Agostino Pr´Astaro Edition April 2014, 1-60 CURRICULUMVITAE

Curriculum Vitae Agostino Pr´astaro Edition April 2014, 1-60 CURRICULUMVITAE

CURRICULUM VITAE

AGOSTINO PRASTARO´

Fig. 1. Agostino Pr´astaro

UNIVERSITY OF ROMA LA SAPIENZA - ROMA - ITALY

1 2 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

University address: Department of Methods and Mathematical Models for Applied Sciences(MEMOMAT) University of Roma La Sapienza, Via A.Scarpa, 16-00161 - Roma, Italy. Phone: +39-06-49766723; Fax: +39-06-4957647 E-mails: [email protected]; [email protected]. Home page: http://www.dmmm.uniroma1.it/ agostino.prastaro/HOMEPAGEPRAS.htm

Fig. 2. University of Roma La Sapienza Pictures.

Home address:

Fig. 3. Porta Maggiore.

Via L’Aquila, 29-00176 - Roma - Italy. (Phone & Fax: +39-06-7023432 ) CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 3

Acknowledgments

Fig. 4. Pr´astaro’smathematicians poster 2013-2014.

“...A World Of Mathematicians...For A World With Mathematics... ” 4 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

1. GENERAL INFORMATIONS

• Birth-place and birth-date: Civitavecchia (Roma - Italy), 01-11-1942. • Diploma di maturitàscientifica: Trieste - Italy, July 1960. • Doctor degree in Theoretical Physics: University of Torino, Torino - Italy, July 1967. • Research grant in Theoretical Physics: University of Torino, Torino - Italy, 1967–1968. • Researcher in Mathematical Models and Processes of Materials: Montedison Research Center, Ferrara - Italy, 1968–1975. • School on Tensors and Group Theory Applied to Crystallography - Brooklyn Crystallographic Laboratory, Cambridge, UK, 1970. • Professor (incaricato) in Mathematical Physics, University of Lecce and Uni- versity of Calabria, Cosenza - Italy, 1975–1976. • Professor (incaricato) in Mathematical Physics, University of Calabria, Cosenza - Italy, 1976-1980. • Professor (associate) in Mathematical Physics, University of Calabria, Cosenza - Italy 1980-1990. • Professor (associate) in Mathematical Physics, University of Roma La Sapienza, Roma - Italy, 1990–2013. • Courses taught: Rational Mechanics, Institutions of Mathematical Physics, Mathematical Physics, Mathematical Analysis, Continuum Mechanics, Differential Geometry. • Member in: Italian Mathematical Union (UMI), European Mathematical So- ciety (EMS), National Group of Mathematical Physics (GNFM/INDAM), Ameri- can Mathematical Society (AMS), International Federation of Nonlinear Analysts (IFNA), Mathematical Association of America (MAA), International Mathematical Union (IMU). • Reviewer for: Mathematical Reviews, Zentralblatt Mathematics and some other mathematical journals. • Head of the research-team MIUR-Faculty of Engineering-University of Roma La Sapienza: “Geometry of PDEs and Applications”. • Member in National Project in Algebraic Topology and Differential Geometry (1986–2004). • Member of PhD Committee: “Models and Mathematical Methods for Technol- ogy and Sociology”, University of Roma La Sapienza (1999–2003). • Fields of Research: Geometry of PDE’s (Differential Geometry, Algebraic Ge- ometry and Algebraic Topology); (Co)bordism in PDE’s and quantum PDE’s; Ge- ometry of PDE’s in Continuum Mechanics; Geometry of PDE’s in Quantum Field Theory and Quantum Supergravity; Geometry of PDE’s in Mathematical Physics. • Organizer of many international conferences. The last ones from 2000: WCNA- 2000 Organized Session Functional Equations Geometric Analysis, Catania, Italy (2000); Joint Meeting AMS-UMI Special Session Advances in Differential Geome- try of PDE’s and Applications, Pisa, Italy (2002); ICM-2006 Satellite Conference Advances in PDE’s Geometry, Madrid, Spain (2006); WCNA 2008 - Organized Session: Advances Geometric Analysis, Orlando, FL - USA (2008). CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 5

• (Some) Invited Speaker or Partecipant and Visiting Professor: Department of Mathematics, Univ. P. Sabatier, Toulouse, France (1986); I.H.E.S., Bures- sur-Yvette, France (1988); Department of Mathematics, University of California, Berkely, USA, (1990); University of Torun, Poland (1996); University of Budapest, Hungary (1996); Technical University of Athens, Greece (1996); Third World Con- gress of Nonlinear Analysts (WCNA2000), Catania, Italy (2000); University of De- brecen, Hungary (2000); Department of Mathematics, University of Opava, Czech Republic (2000); University of Bilbao, Spain (2000); Department of Mathematics, University of Atlanta, USA (2003); Department of Mathematics, Florida Institute of Technology, Melbourne, USA (2005); ICM-2006 and Facultad de Matematicas, Universidad Complutense de Madrid, Madrid - Spain (2006); Department of Math- ematics, Morehouse College, Atlanta, USA (2007); WCNA 2008 - Orlando, FL - USA (2008).

• Award Sapienza Ricerca 2010. (Awarded publications [58, 59, 60].) • Member in IFNA Board of Global Advisors World Congress of IFNA, 2012 - Athens, Greece, June 25-July 1, 2012. • Membership in Roma Sapienza Foundation (from 2014 ). • Publications: 96 scientiﬁc publications including 3 monographs as author, 3 as editor and coauthor and 2 patents. (See pages 7, 21 and 27.) [Metric informations from Googlescholar: ”author:Prastaro author:A.”: (i) Works- number: 116; (ii) Citations-number: 912; (iii) H-index: 17.]1

1Please visit Wikipedia for criticism about H-index. 6 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

Fig. 5. School of Engineering, University of Roma La Sapienza. CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 7

2. SUMMARY OF PRINCIPAL RESULTS

• [1]. This work is inserted in the problematic of the Regge’s trajectories in strong interactions. The paper shows a relation between Regge’s trajectories of s- channel with the ones in the t-channel that allows us to obtain general behaviours of diffusion amplitudes. • [2, 3, 4, 5, 95, 96]. These are mathematical models for elongational flows, extrusion and flash-spinning of polymers. These models allow us to obtain optimum industrial conditions of manufacturing. • [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 25]. These works are devoted to build new mathematics that allow us to describe mechanics in an intrinsic and completely covariant way. This is obtained by introducing new spaces, derivative spaces, that are the natural universal spaces for differential calculus and PDEs. This point of view generalizes previous one introduced by Ehresmann and allows us to treat all the differential objects in algebraic way, i.e., as generalized tensor-like objects. In this context a generalized form of the Noether theorem for any PDE, even if of non-Lagrangian type, is obtained. Then this mathematical machinery is applied also to describe the physics of continuum media, gauge theories and supergravity. • [22, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 50]. These works extend the geometric formulation of PDEs, developed in the previous papers, in order to consider also singular solutions and to obtain a geometric formulation of quantization of PDEs. This geometric approach gives an intrinsic formulation of the canonical quantization of PDEs. All this theory is extended also to super PDEs, i.e., PDEs defined in the category of supermanifolds. In particular, a geometric formulation of the canonical quantization of PDEs in supergravity is obtained. • [37, 38, 39, 42, 45, 46, 51, 55, 56, 63, 70, 89]. These works present a general theory of integral (co)bordism in PDEs and develop general methods to calculate the corresponding (co)bordism groups. Such results allow us to characterize also global solutions by means of algebraic topologic methods, and to recognize tunnel effects in such solutions. In particular, in [55] some improvements are given emphasizing the role played by singular and weak-solutions and their relations with the bordism groups for smooth solutions. Tricomi equation, heat equation, Ricci-flow equation and d’Alembert equation on finite dimensional smooth manifolds are considered there as interesting examples where to apply the general theory. Variational PDE’s, and their global solutions characterizations, by means of integral bordism groups, are given in [56]. The methods proposed are constructive, as they give us suitable tools to build solutions. In particular in [39, 42, 44, 45, 46, 63] the Navier-Stokes equation it is also carefully considered and it is proved for such an equation existence of global (smooth) solutions. In this way it is solved an old well known problem in the theory of PDEs, remained open for many decades. (See also [65] where some further improvements are given.)2

2First results on the geometric theory of the Euler equation and the Navier-Stokes equation were obtained in [15, 16, 17, 19]. 8 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

