Geometric Measure Theory in Carnot groups. Differential forms in
Carnot groups and differential operators on forms. Homogenization
for differential operators in Carnot groups. Integral inequalities
in Carnot groups. Mathematical models in medicine
Geometric Measure Theory in Carnot groups: in collaboration with R.
Serapioni and F. Serra Cassano we plan to carry out a complete
theory of intrinsic submanifolds of Carnor groups. We plan also to
attack the study of rectifiable currents in Heisenberg groups.
Differential forms in Carnot groups and differential operators on
forms: in collaboration with A. Baldi and M.C. Tesi we plan to
develop a L^p theory of intrinsic differential forms in Carnot
groups and to study Maxwell system in groups.
Homogenization for differential operators in Carnot groups: in
collaboration with A. Baldi, N. Tchou and M.C. Tesi we plan to
carry on the study of the homogenization of differential operators
in Carnot groups via compensated compactness methods.
Integral inequalities in Carnot groups: I plan to extend to
intrinsic differential forms in Carnot groups a chain of
Poincaré-type inequalities proved recently by Lanzani & Stein
for De Rham differential forms.
Mathematical models in medicine: in collaboration with Maria Carla
Tesi and with a group of neurologists and nuclear physicians of the
AO "S.Maria Nuova" of Reggio Emilia (coordinated by N. Marcello),
we try to develop a mathematical model of the formation of plaques
in Alzheimer disease and to compare the results with clininical
evidences obtained using new specific probes.
