PhD Program in Mathematics and Computer Science

During the period April 26 – May 16, 2023, Prof. Vasile Staicu
(University of Aveiro) has given a PhD course while visiting the
Department of Mathematics and Computer Sciences within the Visiting
Professor Program of INdAM.

The aim of this course is to present some methods and tools used in
the study of nonlinear boundary value problems involving multivalued
maps and nonsmooth functions and illustrate those methods with some
recent results concerning existence and multiplicity of solutions.

We start by introducing some notions of multivalued analysis:
multivalued maps and their continuity. Then we introduce some elements
of nonsmooth analysis that we use to develop nonsmooth critical point
theory succeeded in extending a big part of the smooth theory to
nonsmooth functionals: the main tool in the variational method for
solving boundary value problems, which consists of trying to find
solutions for a given boundary value problem by looking for stationary
points of a real functional defined on a space of functions in which
the solution of the boundary value problem is assumed to lie. 

Degree theory is a basic tool of nonlinear analysis and produces
powerful existence and multiplicity results for nonlinear boundary
value problems. We present a generalization of Brouwer degree theory
to multivalued perturbations of monotone type maps, developed in a
joint paper with Aizicovici and Papageorgiou (Mem. Amer. Math. Soc.,
196, 2008). Then we consider a nonlinear Dirichlet problem with
nonsmooth potential (hemivariational inequality) and use variational
method to prove two constant sign solutions (one positive and another
negative) and then by using degree theoretical approach we prove the
existence of a third one, nontrivial solution, distinct from the
previous two.

We present several existence and multiplicity results, with sign
information for the solutions of nonlinear boundary value problems
driven by the Laplacian, the p-Laplacian and then try to extend such
results to nonlinear boundary value problems driven by fractional
order differential operators and to nonlocal pseudo-differential
inclusions, possibly including obstacles or constraints. It is
reasonable to expect that the pairing of nonlocal diffusion operators
and set-valued reaction will provide a more realistic model for
applications to such problems as quantum mechanics, image restoration,
and financial mathematics, which typically present a high degree of
uncertainty, rather than elliptic equations with smooth, single-valued
reactions.

Lecture notes will be provided and open questions will be considered
and analyzed with the Ph. D. students.

Schedule: 27/4 11-13 28/4 9-11 2/5 11-13 5/5 9-11 9/5 11-13 11/5
11-13 12/5 9-11 16/5 11-13 (room F)

Staicu Lecture Notes

Il calendario delle lezioni è il seguente:

Lunedì 28 novembre dalle 16:00 alle 18:00 Aula B
Martedì 29 novembre dalle 16:00 alle 18:00 Aula A
Mercoledì 30 novembre dalle 16:00 alle 18:00 Aula A
Giovedì 1 dicembre dalle 16:00 alle 18:00 Aula A
Venerdì 2 dicembre dalle 16:00 alle 18:00 Aula A

Many problems in algebra involve the decomposition of certain elements of a ring (or more generally
of a monoid) into a product of certain other elements (hereinafter generically referred to as building
blocks) that are in some sense minimal. The classical theory of factorization investigates factorizations
in which the building blocks are atoms, i.e., non-unit elements of a monoid that are not products of
two non-units. For example, it is well known that every non-zero non-unit of a Dedekind domain (more
generally, of a Noetherian domain) can be written as a finite product of atoms and that in general
such decompositions are not unique. On the other hand, examples of factorizations that lie beyond the
scope of the classical theory include additive decompositions into multiplicative units in rings; cyclic
decompositions of permutations in the symmetric group of degree n; idempotent factorizations of the
“singular elements” of a monoid; and so on.
After an introductory overview of the history and main results in the classical framework, in this
course, we combine the language of monoids and preorders (partial orders that are not necessarily an-
tisymmetric) to make first steps towards the construction of a “unified theory of factorization”. In
particular, we prove an abstract existence theorem that recover, among others, a classical theorem of
Cohn on atomic factorizations in cancellative monoids, a classical result by Anderson and Valdes-Leon
on “irreducible factorizations” in commutative monoids, but also a theorem by Erdos on idempotent fac-
torizations of square singular matrices over fields. We also introduce a notion of “minimal factorization”,
suitable for the new abstract setting, and we qualify and quantify the related non-uniqueness properties
in specific cases but also in some generality. Examples will help to motivate and illustrate the theory.
Prerequisites: Standard knowledge of (basic) algebraic structures and (binary) relations.
Duration of the course: 10 hours.
Exam: Written. Writing of a short essay.
References
[1] L. Cossu and S. Tringali, Abstract Factorization Theorems with Applications to Idempotent Factor-
izations, e-print (Aug. 2021), arXiv:2108.12379.
[2] L. Cossu and S. Tringali, Factorization under Local Finiteness Conditions, e-print (Aug. 2022),
arXiv:2208.05869.
[3] A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations. Algebraic, Combinatorial and An-
alytic Theory, Pure Appl. Math. 278, Chapman & Hall/CRC, Boca Raton (FL), 2006.
[4] S. Tringali, An Abstract Factorization Theorem and Some Applications, J. Algebra 602 (2022), 352–
380.

Il dott. Marco Ortu del DIEE, Università di Cagliari
terrà un corso breve dal titolo:

Leggi di potenza, con applicazioni nell’ingegneria del software

16 Gennaio 2020, ore 9:00 – 14:00 ( ​ Aula C – Dip. di Matematica e Informatica, via Ospedale 72)

Allegata la Locandina