In [51] some applications of previous results on integral (co)bordism groups and canonical quantization of PDEs are considered. In particular fully covariant canonical quantizations of elementary particles interacting with the gravitational field, also for massive neutrinos, are obtained. (See also [89].) • [32, 33, 34, 40, 41]. Interesting applications of the geometric theory of PDE’s, and integral bordism group theory for PDE’s, as formulated by A.Prástaro,are considered to characterize local and global solutions of the d’Alembert equation and its generalizations on Rn. For such equations integral bordism groups are explicitly calculated and global solutions built. Sophisticated solutions with tunnel effects are recognized. • [37, 43, 44, 49, 54, 57, 58, 59, 60, 76, 78]. Here the new concepts of quantum manifolds and quantum PDEs are introduced, and for such structures it is formulated a new noncommutative mathematics, (differential geometry, algebraic geometry, geometric theory of quantum PDEs and determination of their integral (co)bordism groups), that extends previous one for commutative PDEs. Applica- tions to many interesting quantum PDEs are given. In particular, in [44] the concept of category of quantum quaternionic manifolds is introduced and for PDEs built in such a category theorems of existence of local and global solutions are proved. Extensions to the quantum Navier-Stokes equation of previous results, obtained in [39, 42, 44, 45, 46, 63] for the Navier-Stokes equation, are given in [49]. Moreover, in [54, 57, 58, 59, 60, 69] the theory of quantum PDEs is extended to quantum super PDEs and applied to quantum Yang-Mills PDEs and quantum supergravity PDEs. These papers extend previous results of the integral (co)bordism groups theory for quantum PDE’s, to super quantum PDE’s. Furthermore, quantized (super) PDE’s are identified with quantum (super) PDE’s in a canonical way. A proof that quantum (super) PDE’s are a more general approach in the description of quantum physics, that goes beyond the point of view of the quantization of (super) classical theories (PDE’s), is given. Similarly to the commutative case, integral (co)bordism groups for quantum (super) PDE’s, are related to weak, singular and smooth solutions, showing algebraic relations between such groups. Such a theory is applied to many interesting PDE’s, and in particular, to quantum super Yang-Mills equations. Characterizations of important and very sophisticated solutions, as quantum tunnel effects and quantum black holes evaporation processes, are obtained by means of integral bordism groups. The new algebraic topologic methods introduced allowed also to solve some important problems in Mathematics. In particular, in [54, 60] the open problem to prove existence of global smooth solutions for quantum (super)Yang-Mills equations with mass-gap is solved. • [47, 48]. Here some interesting applications of the geometric theory of quantum PDE’s are considered. In fact, generalizations in the category of quantum manifolds of previous results on d’Alembert equation [32, 33, 34], are given. • [89, 91]. These monographs present, for the first time, a systematic formulation of the geometric theory of noncommutative PDE’s which is suitable enough to be used for a mathematical description of quantum dynamics and quantum field theory. A geometric theory of supersymmetric quantum PDE’s is also considered, in order to describe quantum supergravity. Covariant and canonical quantizations of (super) PDE’s are shown to be founded on the geometric theory of PDE’s and to produce quantum (super) PDE’s by means of functors from the category of CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 9 commutative (super) PDE’s to the category of quantum (super) PDE’s. Global properties of solutions to (super) (commutative) PDE’s are obtained by means of their integral bordism groups. In particular the (quantum) Navier-Stokes problem and the (quantum)Yang-Mills problem are considered showing that their solutions can be obtained in the framework of the integral bordism groups for such equations. • [61, 62]. In these two papers the theory on the integral bordism groups for PDE’s, formulated by Prástaro,is applied to some interesting PDE’s. In particular, in the first part a new general theory is developed that recognizes natural webs structures on PDE’s. These structures are important to solve (generalized) Cauchy problems. Some applications interesting PDE’s of the Mathematical Physics are also given (wave equation, Korteweg de Vries equation). In the second part applications of the general theory of webs on PDE’s is applied to some problems concerning the Riemannian geometry. In particular are considered generalized Yamabe problems and the proof of the Poincaréconjecture for 3-dimensional compact Riemann- ian manifolds, via the Ricci-flow equation. The new algebraic topologic methods introduced by A.Prástaroin the theory of PDE’s, allowed to give a new proof that the Poincaréconjecture is true. • [64, 65, 66, 67, 68]. In these papers an unified geometric theory of stability for PDE’s and solutions of PDE’s, is formulated in the framework of Prástaro’s geometric theory of partial differential equations, i.e., by using integral bordism groups. Relations with the Ljapunov’s stability theory, and the classic Ulam problem for approximate homomorphisms are stressed. (See also [52, 53].) Examples for some important PDE’s are given. In particular the theory is applied to new anisotropic MHD-PDE’s encoding dynamics for incompressible plasma fluids with nuclear energy production. Such PDE’s are related, on the ground of their integrability properties, to crystallographic groups, (extended crystal PDE’s). • [69, 70, 71]. In these papers one further extends the theory of quantum (super) PDE’s, previously developed in [37, 43, 44, 49, 54, 57, 58, 59, 60, 76, 78] to systematically adapt the algebraic topologic surgery techniques to quantum supermanifolds and to solutions of quantum super manifolds. In particular in [70], by using also the Prástaro’sgeometric theory of quantum super PDE’s, one formulates and proves the quantum Poincaréconjecture. This generalizes to the category of quantum super PDE’s, the well known Poincaréconjecture for 3-dimensional closed Riemannian manifolds. (See also [62] for a proof of this conjecture that is different by one by G. Perelman.) Furthermore in [71] one generalizes to the category of quantum supermanifolds QS, the variational calculus for variational problems, constrained by PDE’s, as formulated in [56]. In this way one formulates also a new quantum gravity theory that extends to the category QS the Einstein’s General Relativity. There, and in [69] solutions of quantum gravity equations, interpreting nuclides and quantum black-holes, are considered in details. In particular quantum black-holes are solutions that describe very high energy level production of particles, where the effects of strong-quantum-gravity become dominant. A new concept of quantum propagator is introduced that allows us to recognize Green kernels as “linear approximations” of such non-linear quantum propagators. More precisely in [60] we have seen how to identify solutions of quantum super Yang-Mills equations that come from the Dirac quantization of super classic counterpart of such equations, i.e. (classic Dirac solutions). Here we go beyond classic Dirac solutions 10 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 and recognize solutions related to a new nonlinear geometric definition of quantum super Yang-Mills equation propagators. • [72, 73]. In these papers one gives a geometric formulation of singular PDE’s theory on the ground of the PDE’s geometric theory by A.Prástaro,i.e. by considering integral bordism groups and surgery techniques. Many interesting applications are given by explicitly solving (singular) boundary value problems of physical relevance. A particular attention is reserved to singular ODE’s, where a general characterization of their singular points is obtained. Many examples of (singular) boundary value problems for ODE’s are given too. Relations with the stability theory for PDE’s and PDE’s solutions are given by utilizing the recent geometric stability theory given by A.Prástaroin [64, 65, 66, 67, 68]. • [74]. In this paper we show that between PDE’s and crystallographic groups there is an unforeseen relation. In fact we prove that integral bordism groups of PDE’s can be considered extensions of crystallographic subgroups. In this respect we can consider PDE’s as extended crystals. Then an algebraic-topological obstruction (crystal obstruction), characterizing existence of global smooth solutions for smooth boundary value problems, is obtained. These results, are also extended to singular PDE’s, introducing (extended crystal singular PDE’s). An application to singular MHD-PDE’s, is given extending some our previous results on such equations, and showing existence of (finite times stable smooth) global solutions crossing critical zone nuclear energy production • [75, 76]. Aim of the second paper (partially announced in [75]) is to specialize our study to quantum supergravity Yang-Mills PDE’s (quantum SG-Yang-Mills PDE’s). This type of equations have been previously introduced by us in some recent works and appears very useful to encode quantum dynamics unifying, just at quantum level, gravity with the other fundamental forces of Nature, i.e., electromagnetic, weak and strong forces. These equations extend, at quantum level, some superclassical ones, well known in literature about supergravity. In fact supergravity, as has been usually considered, is a classical field theory, that, in some sense comes from a generalization of Charles Ehresmann and Elie` Cartan’s differential geometry. Then classical supergravity requires to be quantized. But in this way one discards nonlinear phenomena. In fact this quantization is obtained by means of so-called quantum propagators, that are just associated to linearizations of classical PDE’s. Our formulation, instead, works directly on noncommutative manifolds (quantum supermanifolds), and the quantization is not more necessary. In fact, whether it is performed in this noncommutative framework, it can bee seen as a linear approximation of a more general nonlinear integration. In some previous papers this important aspect has been carefully proved. Here we characterize quantum super PDE’s like extended crystals, in the sense that their integral bordism groups can be considered as extensions of crystallographic subgroups. This approach generalizes our previous one for commutative PDE’s, and allows us to identify an algebraic topologic obstruction to the existence of global smooth solutions for PDE’s in the category QS. Furthermore, for such solutions we study their stability properties from a geometric point of view. Section 3. Here we consider “quantum gravity” in the category QS, and encoded by suitable quantum Yang-Mills equations (quantum SG-Yang-Mills PDE’s), say (\YM). In this way we are able to characterize quantum (super)gravity like a secondary object, associated to some geometric fundamental objects (fields), solutions of (\YM). Then the CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 11 mass properties of such solutions are directly pointed-out, without the necessity to directly assume symmetry breaking Higgs-mechanisms. However, we recognize a constraint in (\YM), that gives a pure quantum geometrodynamic mechanism able to justify mass acquisition (or loss) to a quantum solution of (\YM). Furthermore, nuclear particles and nuclides can be seen as suitable p-chain solution of (\YM), and their energetic, thermodynamic and stability properties characterized. • [77, 78, 79, 80]. The famous Poincaré’sconjecture is about n-dimensional manifolds, with n = 3, (R. S. Hamilton, G. Perelman, A. Prástaro[74, 62]), but there are also generalizations of this conjecture for higher dimension manifolds. For n = 4 the Poincaréconjecture has been proved by M. Freedman and for n ≥ 5, by S. Smale. More recently has been given a generalization for quantum supermanifolds by A. Prástaro,that has also proved it in [70]). Nowadays, one can state that a generalized Poincaréconjecture can be proved, or disproved, depending on the particular category C in which it is formulated. This problem aroses in the framework of the geometric topology, but in order to be solved it was necessary to go outside that framework and recast the problem in a theory of PDE’s. But the more recent results by A. Prástaro[70, 74, 62]), have proved that it was necessary to return inside the algebraic topology framework, applied to the PDE’s geometric theory. Really, it was soon evident that remaining in a pure algebraic topologic approach it was not enough to solve this conjecture. In fact, a fundamental idea to solve this problem is to ask whether it is possible find a smooth manifold, V , that without singular points bords a 3-dimensional compact, closed, smooth, sim- 3 3 ply connected manifold N with S , when N is homotopy equivalent to S .∪ The bordism theory is able to state that a smooth manifold V such that ∂V = N S3, there exists, since the nonoriented and oriented 3-dimensional bordism groups Ω3 + + and Ω3 respectively, are both trivial: Ω3 = Ω3 = 0. However, by simply looking to the above bordism groups it is impossible to state if V has singular points (i.e., has holes) or it is a cylinder. By the way, more informations can be obtained by the h-cobordism theory. More precisely the h-cobordism theorem in a category C of manifolds, states that if the compact manifold V has ∂V = N0 ⊔ N1, such that the inclusion maps N ,→ V , i = 0, 1, are homotopy equivalences, (i.e., V is a h- i ∼ cobordism), and π1(Ni) = 0, then V =C N0 × [0, 1]. This theorem holds for n ≥ 5 in the category of smooth manifolds (S. Smale) and for n = 4 in the category of topological manifolds (M. Freedman). But it does not work for n = 3 ! A very important angular stone, in the long history about the solution of the Poincaréconjecture, has been the introduction, by R. S. Hamilton, of a new approach recasting the problem in to solving a PDE, the Ricci flow equation, and asking for nonsingular solutions there, that starting from a Riemannian manifold (N, γ) arrive to the 3-dimensional sphere S3, respectively identified with initial and final Cauchy manifolds in the Ricci flow equation. In that occasion the Mathemat- ical Analysis, or more precisely the Functional Analysis, entered in the Poincaré conjecture problem. This approach has had many improvements until the papers by G. Perelman. More recently, A. Prástaro,by using his algebraic topologic theory of PDE’s was able to give a pure geometric proof of the Poincaréconjecture. Let us emphasize that the usual geometric methods for PDE’s (Spencer, Cartan), were able to formulate for nonlinear PDE’s, local existence theorems only, until the introduction, by A. Prástaro,of the algebraic topologic methods in the PDE’s geometric theory. These give suitable tools to calculate integral bordism groups 12 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 in PDE’s, and to characterize global solutions. Then, on the ground of integral bordism groups, a new geometric theory of stability for PDE’s and solutions of PDE’s has been built. These general methodologies allowed to A. Prástaroto solve fundamental mathematical problems too, other than the Poincaréconjecture and some of its generalizations, like characterization of global smooth solutions for the Navier-Stokes equation and global smooth solutions with mass-gap for the quantum Yang-Mills superequation. (See [38, 45, 46, 54, 60, 63, 70, 71, 74, 91, 81, 62].) The main purpose of this paper is to emphasize some problems related to exotic heat PDE’s recently focused.3 The following theorem is a direct issue from results contained in [74]. Theorem 2.1. Any 3-dimensional compact, closed simply connected smooth manifold, homotopy equivalent to S3, is diffeomorphic to S3.4 The main purpose of [78] is to show how, by using the PDE’s algebraic topology, introduced by A. Prástaro,one can prove the Poincaréconjecture in any dimension for the category of smooth manifolds, but also to identify exotic spheres. In the framework of the PDE’s algebraic topology, the identification of exotic spheres is possible thanks to an interaction between integral bordism groups of PDE’s, conservation laws, surgery and geometric topology of manifolds. With this respect we shall enter in some details on these subjects, in order to well understand and explain the meaning of such interactions. So the paper splits in three sections other than this Introduction. 2. Integral bordism groups in Ricci flow PDE’s. 3. Morse theory in Ricci flow PDE’s. 4. h-Cobordism in Ricci flow PDE’s. The main result is contained just in this last section and it is Theorem 2.2 that by utilizing the previously considered results states (and proves) the following.5 Theorem 2.2. The generalized Poincaréconjecture, for any dimension n ≥ 1 is true, i.e., any n-dimensional homotopy sphere M is homeomorphic to Sn: M ≈ Sn. ∼ For 1 ≤ n ≤ 6, n ≠ 4, one has also that M is diffeomorphic to Sn: M = Sn. But for n > 6, it does not necessitate that M is diffeomorphic to Sn. This happens when the Ricci flow equation, under the homotopy equivalence full admissibility hypothesis, (see below for definition), becomes a 0-crystal. Moreover, under the sphere full admissibility hypothesis, the Ricci flow equation becomes a 0-crystal in any dimension n ≥ 1.

