Angeleri Ministerio de Ciencia e Innovación 1/1/09 36
Grup de Recerca en Teoria de Anells 2005-2008 Lidia Angeleri Generalitat de Catalunya 1/1/05 36
Grup de Recerca en Teoria de Anells 2009-2013 Lidia Angeleri Generalitat de Catalunya 1/1/09 60
Strutture algebriche e loro applicazioni: categorie abeliane e derivate, entropia algebrica e rappresentazioni di algebre Francesca Mantese, Lidia Angeleri Fondazione CARIPARO 10/1/12 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 9/22/08 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 1/30/06 24
Teoria tilting, localizazzione e purità in categorie di moduli e categorie derivate (PRIN 2009) Lidia Angeleri PRIN VALUTATO POSITIVAMENTE 7/15/11 12
TTinDMod (FP7-PEOPLE-2012-IEF) Lidia Angeleri, Jorge Nuno Dos Santos Vitoria Unione Europea 9/2/13 24

Activities

Research facilities