FIELDS OF RESEARCH: Geometry of PDE's (Differential Geometry, Algebraic Geometry and Algebraic Topology); (Co)bordism in PDE's and quantum PDE's; Geometry of PDE's in
Continuum Mechanics; Geometry of PDE's in Quantum Field Theory and
Quantum Supergravity; Geometry of PDE's in Mathematical Physics.
SOME RECENT RESULTS: New points of view were recently introduced in
the geometric theory of PDE's, by adopting some algebraic
topological approaches. In particular, integral (co)bordism
groups are seen very useful to characterize global solutions. The
methods developed, in the category of (non)commutative
PDE's, to find integral bordism groups, allowed us to obtain, as a
by-product, existence theorems for global solutions in a pure
geometric way. Another result that is directly related to the
knowledge of integral bordism groups of PDE's, is the possibility
to characterize PDE's by means of some important algebras, related
to the conservation laws of these equations (Hopf algebras of
PDE's). Objects of these algebras identify invariants of global
solutions. Moreover, thanks to an algebraic characterization of
PDE's, one has also a natural way to recognize quantized PDE's as
objects of the category of quantum PDE's. These results have
opened a new sector in Algebraic Topology, that we can formally
define the Algebraic Topology of PDE's.
The characterization of global solutions, made by means of integral bordism groups, has allowed
to obtain applications in some important PDE's of the Riemannian Geometry and Mathematical
Physics, as Ricci-flow equation, Navier-Stokes equation and quantum Yang-Mills equations. Quantum super
Yang-Mills equations, are discussed in the framework of quantum
supermanifolds, obtaining a new approach to unify the four fundamental forces,
(gravitational, electromagnetic, weak-nuclear, strong-nuclear), in
an unique geometric structure at the quantum level. The geometric theory of PDE's,
built in the category of quantum supermanifolds,
gives us a dynamic theory to describe quantum phenomena also at
very high energy levels, where quantum-gravity becomes
dominant. Quantum black holes are interpreted as solutions of
quantum super Yang-Mills equations with quantum-(super)gravity in
action.
By means of the Algebraic Topology of PDE's some fundamental problems (some Millennium Problems) in Mathematics are solved. More precisely, solutions for the following problems: 1) Poincaré conjecture, 2) Navier-Stokes existence and smoothness, 3) Yang-Mills existence and mass-gap, are contained in some works ([62, 70, 74, 77, 78, 80, 81], [39, 42, 45, 46, 63, 74], [54, 60, 69, 71, 75, 76],) quoted below in ''Publications''. (For more details see also CV and works quoted there.)
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Award Sapienza Ricerca 2010 - (See CV) 
...For A World
With Mathematics...