3The Ricci flow equation can be considered a generalization of the classical Fourier’s heat equation ut − κuxx = 0. In this paper we call exotic heat equations, PDE’s that, like the Ricci j ≡ j − j i ≤ ≤ | | flow equation, are of the type F ut f (uk) = 0, 1 i, j m, with the length k of the multi- index k ∈ {1, ··· , n}, given by 0 ≤ |k| ≤ s ≤ r, where F j : Jr(W ) → R are analytic functions of order r ≥ 0 on a fiber bundle π : W ≡ R × E(M) → R × M, where E(M) is a vector bundle over M, with M an analytic manifold of dimension n. Let us emphasize that the structure of exotic heat equation is the more suitable to use in order to prove (generalized) Poincaréconjectures. In fact, the idea to use PDE’s to solve the Poincaré’sconjecture, was the initial motivation to introduce and study the well-known Yamabe equation. But that road did not turn out a lucky choice to prove the conjecture, even if the Yambe equation is a very important equation to study conformal problems in Riemannian geometry. (See, e.g., [62].) 4This last result agrees with the Hauptvermuntung conjecture that was proved for (n = 2, 3)- dimensional manifolds and disproved for (n ≥ 4)-dimensional manifolds. (See A. Casson, J. Milnor, E. Moise, T. Radó,D. Sullivan, W. Tuschmann.) 5Results of this paper agree with previous ones by J. Cerf, M. Freedman, M. A. Kervaire and J. W. Milnor, E. Moise and S. Smale, and with the recent proofs of the Poincaréconjecture by R. S. Hamilton, G. Perelman, and A. Prástaro. CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 13

In [79] a theorem characterizing global solutions of exotic 8-d’Alembert equation is given. This theorem allows us to state that two diffeomorphic exotic 7-sphere, ′ 8 identified with two Cauchy manifolds in (d A)8 over R , bound singular solutions only, but they cannot bound smooth solutions. (Compare with the situation in the Ricci flow equation on compact, simply connected 7-dimensional Rimennian manifolds [78].) In [80] are generalized results of the previous three papers to any PDE and relating them to general algebraic topological properties of PDE’s, like integral bordism groups, conservation laws, spectral sequences, algebraic topological spectra. In particular it is emphasized their relations with exotic Cauchy manifolds, that motivate the name “exotic” given to these equations. Theorem 2.2 is generalized to any PDE. The main results in this paper is just the generalization to dimension n = 4, of such theorem, obtained by means the proof of the smooth Poincaréconjecture. This is the generalization of the Poincaréconjecture in the category of smooth manifolds in dimension n = 4. This was a very important open problem, remained unsolved also after the solution of the famous Poincaréconjecture. • [81]. This paper aims to further develop the A. Prástaro’sgeometric theory of quantum PDE’s, by considering three different (even if related) subjects in this theory. The first is a way to characterize quantum PDE’s by means of suitable Heyting algebras. Nowadays these algebraic structures are considered important in order to characterize quantum logics and quantum topoi. Really we prove that to any quantum PDE can be associated a Heyting algebra, naturally arising from the algebraic topologic structure of the PDE’s and that encodes its integral bordism group. Another aspect that we shall consider is the extension of the category Q of quantum manifolds, or QS of quantum supermanifolds, to the ones Qhyper of quantum hypercomplex manifolds. These generalizations are obtained by extending a quantum algebra A, in the sense of A. Prástaro,by means of Cayley-Dickson algebras. In this way one obtains a new category of noncommutative manifolds, that are useful in some geometric and physical applications. In fact there are some fashioned research lines, concerning classical superstrings and classical super-2-branes, where one handles with algebras belonging just to some term in the Cayley-Dickson construction. Thus it is interesting to emphasize that the Prástaro’sgeometric theory of quantum PDE’s can be directly applied also to PDE’s for such quantum hypercomplex manifolds. This allows us to encode quantum micro-worlds, by a general theory that goes beyond the classical simple description of classical extended objects, and solves also the problem of their quantization. Let us emphasize that in some previous works we have formulated a geometric theory of quantum manifolds that are noncommutative manifolds, where the fundamental algebra is a suitable associative noncommutative topological algebra, there called quantum algebra. Extensions to quantum supermanifolds and quantum- quaternionic manifolds are also considered too. Furthermore, we have built a geometric theory of quantum PDE’s in these categories of noncommutative manifolds, that allows us to obtain theorems of existence of local and global solutions, and constructive methods to build such solutions too. On the other hand it is well known that the sequences R ⊂ C ⊂ H ⊂ O ⊂ S, where S is the sedenionic algebra, fit in the so-called Cayley-Dikson construction, R ⊂ C ⊂ H ⊂ O ⊂ S ⊂ · · · ⊂ Ar ⊂ · · · , where Ar is a Cayley algebra of 14 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

r A ∼ R2r ≥ A R dimension 2 : r = ⊗, r 0, 0 = . Thus we can also consider quantum A ≤ ≤ ∞ hypercomplex algebras A r, 1 r , where A is⊗ a quantum algebra⊗ in the sense of Prástaro,obtaining the natural inclusions A R Ar ,→ A R Ar+1. It is important to note that the Cayley algebras Ar are not associative for r ≥ 3, (even if they are exponential associative6) hence also the corresponding quantum hypercomplex algebras are non-associative for r ≥ 3, despite the associativity of the quantum algebra A. This fact introduces some particularity in the theory of PDE’s on such algebras. The purpose of this paper is just to study which new behaviours have PDE’s on the category Qhyper of quantum hypercomplex manifolds. Let us emphasize that a first justification to use the category of quantum (super) manifolds to formulate PDE’s that encode quantum physical phenomena, arises from the fact that quantized PDE’s can be identified just with quantum (super) PDE’s, i.e., PDE’s for such noncommutative manifolds. However, to formulate equations just in the category Q (or QS) allows us to go beyond the point of view of quantization of classical systems, and capture more general nonlinear phenomena in quantum worlds, that should be impossible to characterize by some quantization process. (See Refs. [58, 59, 60, 70, 71, 75, 76].) In fact the concept of quantum algebra (or quantum superalgebra) is the first important brick to put in order to build a theory on quantum physical phenomena. In other words it is necessary to extend the fundamental algebra of numbers, R, to a noncommutative algebra A, just called quantum algebra. The general request on such a type of algebra can be obtained on the ground of the mathematical logic. (See [89, 91].) In fact, we have shown that the meaning of quantization of a classical theory, encoded by a PDE Ek, in the category of commutative manifolds, is a representation of the logic L(Ek) of the classic theory, into a quantum logic Lq. More precisely, L(Ek) is the Boolean algebra of subsets of the set Ω(Ek)c of solutions of Ek: L(Ek) ≡ P(Ω(Ek)c). (The infinite dimensional manifold Ω(Ek)c is called also the classic limit of the quantum situs of Ek.) Furthermore, Lq is an algebra A of (self-adjoint) operators on a locally convex (or Hilbert) space H: Lq ≡ A ⊂ L(H). Then to quantize a PDE Ek, means to define a map L(Ek) → Lq, or an homomorphism of Boolean algebras q : P(Ω(Ek)c) → Pr(H), where Pr(H) is a Boolean algebra of projections on H. This construction allowed us also to prove that a quantization of a classical theory, can be identified by a functor relating the category of differential equations for commutative (super)manifolds, with the category of quantum (super) PDE’s. We can also extend a quantum algebra A, when the particular mathematical (or physical) problem requires it useful. Then the extended algebra does not necessitate to be associative. This is, for example, the case when the extension is made by means of some algebra in the Cayley-Dickson construction, obtaining a quantum- Cayley-Dickson construction: / / / / / Q0 Q1 Q2 · ·· Qr · ·· ⊗ ⊗ ⊗ where Qr ≡ A R Ar, hence Q0 = A, Q1 = A R C and Q2 = A R H, etc. The main of this paper is just to show that the Prástaro’salgebraic topologic theory of quantum PDE’s, formulated starting from 80s, directly applies to these non-associative quantum algebras arising in the above quantum-Cayley-Dickson construction.

6 n+m n m i.e., z = z z , ∀z ∈ Ar and n, m ∈ N. CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 15

