Research activity of
Pierluigi Benevieri
My research activity concerns mainly the following fields :
Topological degree theory.
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Topological methods in differential delay equations.
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Differential topology in general economic equilibrium theory.
Topological degree theory
The research in this area is in collaboration with
Prof. Massimo Furi and Prof. Maria Patrizia Pera.
The most important result obtained is the construction of a concept of
oriented degree for nonlinear Fredholm maps of indez zero between
differentiable manifolds modeled on Banach spaces.
The starting point of this construction is a (merely algebraic) notion
of orientation for linear Fredholm maps between real vector spaces.
This notion allows to define an orientation for nonlinear Fredholm
maps of index zero between Banach manifolds and the degree is defined
for the so called orientable maps.
This concept of degree is connected with other theories.
You can see in particular papers of
Elworthy-Tromba and
Fitzpatrick-Pejsachowicz-Rabier.
Topological methods in differential delay equations
This topic has been studied during three months in Rutgers University,
NJ, USA, with the supervision of Prof. Roger Nussbaum.
The research in this area started with some `state dependent time lag'
equations.
The purpose is the application of topological tools as Fuller index
and recent extensions of Fenske.
Differential topology in general economic equilibrium theory.
This is a project in collaboration with Prof. Andrea
Battinelli of University of Siena, Dott. Laura Carosi of
University of Pisa and Prof. Antonio
Villanacci of University of Firenze.
Recent publications
1
P. Benevieri, Recenti sviluppi nella teoria del grado
topologico, Boll. UMI, 1-A (8) Suppl. (1998), (fascicolo
speciale dedicato alle tesi di dottorato) 81-84
(file PDF).
2
P. Benevieri and M. Furi, A simple notion of orientability
for Fredholm maps between Banach manifolds and degree theory, Annales
des sciences mathémathiques du Québec, 22 n. 2 (1998),
131-148
(file PDF).
3
P. Benevieri and M. Furi, On the concept of orientability for
Fredholm maps between real Banach manifolds, Topol. Methods
Nonlinear Anal, 16 (2000), no. 2, 279-306
(file PDF).
4
P.Benevieri, M. Furi and M. P. Pera The Invariance of
Domain for C1 Fredholm maps of index zero, Recent trends in
nonlinear analysis, Progr. Nonlinear Differential Equations Appl., 40,
Birkhäuser, Basel, 2000, 35-39
(file PDF).
5
P.Benevieri Orientation and degree for Fredholm maps of
index zero between Banach spaces, Nonlinear analysis and its applications to
differential equations (Lisbon, 1998), 201-213, Progr. Nonlinear Differential
Equations Appl., 43, Birkhäuser Boston, Boston, MA, 2001
(file PDF).
6
P. Benevieri, M. Furi, Bifurcation results for families
of Fredholm maps of index zero between Banach spaces, Nonlinear analysis and
its applications (St. John's, NF, 1999). Nonlinear Anal. Forum 6 (2001), no. 1,
35-47
(file PDF).
7
P. Benevieri, M. Furi and M. P. Pera, On the
Product Formula for the oriented degree for Fredholm maps of index zero
between Banach manifolds, Nonlinear Anal. 48 (2002), no. 6, Ser. A: Theory
Methods, 853-867
(file PDF).
8
A. Villanacci, A. Battinelli, P. Benevieri and L. Carosi, Differential
Topology and General Equilibrium with Complete and Incomplete
Markets, Kluwer Academic Publishers, Dordrecht, 2002.
9
P. Benevieri, M. Furi, M. P. Pera and M. Spadini, About the
Sign of Oriented Fredholm Operators between Banach Spaces,
Z. Anal. Anwendungen, 22 (2003), 3, 619-654
(file PDF).
10
P. Benevieri, A. Gavioli and M. Villarini,
Existence of Periodic Orbits
for Vector Fields via Fuller Index and
the Averaging Method, preprint.