Finally, the last purpose of this paper is to extend the concept of exotic PDE’s, recently introduced by A. Prástaro,for PDE’s in the category of commutative manifolds, also for the ones in the category of quantum PDE’s. In particular, we prove also smooth versions of quantum generalized Poincaréconjectures. In the following we list the main results of this paper, assembled for sections. 2. It ˆ ⊂ ˆk is shown that to any quantum hypercomplex PDE Ek Jn (W ), can be associated a topological spectrum (integral spectrum) and a Heyting algebra (integral Heyting algebra) encoding some algebraic topologic properties of such a PDE. This allows us to give a new constructive point of view to the actual approach to consider quantum logic in field theory by means of topoi. 3. Here the Prástaro’sformal geometric theory of PDE’s is extended from the category of quantum (super)manifolds to the ones for quantum hypercomplex manifolds. Global solutions of PDE’s in the category Qhyper, are characterized by means of suitable bordism groups. 4. An algebraic topologic characterization of singular PDE’s in the category Qhyper is given. 5. The concept of exotic PDE’s, perviously introduced by A. Prástaro for PDE’s in the category of commutative manifolds, is extended to the category Qhyper . Global solutions for exotic PDE’s in the category Qhyper, that allow to classify smooth solutions starting from quantum homotopy spheres are classified. In particular, an integral h-cobordism theorem in quantum Ricci flow PDE’s is proved. • [82, 83, 84]. In order to encode strong reactions of the high energy physics, by means of quantum nonlinear propagators in the Prástaro’sgeometric theory of quantum super PDE’s, some related geometric structures are further developed and characterized. In particular super-bundles of geometric objects in the category QS of quantum supermanifolds are considered and quantum Lie derivative of sections of super bundle of geometric objects are calculated. Quantum supermanifolds with classic limit are classified with respect to the holonomy groups of these last commutative manifolds. A theorem characterizing quantum super manifolds with structured classic limit as super bundles of geometric objects is obtained. A theorem on the characterization of chi-flow on suitable quantum manifolds is proved. This solves a previous conjecture too. Quantum instantons and quantum solitons are defined are useful generalizations of the previous ones, well-known in the literature. Quantum conservation laws for quantum super PDEs are characterized. Quantum conservation laws are proved work for evaporating quantum black holes too. Characterization of observed quantum nonlinear propagators, in the observed quantum super Yang-Mills PDE, by means of conservation laws and observed energy is obtained. Some previous results by A. Prástaroabout generalized Poincaré conjecture and quantum exotic spheres, are generalized to the category Qhyper,S of hypercomplex quantum supermanifolds. (This is the first part of a work divided in two parts. For part II see [83].) In the second part decomposition theorems of integral bordisms in quantum super PDEs are obtained. In particular such theorems allow us to obtain representations of quantum nonlinear propagators in quantum super PDE’s, by means of elementary ones (quantum handle decompositions of quantum nonlinear propagators). These are useful to encode nuclear and subnuclear reactions in quantum physics. Prástaro’sgeometric theory of quantum PDE’s allows us to obtain constructive and dynamically justified answers to some important open problems in high energy physics. In fact a Regge-type relation between reduced quantum 16 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 mass and quantum phenomenological spin is obtained. A dynamical quantum Gell- Mann-Nishijima formula is given. An existence theorem of observed local and global solutions with electric-charge-gap, is obtained for quantum super Yang-Mills \ \ \ PDE’s, (YM)[i], by identifying a suitable constraint, (YM)[i]w ⊂ (YM)[i], quantum electromagnetic-Higgs PDE, bounded by a quantum super partial differential \ \ relation (Goldstone)[i]w ⊂ (YM)[i], quantum electromagnetic Goldstone-boundary. An electric neutral, connected, simply connected observed quantum particle, iden- \ \ tified with a Cauchy data of (YM)[i], it is proved do not belong to (YM)[i]w. Existence of Q-exotic quantum nonlinear propagators of (\YM)[i], i.e., quantum nonlinear propagators that do not respect the quantum electric-charge conservation is obtained. By using integral bordism groups of quantum super PDE’s, a quantum crossing symmetry theorem is proved. As a by-product existence of massive photons and massive neutrinos are obtained. A dynamical proof that quarks can be broken-down is given too. A quantum time, related to the observation of any quantum nonlinear propagator, is calculated. Then an apparent quantum time estimate for any reaction is recognized. A criterion to identify solutions of the quantum super Yang-Mills PDE encoding (de)confined quantum systems is given. Supersymmet- ric particles and supersymmetric reactions are classified on the ground of integral bordism groups of the quantum super Yang-Mills PDE (\YM). Finally, existence of the quantum Majorana neutrino is proved. As a by-product, the existence of a new quasi-particle, that we call quantum Majorana neutralino, is recognized made e by means of two quantum Majorana neutrinos, a couple (νee, ν¯e), supersymmetric partner of (νe, ν¯e), and two Higgsinos. (Part I and Part II are unified in arXiv.) in the third part quantum nonlinear propagators in the observed quantum super Yang-Mills PDE, (\YM)[i], are further characterized. In particular, a criterion that assures the zero lost quantum electric-charge is obtained. In a previous work [?], we have characterized observed quantum nonlinear propagators V of the observed quantum super Yang-Mills PDE, (\YM)[i], proving that the total quantum electric- charge of incident particles in quantum reactions does not necessitate to be the same of the total quantum electric-charge of outgoing particles. This allowed us to define Q-exotic quantum nonlinear propagators ones where there is a non-zero lost quantum electric-charge, Q[V ] ∈ A, in the corresponding encoded reactions. (A is the fundamental quantum superalgebra in (\YM)[i]) This important phenomena, that is related to the gauge invariance of (\YM)[i], was non-well previously understood, since the gauge invariance was wrongly interpreted. Really just the gauge invariance is the main origin of such phenomenon, beside the structure of the quantum nonlinear propagator. This fundamental aspect of quantum reactions in (\YM)[i], gives strong theoretical support to the guess about existence of quantum reactions where the “electric-charge” is not conserved. This was quasi a dogma in particle physics. However, there are in the world many heretical experimental efforts to prove existence of decays like the following e− → γ + ν, i.e. electron decay into a photon and neutrino. In this direction some first weak experimental evidences were recently obtained. Some other exotic decays were also investigated, as for example the exotic neutron’s decay: n → p + ν +ν ¯. (See References quoted in the paper.) With this respect, one cannot remark the singular role played, in the history of the science in these last 120 years, by the electron, a very small and light CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 17 particle. In fact, at the beginning of the last century was just the electron to produce a break-down in the Maxwell and Lorentz physical picture of the world, until to produce a completely new point of view, i.e. the quantum physics. Now, after 120 years the electron appears to continue do not accept the place that physicists have reserved to it in the world-puzzle. Aim of the third paper is to further characterize a criterion to recognize under which constraints quantum nonlinear propagators preserve quantum electric charges between incoming and outgoing particles. The main result of this third \^ \ part is the existence of a sub-equation (YM)[i]• ⊂ (YM)[i], were live solutions strictly respecting the conservation of the quantum electric charge. Instead, so- \^ \ lutions bording Cauchy data contained in the sub-equation (YM)[i]• ⊂ (YM)[i], \^ that are globally outside (YM)[i]•, can violate the conservation of the quantum electric charge. This effect is interpreted caused by the quantum supergravity. The action of the quantum supergravity is able to guarantee existence of such more general quantum nonlinear propagators in quantum super Yang-Mills PDEs. In fact quantum supergravity can deform quantum nonlinear propagators in order that they can produce such exotic solutions of (\YM)[i]. In other words, quantum exotic strong reactions exist as a by-product of quantum supergravity that produces non-flat quantum nonlinear propagators. In the standard model quantum supergravity is completely forgotten. Without quantum supergravity, exotic quantum propagators cannot be realized ! The main results are the following. • A criterion to recognize under which constraints quantum nonlinear propagators have zero lost quantum electric-charge. Our main result is the identification of a sub-equation \^ \ (YM)[i]• ⊂ (YM)[i], that is formally integrable and completely integrable, such that all quantum reactions encoded there, are characterized by non-Q-exotic quantum nonlinear propagators. • A theorem proving that Q-exotic quantum nonlinear propagators of (\YM)[i] are solutions with exotic-quantum supergravity, i.e., hav- ejα • ing non zero observed quantum curvature components RK . A justification of the so-called quantum entanglement phenomenon on the algebraic topologic structure of quantum nonlinear propagators. We prove that the EPR paradox is completely solved in the framework of the Algebraic Topology of quantum PDE’s, as formulated by A. Prástaro. In fact, EPR paradox is related to a macroscopic model of physical world (Einstein’s General Relativity (GR)). In order to reconcile this model with quantum mechanics, it is necessary to extend GR to a noncommutative geometry, as made by the Prástaro’sAlgebraic Topology of quantum (super) PDE’s. In fact the logic of microworlds is not commutative, hence it is natural that macroscopic mathematical models cannot justify quantum dynamics. In other words, the incompleteness of quantum mechanics (QM) is the complementary incompleteness of the GR, since they talk at different levels: microscopic the first (QM), macroscopic the second (GR). These different points of view can be recon- ciled by introducing the noncommutative logic of the QM in the geometric point of view of the GR. But this must be made at the dynamic level ! It is not enough to formulate some noncommutative geometry to encode microworlds ! Therefore the necessity to formulated a geometric theory of quantum PDE’s as has been realized by A. Prástaro. • Existence of solutions of (\YM)[i] admitting negative (absolute) temperature and their relations with quantum entanglement is given. 18 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

• [85]. The well known Goldbach’s conjecture in number theory, remained unsolved up to now, was one of the most famous example of the Gödel’s incompleteness theorem. In this paper we give a direct proof of this conjecture. Some useful applications regarding geometry and quantum algebra are also obtained. Our proof is founded on the experimental observation that fixed an even integer, say 2n, n ≥ 1, and considered the highest prime number p1 ∈ P , that does not exceed 2n, the difference 2n − p1 is often a prime number, or if not, we can pass (1) − (1) to consider the next prime number, say p1 < p1, and find that 2n p1 is just a prime number. (We denote by P the set of prime numbers.) Otherwise, we can − (s) (s) continue this process, and after a finite number of steps, obtain that 2n p1 = p2 , (s) ∈ (s) where p2 P . This process gives us a practical way to find two primes p1 and (s) (s) (s) p2 , such that 2n = p1 + p2 , hence satisfy the Goldbach’s conjecture. Of course the question is ”Does this phenomenon is a law and why ?” The main result of this paper is to prove that our criterion, is mathematically justified. • [86]. In this paper we consider some problems in Number Theory called Lan- dau’s problems listed by Edmund Landau at the 1912 International Congress of Mathematicians. These problems are the following. 1. Goldbach’s conjecture. 2. Twin prime conjecture. 3. Legendre’s conjecture. 4. Are there infinitely many primes p such that p − 1 is a perfect square ? In [85] the proof of the Goldbach’s conjecture has been already given. In this paper we show that by utilizing some algebraic topologic methods introduced in [85], some Landau’s problems can be proved too. Furthermore, for the above fourth Landau’s problem a Euler-Riemann zeta function estimate is given and settled the problem negatively by evaluating the cardinality of the set of solutions of a suitable Diophantine equation of Ramanujan- Nagell-Lebesgue type. • [87]. The Riemann hypothesis is the conjecture concerning the zeta Riemann function ζ(s), given by B. Riemann (1892). The difficulty to prove this conjecture is related to the fact that ζ(s) has been formulated in a some cryptic way as complex continuation of hyperharmonic series and characterized by means of a functional equation that in a sense caches its properties about the identifications of zeros. Our approach to solve this conjecture has been to recast the zeta Riemann function ζ(s) to a quantum mapping between quantum-complex 1-spheres, i.e., working in the category Q of quantum manifolds as introduced by A. Prástaro.(See on this subject References [70, 81] and related works by the same author quoted therein.) More precisely the fundamental quantum algebra is just A = C, and quantum-complex manifolds are complex manifolds, where the quantum class of differentiability is the holomorphic class. In this way one can reinterpret all the theory on complex manifolds∪ as a theory on quantum-complex manifolds. In particular the Riemann sphere C {∞} can be identified with the quantum-complex 1-sphere Sˆ1, as considered in [70, 81]. The paper splits in two more sections. In Section 2 we resume some fundamental definitions and results about the Riemann zeta function ζ(s). In Section 3 the main result, i.e., the prof that the Riemann hypothesis is true, is contained in Theorem 3.1. This is made splitting the proof in some steps (lemmas). It is important to emphasize the central role played by Lemma 3.7. This focuses the attention on the completed Riemann zeta function, ζ˜(s), that symmetrizes the role between poles, with respect to the critical line of C, and between zeros, with respect to the x = ℜ(s)-axis. Finally the conclusion can be obtained by extending ζ˜(s) to a quantum-complex mapping ζˆ(s), between quantum-complex 1-spheres. CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 19

Then by utilizing the properties of meromorphic functions between compact Rie- mann spheres, identified with quantum-complex 1-spheres, we arrive to prove that all (non-trivial) zeros of ζ(s) must necessarily be on the critical line. In fact, the extension of ζ˜(s) to ζˆ(s), reduces zeros of this last meromorphic function to have have two simple zeros, symmetric with respect to the equator, and two simple poles, symmetric with respect to the critical line. For symmetry properties, this implies that also ζ˜(s) cannot have zeros outside the critical line, hence the same must happen for ζ(s) for non-trivial zeros. • [88]. Aim of this paper is to utilize previous Prástaro’sresults on quantum supergravity to prove that the geometric structure of quantum propagator encoding Universe at the Planck epoch is the cause of the Universe’s expansion.7 This expansion is not caused by a strange exoteric force, but it is the boundary-effect of the quantum nonlinear propagator encoding the Universe. In fact this propagator has a boundary with thermodynamic quantum exotic components. The presence of such exotic components produces an increasing of energy contents in the Universe, seen as transversal sections of such a quantum nonlinear propagator. To the increasing of energy corresponds an increasing in the expansion of the Universe. Such expansion has produced the passage of the Universe from the Higgs-universe, the first massive universe at the Planck epoch, to the actual macroscopic one. How- ever, yet in such a macroscopic age the expansion of the Universe can be justified by using the same philosophy. This will be illustrated by adopting the Einstein’s General Relativity equations, but taking into account the effect of its quantum origin (Planck-epoch-legacy). With this respect one can state that the so-called dark-energy-matter, is nothing else than the increasing in energy produced by the thermodynamic exotic boundary encoding the Universe. Therefore it is a pure geometrodynamic bordism effect that produces an expansion of our Universe also at the Einstein epoch. Paradoxically this is a consequence of the energy conservation law that continues to work whether at the Planck epoch or at the Einstein age. This Prástarotheory gives also a precise mathematical support to some early conjectures on the continuous creation of matter. (See, e.g., P. A. Dirac (1974), F. Hoyle (1949) and F. Hoyle and J. V. Narlikar (1963).

7Georges Lematre (1927) and Edwin Hubble (1929) ﬁrst proposed that the Universe is ex- panding. Lemaitre used Einstein’s General Relativity equations and Hubble estimated value of the rate of expansion by observed red-shifts. The most precise measurement of the rate of the Universe’s expansion, has been obtained by NASA’s Spitzer Space Telescope and published in October 2012. Very recently (March 17, 2014) some scientists of the Harvard-Smithsonian Center for Astrophysics, announced that observations with the telescope Bicep2 (Background Imaging of Cosmic Extragalactic Polarization), located at the South Pole, allowed to give an experimental proof of the Big Bang. (See the following link First Direct Evidence of Cosmic Inﬂation.) 20 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

Fig. 6. Florida Institute of Technology - Melbourne, FL, USA - 2005. CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 21

3. PUBLICATIONS

References

PAPERS

[1] L. Corgnier, A. D’Adda & A. Prástaro, Regge pole vs.resonance duality and boostrap calculations, Nuovo Cimento 57 A(4)(1968), 881–885. DOI: 10.1007/BF02751394. [2] A. Prástaro& P. Parrini, A mathematical model for spinning molten polymer and conditions of spinning, Tex. Res. J. 15(1975), 118–127. DOI: 10.1177/004051757504500206. [3] A. Prástaro& P. Parrini, Ein mathematisches Model fürdas Verspinnen geschmolzener Polymerer und fürdie Spinnbedingungen, Colloid & Polymer Science 255(2)(1977), 624– 633. DOI: 10.1007/BF01549886. [4] A. Prástaro, A mathematical model for spinning viscoelastic molten polymers, Riv. Mat. Univ. Parma 4(2)(1976), 295–313. Zbl 0375.73034. [5] A. Prástaro, Modello matematico sulla formazione della melt-fracture nei polimeri fusi, Quad. Ing. Chim. Ital. Suppl. 13(3-4)(1977), 37–44. (Chemical Abstracts: Collective In- dex 2000.) [6] A. Prástaro, Geometrodynamics of some non-relativistic incompressible fluids, Stochastica 3(2)(1979), 15–31. MR0556645(81b:76014); Zbl 0427.76003. [7] A. Prástaro, Spazi derivativi e fisica del continuo in relativitàgenerale, Atti Accad. Sci. Torino Suppl. 114(1980/81), 289–292. MR0670263(83h:58013). [8] A. Prástaro, On the general structure of continuum physics.I: Derivative spaces, Boll. Unione Mat. Ital. (5)17-B(1980), 704–726. MR0590551(81m:73012); Zbl 0438.58004. [9] A. Prástaro, On the general structure of continuum physics.II: Differential operators, Boll. Unione Mat. Ital. (5)S.-FM(1981), 69–106. MR0641760(83c:73002a); Zbl 0478.58004 [10] A. Prástaro,On the general structure of continuum physics.III: The physical picture, Boll. Unione Mat. Ital. (5)S.-FM(1981), 107–129. MR0641761(83c:73002b); Zbl 0478.58005. [11] A. Prástaro, On the intrinsic expression of Euler-Lagrange operator, Boll. Unione Mat. Ital. (5)18-A(1981), 411–416. MR0633674(842:58049); Zbl 0471.58012. [12] A. Prástaro, Dynamic conservation laws and the Korteweg-De Vries equation, Atti convegno su onde e stabilitànei mezzi continui, Catania 1981, Quaderni CNR-GNFM, Catania (1982), 272–274. [13] A. Prástaro, Spinor super bundles of geometric objects on spinG space-time structures, Boll. Unione Mat. Ital. (6)1-B(1982), 1015–1028. MR0683489(84c:53036); Zbl 0501.53023. [14] A. Prástaro, Gauge geometrodynamics, Riv. Nuovo Cimento 5(4) (1982), 1–122. DOI: 10.1007/BF02740593. MR0693882(84e:83045); Zbl 0695.58028. [15] A. Prástaro, Geometry and existence theorems for incompressible fluids, Geometro- dynamics Proceedings 1983, A. Prástaro (ed.), Pitagora Ed., Bologna (1984), 65–90. MR0823718(87g:58034). [16] A. Prástaro, Geometrodynamics of non-relativistic continuous media.I: Space- time structures, Rend. Sem. Mat. Univ. Politec. Torino40(2)(1982), 89–117. MR0724201(85e:53095); Zbl 0525.53036. [17] A. Prástaro, Geometrodynamics of non-relativistic continuous media.II: Dynamic and constitutive structures, Rend. Sem. Mat. Univ. Politec. Torino 43(1)(1985), 91–116. MR0859851(87m:53091); Zbl 0609.53042. [18] A. Prástaro, A geometric point of view for the quantization of non-linear field theories, Atti VI Convegno Nazionale di RelativitàGenerale e Fisica della Gravitazione, Firenze 1984, R. Fabbri and M. Modugno (eds.), Pitagora Ed., Bologna (1986), 289–292. [19] A. Prástaro, Dynamic conservation laws, Geometrodynamics Proceedings 1985, A. Prástaro (ed.), World Scientific Publishing, Singapore (1985), 283–420. MR0825801(87g:53109); Zbl 0645.58038. [About “Geometrodynamics” see also Wikipedia.] [20] A. Prástaro& T. Regge, The group structure of supergravity, Ann. Inst. H. PoincaréPhys. Théor. 44(1)(1986), 39–89. MR0834019(87i:83104); Zbl 0588.53066. [21] V. Marino & A. Prástaro, On the geometric generalization of the Noether theorem, Lecture Notes in Math. 1209, Springer-Verlag, Berlin (1986), 222–234. DOI: 10.1007/BFb0076634. MR0863759(88j:58142); Zbl 0603.53058. 22 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

[22] A. Prástaro, Quantum gravity and group model gauge theory, Journées Relativistes, Toulouse, France 1986, A. Crumeyrolle (ed.), Univ. Paul Sabatier Toulouse (1986), 213–222. [Recent results on the quantum geometrodynamics. Proceedings General Relativity and Grav- itation, vol.1. July, 4-9, 1983, Padova - Italy. (B. Bertotti, F. de Felice & A. Pascolini eds.) CNR - Roma (1983), 1145.] [23] V. Marino & A. Prástaro, On the conservation laws of PDE’s, Rep. Math. Phys. 26(2)(1987/8), 211–225. DOI: 10.1016/0034-4877(88)90024-9. MR0991720(91b:58097); Zbl 0695.58029. [24] A. Prástaro, On the quantization of Newton equation, Atti IX Congresso AIMETA, Bari 1988, AIMETA(1988), 13–16. [25] A. Prástaro, Wholly cohomological PDE’s, International Conference on Differential Geometry and Applications, Dubrovnick (Yu), 1988, Univ. Beograd & Univ. Novi Sad (1989), 305–314. MR1040078(91c:58151); Zbl 0695.58030. [26] A. Prástaro, Geometry of quantized PDE’s, Differential Geometry and Applications, J. Janyska & D. Krupka (eds.), World Scientific Publishing, Teanek, NJ, (1990), 392–404. MR1062046(91m:58175); Zbl 0796.35006. [27] A. Prástaro, On the singular solutions of PDE’s, Atti X Congresso Nazionale AIMETA, Pisa 1990, AIMETA(1990), 17–20. [28] A. Prástaro, Cobordism of PDE’s, Boll. Unione Mat. Ital. (7)5-B(1991), 977–1001. MR1146763(93a:57037); Zbl 0746.57015. [29] A. Prástaro, Quantum geometry of PDE’s, Rep. Math. Phys. 30(3) (1991), 273–354. DOI:10.1016/0034-4877(91)90063-S. MR1198655(94e:58150); Zbl 0771.58024. [30] A. Prástaro, Geometry of super PDE’s, Geometry of Partial Differential Equations, A. Prástaro& Th. M. Rassias (eds.), World Scientific Publishing, River Edge, NJ, (1994), 259– 315. MR1340222(96g:58025); Zbl 0879.58080. [31] V. Lychagin & A. Prástaro, Singularities for Cauchy data, characteristics, cocharacteristics and integral cobordism, Differential Geom. Appl. 4(3)(1994), 283–300. DOI: 10.1016/0926- 2245(94)00017-4. MR1299399(96b:58122); Zbl 0808.58039. [32] A. Prástaro& Th. M. Rassias, On a geometric approach to an equation of J. D’Alembert, Ge- ometry in Partial Differential Equations, A. Prástaro& Th. M. Rassias (eds.), World Scientific Publishing, River Edge, NJ, (1994), 316–322. MR1340223(96g:35131); Zbl 0879.35038. [33] A. Prástaro,Th. M. Rassias & J. Simˇsa,ˇ Geometry of the J. D’Alembert equation, in Finite Sums Decompositions in Mathematical Analysis, Th. M. Rassias & J. Simˇsa(eds.),ˇ J. Wiley (1995), 133–159. MR96k:26006; Zbl 0859.26005. [34] A. Prástaro & Th. M. Rassias, A geometric approach to an equation of J. D’Alembert, Proc. Amer. Math. Soc. 123(5)(1995), 1597–1606. DOI: 10.2307/2161153. MR1232143(95f:58007); Zbl 0839.58068. [35] A. Prástaro, Geometry of quantized super PDE’s, The Interplay Between Differential Geom- etry and Differential Equations, V. Lychagin (ed.), Amer. Math. Soc. Transl. 2/167(1995), 165–192. MR1343988(96d:58159); Zbl 0844.58012. [36] A. Prástaro, Quantum geometry of super PDE’s, Rep. Math. Phys. 37(1)(1996), 23–140. DOI: 10.1016/0034-4877(96)88921-X. MR1394861(97e:58235); Zbl 0887.58064. [37] A. Prástaro, (Co)bordism in PDEs and quantum PDEs, Rep. Math. Phys. 38(3)(1996), 443– 455. DOI: 10.1016/S0034-4877(97)84894-X. MR1437641(97m:58004); Zbl 0885.58094. [38] A. Prástaro, Quantum and integral (co)bordisms in partial differential equations, Acta Appl. Math. 51(3) (1998), 243–302. DOI: 10.1023/A:1005986024130. MR1437641(99d:58183); Zbl 0924.58103. [39] A. Prástaro, Quantum and integral bordism groups in the Navier-Stokes equation, New De- velopments in Differential Geometry, Budapest 1996, J. Szenthe (ed.), Kluwer Academic Publishers, Dordrecht (1998), 343–360. MR1670467(2000h:58065); Zbl 0937.35133. [40] A. Prástaro& Th. M. Rassias, On the set of solutions of the generalized d’Alembert equation, C. R. Acad. Sci. Paris 328(I-5)(1999), 389–394. DOI: 10.1016/S0764-4442(99)80177-3. MR1678135(2000a:3508); Zbl 0931.35031. [41] A. Prástaro& Th. M. Rassias, A geometric approach of the generalized d’Alembert equation, J. Comput. Appl. Math. 113(1-2)(2000), 93–122. DOI: 10.1016/S0377-0427(99)00247- 2. MR1735816(2001c:58079); Zbl 0936.35011. [42] A. Prástaro, (Co)bordism groups in PDEs, Acta Appl. Math. 59(2) (1999), 111–201. DOI: 10.1023/A:1006346916360. MR1741657(2001m:58046); Zbl 0949.35011. CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 23

[43] A. Prástaro, (Co)bordism groups in quantum PDEs, Acta Appl. Math. 64(2)(2000), 111–217. DOI: 10.1023/A:1010685903329. MR1826643(2002e:58037); Zbl 0978.58016. [44] A. Prástaro, Theorems of existence of local and global solutions of PDEs in the category of noncommutative quaternionic manifolds, Quaternionic Structures in Mathematics and Physics, S. Marchiafava, P. Piccinni & M. Pontecorvo (eds.), World Scientific Publishing, Singapore (2001), 329–337. MR1848873(2002f:58007); Zbl 0978.81038. [45] A. Prástaro, Local and global solutions of the Navier-Stokes equation, Steps in Differential Geometry, Proceedings of the Colloquium on Differential Geometry, 25–30 July, 2000, Debre- cen, Hungary, L. Kozma, P. T. Nagy & L. Tomassy (eds.), Univ. Debrecen (2001), 263–271. MR1859305(2002d:53008); Zbl 0983.35105. [46] A. Prástaro, Navier-Stokes equation: Global existence and uniqueness. (A geometric way to solve the “(NS)-problem”.), published as: Addendum I: Bordism Groups and the (NS)- Problem, in Quantized. Partial Differential Equations, World Scientific Publishing, Singapore, (2004), 377–434. [47] A. Prástaro& Th. M. Rassias, A geometric approach to a noncommutative generalized d’Alembert equation, C. R. Acad. Sc. Paris 330(I-7)(2000), 545–550. DOI: 10.1016/S0764- 4442(00)00238-X. MR1760436(2001d:58026); Zbl 0966.35105. [48] A. Prástaro& Th. M. Rassias, Results on the J. d’Alembert equation, Ann. Acad. Paed. Cracoviensis. Studia Math. 1(2001)117–128. Zbl 1137.58308. [49] A. Prástaro, Quantum manifolds and integral (co)bordism groups in quantum partial differential equations, Nonlinear Anal. Theory Methods Appl. 47/4(2001), 2609–2620. DOI: 10.1016/S0362-546X(01)00382-0. MR1972386(2004c:35343); Zbl 1042.35610. [50] A. Prástaro, Dirac quantization, Encyclopaedia Math. Suppl.III., M. Hazwinkel (ed.), Kluwer Academic Publishers, Dordrecht (2002), 127–129. DOI: 10.1007/978-0-306-48373-8. [51] A. Prástaro, Integral bordisms and Green kernels in PDEs, Cubo 4(2)(2002), 316–370. MR1928829(2003g:58056). [52] A. Prástaro& Th. M. Rassias, On the Ulam stability in geometry of PDE’s , Functional Equations Inequality and Applications, Th. M. Rassias (ed.), Kluwer Academic Publishers, Dordrecht (2003), 139–147. MR2042561(2004k:58031); Zbl 1059.39024. [53] A. Prástaro& Th. M. Rassias, Ulam stability in geometry of PDE’s, Nonlinear Funct. Anal. Appl. 8(2)(2003), 259–278. MR1994707(2004g:35179); Zbl 1096.39028. [54] A. Prástaro, Quantum super Yang-Mills equations: Global existence and mass-gap, Dynamic Syst. Appl. 4(2004), 227–232. (Eds. G. S. Ladde, N. G. Madhin and M. Sambandham), Dy- namic Publishers, Inc., Atlanta, USA. ISBN:1-890888-00-1. MR2117787(2005m:81203); Zbl 1067.81097. [55] A. Prástaro, Geometry of PDE’s.I: Integral bordism groups in PDE’s, J. Math. Anal. Appl. 319(2)(2006), 547–566. DOI: 10.1016/j.jmaa.2005.06.044. MR2227923(2007d:58031); Zbl 1100.35007. [56] A. Prástaro, Geometry of PDE’s.II: Variational PDE’s and integral bordism groups, J. Math. Anal. Appl. 321(2)(2006), 930–948. DOI: 10.1016/j.jmaa.2005.08.037. MR2241487(2007d:58032); Zbl 1160.58301. [57] A. Prástaro, Conservation laws in quantum super PDE’s, Proceedings of the Conference on Differential & Difference Equations and Applications (eds. R. P. Agarwal & K. Perera), Hindawi Publishing Corporation, New York (2006), 943–952. MR23049427(2008b:58041); Zbl 1131.35381. [58] A. Prástaro, (Co)bordism groups in quantum super PDE’s.I: Quantum supermanifolds, Non- linear Anal. Real World Appl. 8(2)(2007), 505–538. DOI: 10.1016/j.nonrwa.2005.12.008. MR2289563(2008j:58052); Zbl 1152.58313. [59] A. Prástaro, (Co)bordism groups in quantum super PDE’s.II: Quantum super PDE’s, Non- linear Anal. Real World Appl. 8(2)(2007), 480–504. DOI: 10.1016/j.nonrwa.2005.12.007. MR2289562(2008j:58053); Zbl 1152.58312. [60] A. Prástaro, (Co)bordism groups in quantum super PDE’s.III: Quantum super Yang- Mills equations, Nonlinear Anal. Real World Appl. 8(2)(2007), 447–479. DOI: 10.1016/j.nonrwa.2005.12.006. MR2289561(2009b:58082); Zbl 1152.58311. [61] R. Agarwal & A. Prástaro, Geometry of PDE’s.III(I): Webs on PDE’s and integral bordism groups. The general theory, Adv. Math. Sci. Appl. 17(1)(2007), 239–266. MR237378(2009j:58026); Zbl 1143.53017. 24 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

[62] R. Agarwal & A. Prástaro, Geometry of PDE’s.III(II): Webs on PDE’s and integral bordism groups. Applications to Riemannian geometry PDE’s, Adv. Math. Sci. Appl. 17(1)(2007), 267–285. MR2337379(2009j:58027); Zbl 1140.53005. [63] A. Prástaro, Geometry of PDE’s.IV: Navier-Stokes equation and integral bordism groups, J. Math. Anal. Appl. 338(2)(2008), 1140–1151. DOI:10.1016/j.jmaa.2007.06.009. MR2386488(2009j:58028); Zbl 1135.35064. [64] A. Prástaro, (Un)stability and bordism groups in PDE’s, Banach J. Math. Anal. 1(1)(2007), 139–147. MR2350203(2009e:58036); Zbl 1130.58014. [65] A. Prástaro, Extended crystal PDE’s stability.I: The general theory, Math. Comput. Modelling. (2008). DOI: 10.1016/j.mcm.2008.07.020. MR2532085(2011b:58041); Zbl 1171.35322. [66] A. Prástaro, Extended crystal PDE’s stability.II: The extended crystal MHD-PDE’s, Math. Comput. Modelling. (2008). DOI: 10.1016/j.mcm.2008.07.021. MR2532086(2011b:58042); Zbl 1171.35323. [67] A. Prástaro, On the extended crystal PDE’s stability.I: The n-d’Alembert extended crystal PDE’s, Appl. Math. Comput. 204(1)(2008), 63–69. DOI: 10.1016/j.amc.2008.05.141. MR2458340(2010h:58058); Zbl 1161.35054. [68] A. Prástaro, On the extended crystal PDE’s stability.II: Entropy-regular-solutions in MHD-PDE’s, Appl. Math. Comput. 204(1)(2008), 82–89. DOI: 10.1016/j.amc.2008.05.142. MR2458342(2010h:58059); Zbl 1161.35462. [69] A. Prástaro, On quantum black-hole solutions of quantum super Yang-Mills equations, Dynamic Syst. Appl. 5(2008), 407–414. (Eds. G. S. Ladde, N. G. Madhin C. Peng & M. Sambandham), Dynamic Publishers, Inc., Atlanta, USA. ISBN: 1-890888-01-6. MR2468173(2010g:83040). [70] A. Prástaro, Surgery and bordism groups in quantum partial differential equations.I: The quantum Poincaréconjecture, Nonlinear Anal. Theory Methods Appl. 71(12)(2009), 502– 525. DOI: 10.1016/j.na.2008.11.077. MR2671857(2012b:58057); Zbl 1238.58025. [71] A. Prástaro, Surgery and bordism groups in quantum partial differential equations.II: Varia- tional quantum PDE’s, Nonlinear Anal. Theory Methods Appl. 71(12)(2009), 526–549. DOI: 10.1016/j.na.2008.10.063. MR2671858(2012b:58058); Zbl 1238.58026. [72] R. P. Agarwal & A. Prástaro, Singular PDE’s geometry and boundary value problems, J. Non- linear Conv. Anal. 9(3)(2008), 417–460. MR2478974(2010b:58030); Zbl 1171.35006. [73] R. P. Agarwal & A. Prástaro, On singular PDE’s geometry and boundary value problems, Appl. Anal. 88(8)(2009), 1115–1131. DOI: 10.1080/00036810902943612. MR2568427(2010k:58033); Zbl 1180.35012. [74] A. Prástaro, Extended crystal PDE’s. Mathematics Without Boundaries: Surveys in Pure Mathematics. P. M. Pardalos and Th. M. Rassias (Eds.) Springer-Heidelberg New York Dor- drecht London, (to appear). arXiv:0811.3693[math.AT]. [75] A. Prástaro, Quantum extended crystal PDE’s, Nonlinear Studies 18(3)(2011), 447–485. arXiv:1105.0166[math.AT]. MR2012k:57043; Zbl 1253.35135. [76] A. Prástaro, Quantum extended crystal super PDE’s. Nonlinear Anal. Real World Appl. 13(6)(2012), 2491–2529. DOI: 10.1016/j.nonrwa.2012.02.014. arXiv:0906.1363[math.AT]. MR2927202; Zbl 1258.81064. [77] A. Prástaro, Exotic heat PDE’s, Commun. Math. Anal. 10(1)(2011), 64–81. arXiv:1006.4483[math.GT]. MR2825954; Zbl 06008771. [78] A. Prástaro, Exotic heat PDE’s.II. Essays in Mathematics and its Applications. In Honor of Stephen Smale’s 80th Birthday. P. M. Pardalos and Th. M. Rassias (Eds.) Springer-Heidelberg New York Dordrecht London (2012), 369–419. ISBN 978-3-642-28820-3 (Print) 978-3-28821-0 (Online). DOI: 10.1007/978-3-642-28821-0. arXiv: 1009.1176[math.AT]. MR2975595. [79] A. Prástaro, Exotic n-d’Alembert PDE’s and stability. Nonlinear Analysis. Stability, Approx- imation and Inequalities. Series: Springer Optimization and its Applications Vol 68. P. M. Pardalos, P. G. Georgiev and H. M. Srivastava (Eds.). Springer Optimization and its Ap- plications Volume 68 (2012), 571–586. ISBN 978-1-4614-3498-6. arXiv:1011.0081[math.AT]. Zbl 06073130. [80] A. Prástaro, Exotic PDE’s. Mathematics Without Boundaries: Surveys in Interdisciplinary Research. P. M. Pardalos and Th. M. Rassias (Eds.) Springer-Heidelberg New York Dordrecht London, (to appear). arXiv:1101.0283[math.AT]. CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 25

[81] A. Prástaro, Quantum exotic PDE’s. Nonlinear Anal. Real World Appl. 14(2)(2013), 893– 928. DOI: 10.1016/j.nonrwa.2012.04.001. arXiv:1106.0862[math.AT]. MR2991123. ; Zbl 06142846. [82] A. Prástaro, Strong reactions in quantum super PDE’s. I: Quantum hypercomplex exotic super PDE’s. arXiv:1205.2984[math.AT]. (Part I and Part II are unified in arXiv.) [83] A. Prástaro, Strong reactions in quantum super PDE’s. II: Nonlinear quantum propagators. arXiv:1205.2984[math.AT]. [84] A. Prástaro, Strong reactions in quantum super PDE’s. III: Exotic quantum supergravity. arXiv:1206.4956[math.AT]. [85] A. Prástaro, The Landau’s problems.I: The Goldbach’s conjecture proved. arXiv:1208.2473[math.GM]. [86] A. Prástaro, The Landau’s problems.II: Landau’s problems solved. arXiv:1208.2473[math.GM]. (Part I and Part II are unified in arXiv.) [87] A. Prástaro, The Riemann hypothesis proved. arXiv:1305.6845[math.GM]. [88] A. Prástaro, Quantum Geometrodynamic Cosmology. (Submitted for publication.)

MONOGRAPHS AND TEXTS

[89] A. Prástaro, Geometry of PDEs and Mechanics, World Scientific Publishing, River Edge, NJ, 1996, 760 pp. ISBN 9810225202. MR1412798(98e:55182); Zbl 0866.35007. [90] A. Prástaro, Elementi di Meccanica Razionale, Edizione 2010, Aracne Editrice, Roma, 2010, 446 pp. ISBN 978-88-548-3601-3. [91] A. Prástaro, Quantized Partial Differential Equations, World Scientific Publishing, River Edge, NJ, 2004, 500 pp. ISBN 981-238-764-1. MR2086084(2005f:58036); Zbl 1067.58022.

BOOKS (EDITOR AND COAUTHOR)

[92] A. Prástaro, Geometrodynamics Proceedings 1983 , Pitagora Ed., Bologna 1984. ISBN 88- 371-0286-0. MR0823711(86m:58007). [93] A. Prástaro, Geometrodynamics Proceedings 1985, World Scientific Publishing, Sin- gapore 1985. ISBN 9971-978-63-6. BookSG-Contents. MR0825784(86m:58008); Zbl 0637.00006. [94] A. Prástaro& Th. M. Rassias, Geometry in Partial Differential Equations, World Scientific Publishing, River Edge, NJ, 1994. ISBN 978-981-02-14-07-4. MR1340208(96a:58003); Zbl 0867.00017.

PATENTS

[95] A. Prástaroet al., 28/5/1975 - No 23.797 A/75. Process of production of fibrous structures with high degree of birefringence. [96] A. Prástaroet al., 11/7/1975 - No 25.334 A/75. Process of production of plexus-filament of synthetic polymers by means of flash-spinning of polymers solutions. 26 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

FIG. 7. Poster: ICM 2006 Satellite Conference “Advances PDE’s Geometry”, August 31 - September 2, 2006, Madrid - Spain. CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 27

4. ABSTRACTS OF PUBLICATIONS

ABSTRACTS OF PAPERS (1) L. Corgnier, A. D’Adda & A. Prástaro, Regge pole vs. resonance duality and boostrap calculations, Nuovo Cimento 57 A(4)(1968), 881–885. Abstract. In this note we use the Regge pole vs. resonance duality to make bootstrap calulations by comparing the behaviour at fixed scattering angle of Regge like amplitude with that of a sum of resonance contributions. (2) A. Prástaro& P. Parrini, A mathematical model for spinning molten polymer and conditions of spinning, Tex. Res. J. 45(1975), 118–127. Abstract. The equation of spinning of molten polymers in the stationary non-isothermal state have been solved by an analytical numerical method so as to obtain temperature T (s) and section A(z) profiles along the spinning axis z. T (z) and A(z) are thus correlated with the molecular parameters of the molten polymer: viscosity, density, and extrusion temperature, and polymer mass-flow rate. Furthermore, the correlation has been obtained between fiber quality and steady-state solutions. A critical collection rate and the critical extrusion output have been deduced from such correlations. Above such critical values, breakage of the molten polymer takes place. (3) A. Prástaro& P. Parrini, Ein mathematisches Model fürdas Verspinnen geschmolzener Polymerer und fürdie Spinnbedingungen, Colloid & Polymer Science 255(2)(1977), 624–633. Abstract. This is a translated German version of the previous article. (4) A. Prástaro, A mathematical model for spinning viscoelastic molten polymers, Riv. Mat. Univ. Parma (4)(1976), 295–313. Abstract. A mathematical model for spinning viscoelastic materials is proposed. This work can be considered the continuation of the papers [2, 3] which treated the case of newtonian materials. The viscoelastic system, as more differs from the newtonian as the elastic component is present; thus the viscoelastic mathematical model can be not be inferred from the analysis of the previous paper; on the contrary the viscoelastic model in- cludes, as particular case, the newtonian model. The spinning process was analysed by adding the rheological equation for viscoelastic materials to the set of simultaneous partial differential equations describing a general molten spinning process. We gave the steady-state numerical solutions, i.e. the filament croos-section A(z), filament temperature T (z) and filament tension F (z), as function of position z and we related them to the parameters which influence the process of spinning: material parameters and spinning conditions parameters. We related the yarn birefringence ∆n to the same parameters also. Moreover, we proposed to investigate which bounds impose the spinnability criterion on the viscoelastic paramaters and which conditions realize the maximum yarn production with fixed denier and section. (5) A. Prástaro, Modello matematico sulla formazione della melt-fracture nei polimeri fusi, Quad. Ing. Chim. Ital. Suppl. 13(3-4)(1977), 37–44. 28 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

Abstract. The melt-fracture is one of the most serious problem in the extrusion of thermoplastic materials. This is a flow instability that appears as an almost regular distorsion of the extruded material. After a panorama on the known experimental facts, a mathematical model is given in such a way to emphasize some essential parameters controlling the start of this phenomenon. In fact has been decovered an adimensional number, Wc, (Weissenberg number), characterizing the start of flow instability in all melt polymers. This allows us to determine the extrudibility characteristics of polymeric materials. (6) A. Prástaro, Geometrodynamics of some non-relativistic incompressible fluids, Stochastica 3(2)(1979), 15–31. Abstract. In some papers [7, 8, 9, 10] we proposed a geometric formulation of continuum mechanics, where a continuum body is seen as a suitable differentiable fiber bundle C on the Galilean space-time M, beside a differential equation of order k, Ek(C), on C and the assignement of a frame ψ on M. In the present paper we apply this general theory to some incompressible fluids. The scope is to demonstrate that also for these more simple materials our theory is a suitable tool in order to understand better the fundamental principles of continuum mechanics. (7) A. Prástaro, Spazi derivativi e fisica del continuo in relativitàgenerale, Atti Accad. Sci. Torino Suppl. 114(1980/81), 289–292. Abstract. In order to give an axiomatic description of continuum physics, a derivative space, D k(V,W ), is introduced which allows us to describe the derivative of order k of a differentiable map f : V → W as section of the fiber bundle D k(V,W ). Derivative operators and functional differential operators are seen as useful generalizations of usual differential operators. With this language we recognize a structural order to all physical entities which are characteristic in continuum physics. (8) A. Prástaro, On the general structure of continuum physics.I: Derivative spaces, Boll. Unione Mat. Ital. (5)17-B(1980), 704–726. Abstract. In order to give an intrinsic and axiomatic formulation of continuum physics, the derivative space D k(V,W ) is introduced. This allows us to describe the k-order derivative of a mapping f : V → W , as a section of the fibre bundle D k(V,W ) → V . This formulation generalizes the concept of jet of a mapping. The corresponding differential calculus is carefully developed. (9) A. Prástaro, On the general structure of continuum physics.II: Differential operators, Boll. Unione Mat. Ital. (5)S.-FM(1981), 69–106. Abstract. In order to give an intrinsic and axiomatic formulation of continuum physics, the differential operators are studied in the language of derivative spaces [7, 8]. Useful generalizations of differential operators are given by introducing derivative operators and functional differential operators. Finally differential equations are considered as suitable subspaces of derivative spaces. (10) A. Prástaro, On the general structure of continuum physics.III: The physical picture, Boll. Unione Mat. Ital. (5)S.-FM(1981), 107–129. Abstract. An axiomatic and intrinsic formulation of continuum systems is given by using tools of the modern differential geometry. In this framework CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 29

we are able to give a structural ordering to all the physical entities of continuum physics. (11) A. Prástaro, On the intrinsic expression of Euler-Lagrange operator, Boll. Unione Mat. Ital. (5)18-A(1981), 411–416. Abstract. An intrinsic representation of the Euler-Lagrange differential operator is given for the global variational calculus on fibered manifolds. This expression is of particular interest to be utilized in the Lagrangian formulation of the field theory, as it gives an explicit derivative dependence by the field. (12) A. Prástaro, Dynamic conservation laws and the Korteweg-De Vries equation, Atti convegno su onde e stabilitànei mezzi continui, Catania 1981, Quaderni CNR-GNFM, Catania,(1982), 272–274. Abstract. Recently we have introduced a geometric method to describe conservation laws more general than Noetherian ones. In particular in [16, 17, 19] we have given a detailed exposition of this method developing the geometric-differential calculus to build dynamic conserved quantities. In this communication we shall give an account of some of these results emphasizing their applications to the Korteweg-de Vries equation. (13) A. Prástaro, Spinor super bundles of geometric objects on spinG space-time structures, Boll. Unione Mat. Ital. (6)1-B(1982), 1015–1028. Abstract. Spinorial fibre bundles are built on space-times manifolds of type spinG that are fully covariant in the sense of [8]. These fibre bundles generalize that introduced in [8] and are useful in a unified field theory. (14) A. Prástaro, Gauge geometrodynamics, Riv. Nuovo Cimento 5(4)(1982), 1–122. Abstract. In this paper we have a self-contained unitary geometric devel- opment of the methods and structures on which the gauge theories are based. We hope that this geometrical framework shall be useful for a more clear understanding of continuum physics. Since the framework is sufficiently generalized, it can be applied to all the situations of physical interest. Contents: Functors and fibre bundles. Derivative spaces and differential equations. Derivative spaces and variational calculus. Connections and derivative spaces. Geometrodynamics of gauge continuum systems and symmetries properties. Classification of gauge continuum systems. Spinor superbundles of geometric objects and dynamics. (15) A. Prástaro, Geometry and existence theorems for incompressible fluids, Ge- ometrodynamics Proceedings 1983, A. Prástaro(ed.), Pitagora Ed., Bologna (1984), 65–90. Abstract. By utilizing a geometric framework to describe the dynamic of a continuous medium [7, 8, 9, 10, 11, 12, 13, 14, 15] we give existence theorems of local and global solutions for an incompressible fluid. The discussion concerns the Euler equation (E) and the non-isothermic Navier- Stokes equation (NS). (16) A. Prástaro, Geometrodynamics of non-relativistic continuous media.I: Space- time structures, Rend. Sem. Mat. Univ. Politec. Torino 40(2)(1982), 89–117. Abstract. In order to formulate the non-relativistic continuum mechanics as a unified field theory on Galilei space-time M, the geometrical structure 30 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

of M is considered and the space time resolution of bundles of geometric objects on M are analysed in detail. In particular, the concept of geometric object gives rigorous meaning to the concept of observed physical quantity. It clarifies the ambiguity of why “frame dependent” quantities are useful, even essential, in the kinematic of description of continuum mechanical bodies. Moreover, it clarifies the paradosical nature of “frame indifferent statements about frame dpendent quantities”. These turn out to be simply statements about fields of geometric objects which are not tensor fields. (17) A. Prástaro, Geometrodynamics of non-relativistic continuous media.II: Dynamic and constitutive structures, Rend. Sem. Mat. Univ. Politec. Torino 43(1)(1985), 89–116. Abstract. An intrinsic formulation of Continuum Mechanics on the affine Galielan space-time M is given, emphasizing the role of the dynamic equation as a geometric structure. In particular, a continuous body is described as a geometric structure on M. Thus, the study of symnmetry properties of this structure allows us to give useful classifications of continuous bodies and to state generalized forms of Noether’s theorem. These considerations are applied to incompressible fluids. Existence and uniqueness theorems for regular solutions are obtained. (18) A. Prástaro, A geometric point of view for the quantization of non-linear field theories, Atti VI Convegno Nazionale di RelativitàGenerale e Fisica della Gravitazione, Firenze 1984, Pitagora Ed., Bologna (1986), 289–292. Abstract. The fundamental geometric structure of any field theory is a k fiber bundle π : W → M beside a PDE Ek ⊂ JD (W ). So we shall recognize in this geometric structure (W, Ek) suitable properties to interpretate the meaning of quantization. This communication shortly describe our new point of view in this field, showing how it is possible to read the meaning of the quantization in the formal properties of PDEs. (19) A. Prástaro, Dynamic conservation laws, Geometrodynamics Proceedings 1985, A. Prástaro(ed.), World Scientific Publishing, Singapore (1985), 283– 420. Abstract. Part A:PDE and Conservation Laws. Basic results on the formal theory of PDE. Pseudogroups and PDE. The Euler-Lagrange operator for Lagrangians of any order. Cartan form and Noether conservation laws for Lagrangian of any order. Symmetry properties and dynamic conservation laws. Part B: Quantized PDE and Conservation Laws. Quantum charges. Quantum situs. Geometric theory of quantized PDE. Quantum cobordism and Dirac’s approach to quantization. Applications: Klein- Gordon equation; Maxwell equation; Dirac equation; Einstein equation; Yang-Mills equation. Appendix. Topological vector spaces, C∗-algebras and spectral theory. Local characterization of some geometric structures related to PDE. (20) A. Prástaro& T. Regge, The group structure of supergravity, Ann. Inst. H. PoincaréPhys. Théor. 44(1)(1986), 39–89. Abstract. An intrinsic description of the ”group manifold approach” to supergravity is given. Emphasis is placed on some geometric structures which allow us to obtain a direct full covariant formulation. In particular, CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 31

the geometric theory of partial differential equations allows us to give a dynamic description of space-time. Some applications to physically interesting situations are discussed in detail. (21) V. Marino & A. Prástaro, On the geometric generalization of the Noether theorem, Lecture Notes in Math. 1209, Springer-Verlag, Berlin (1986), 222–234. Abstract. Task of this paper is to compare some geometric approaches to obtain conservation laws associated to partial differential equations. More precisely we intend to consider the methods developed by A. Prástaroand A. M. Vinogradov. The differential equations are considered from a geometric point of view: namely they are submanifolds of jet-derivative spaces on fiber bundles. To the symmetries of these submanifolds are associated conservation laws that are not necessarily of Noetherian type. (22) A. Prástaro, Quantum gravity and group model gauge theory, JournéesRel- ativistes, Toulouse, France 1986, A. Crumeyrolle (ed.), Univ. Paul Sabatier Toulouse (1986), 213–222. [Recent results on the quantum geometrodynamics. Proceedings General Relativity and Gravitation, vol.1. July, 4-9, 1983, Padova - Italy. (B. Bertotti, F. de Felice & A. Pascolini eds.) CNR - Roma (1983), 1145.] Abstract. A new geometric point of view of quantization of PDE’s, founded on the formal properties of PDE’s, is applied to the quantization of gravity coupled with a Yang-Mills gauge field. The quantization of Einstein equation in the framework of group model gauge theory is discussed. (23) V. Marino & A. Prástaro, On the conservation laws of PDE’s, Rep. Math. Phys. 26(2)(1987/8), 211–225. Abstract. The general methods of obtaining conservation laws for (nonlinear) partial differential equations (PDE’s) introduced by A. Prástaroin [12, 19] and by A. M. Vinogradov in Soviet Math. Dokl. (5)18(1977), 1200– 1204; J. Math. Anal. Appl. (1)100(1984), 1–129, are considered and the general covariance of such methods is studied. In particular, it is shown that Vinogradov’s method fails to be fully covariant in the non-linear case. The relations between the number of conservation laws for PDEs and the Atiyah- Singer index theorem are studied. A criterion for recognize the wholly cohomological character of a PDE is given and the link between spectral sequences and wholly cohomlogical equations is found. Some examples of interesting PDEs which arise in physics are also considered. (24) A. Prástaro, On the quantization of Newton equation, Atti IX Congresso AIMETA, Bari 1988, AIMETA (1988), 13–16. Abstract. It is proved that the method of formal quantization of PDEs introduced by the author in some previous papers [18, 25, 28] allows to obtain in intrinsic way the canonical Dirac quantization for particles of the lassical mechanics. As examples harmonic oscillator and anharmonic oscillator are considered. These considerations can be generalized to any PDE defined on fiber bundles. (25) A. Prástaro, Wholly cohomological PDE’s, International Conference on Dif- ferential Geometry and Applications, Dubrovnick (Yu), 1988, Univ. Beograd & Univ. Novi Sad (1989), 305–314. 32 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

Abstract. PDE’s are geometric objects to which one can associate conservation laws in relation to their symmetry properties [15, 17, 19, 21, 23]. Then, the wholly-cohomological character of a PDE is its possibility to represent any (n − 1)-dimensional cohomological class of the n-dimensional basis manifold by means of a conservation law. In this paper we resume some recent results in this direction obtained by the author [29] and also announce some new further results for PDEs defined in the category of supermanifolds [36]. (26) A. Prástaro, Geometry of quantized PDE’s, Differential Geometry and Ap- plications, J. Janyska & D. Krupka (eds.), World Scientific Publishing, Singapore (1990), 392–404. Abstract. In this paper we resume some recent results in the direction of the formal quantization of PDE’s obtained by the author [24], and also announce some new further results. The categorial meaning of quantization of PDE’s is given. Formal quantization results a canonical functor defined on the category of differential equations. Furthermore, a Dirac-quantization can be interpreted as a covering in the category of differential equations. A quantum (pre-)spectral measure is a functor that can be factorized by means of formal quantization and a (pre-)spectral measure. A relation between canonical Dirac-quantization and singular solutions of PDE’s is given. It is proved also that knowledge of Bäcklund correspondeces, as well conservation laws, can aid the proceeding of canonical quantization of PDE’s. (See [29].) (27) A. Prástaro, On the singular solutions of PDE’s, Atti X Congresso Nazionale AIMETA, Pisa 1990, AIMETA (1990), 17–20. Abstract. In the geometric formal theory of PDE’s we recognize also the problem of existence of singular solutions with singularities of Thom- Boardman type, i.e., singularities that can be resolved by means of prolon- gations. Scope of this paper is to give a short account of some fundamental results in these directions and apply them to some important classic equations of fluid mechanics: Euler equation (E) and Navier-Stokes equation (NS). Quantum tunneling effects can be described by means of such singular solutions. Furthermore, we show also as singular solutions enter in the description of canonical quantization of PDE’s. We shall specialize, for sake of coincision, on equations (E) and (NS). (See [29].) (28) A. Prástaro, Cobordism of PDE’s, Boll. Unione Mat. Ital. (7)5-B(1991), 977–1001. Abstract. Cobordism groups of systems of partial differential equations of any order are considered and their representations by means of suitable homology groups are given. This approach generalizes previous one by J. Eliashberg, given for PDE’s of first order, These results are useful also in order to give an algebraic-topological characterization of the quantum situs or its classic limit, for PDE’s of any order. (See also [19, 70].) (29) A. Prástaro, Quantum geometry of PDE’s, Rep. Math. Phys. 30(3)(1991), 273–354. Abstract. In this paper we present the formal quantization of PDE’s [18, 19, 22, 24, 26, 27] in categorial language. Formal quantization results as a canonical functor defined on the category of differential equations. CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 33

Furthermore, a Dirac-quantization can be interpreted as a covering in the category of differential equations. A quantum (pre-)spectral measure is a functor that can be factorized by means of formal quantization and a (pre- )spectral measure. A relation between canonical Dirac-quantization and singular solutions of PDE’s is given. It is also proved that the knowledge of Bäklundcorrespondences, as well as the conservation laws, can aid the procedure of canonical quantization of PDE’s. Physically interesting examples are considered. In particular, we give the canonical quantization of an anharmonic oscillator. A general theory of quantum tunneling effects in PDE’s is given. In particular, quantum cobordism has been related with Leray-Serre spectral sequences of PDE’s. (30) A. Prástaro, Geometry of super PDE’s, Geometry of Partial Differential Equations, A. Prástaro& Th. M. Rassias (eds.), World Scientific Publish- ing, River Edge, NJ, (1994), 259–315. Abstract. Superspaces and supermanifolds are introduced by using the concept of weak differentiability as usually given for locally convex spaces. This allows us to consider in algebraic way superdual spaces and superderivative spaces. In this way we obtain a good generalization of just known super- structures general enough to develop a formal theory for super PDE’s that directly extends previous ones for standard manifolds of finite dimension. In particular, we give a Goldschmidt-type criterion of formal superintegrability for super PDE’s, and show that a geometric theory of singular super- solutions, with singularities of Thom-Boardman type, can be formulated in the framework of super PDE’s too. Conservation superlaws associated to super PDE’s are considered and related with some spectral sequences and wholly cohomological character of these equations. (31) V. Lychagin & A. Prástaro, Singularities for Cauchy data, characteristics, cocharacteristics and integral cobordism, Differential Geom. Appl. 4(3)(1994), 283–300. Abstract. A generalization of the classical concept of characteristic for partial differential equations (PDE) is given in the framework of the geometric formal theory for PDE’s. In particular, it is given a relation between singularities of Cauchy data and characteristics in order to obtain integral manifolds (solutions) generated by means of characteristics. In this direction it is shown that to any PDE we can associate a ”dual” equation having the same characteristics. These equations can be related by means of a sort of Bäcklund transformation. Furthermore, a criterion that relates characteristics and integral cobordism (or quantum cobordism [19, 29]) is given. Also a relation between quantum cobordism in non-linear PDE’s and Green’s functions is given. (32) A. Prástaro& Th. M. Rassias, On a geometric approach to an equation of J.D’Alembert, Geometry in Partial Differential Equations, A. Prástaro & Th. M. Rassias (eds.), World Scientific Publishing, Singapore (1994), 316–322. Abstract. Here we announce some firt results on the J. D’Alembert equa- ∂2 tion ( ∂x∂y log f) = 0. More precisely, by using a geometric framework we prove that the set of smooth functions of two variables f(x, y), solutions of 34 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

the J. D’Alembert equation, is larger than the set of functions of the form f(x, y) = h(x).g(y). (33) A. Prástaro,Th. M. Rassias & J. Simˇsa,ˇ Geometry of the J. D’Alembert equation, in Finite Sums Decompositions in Mathematical Analysis, Th. M. Rassias & J. Simˇsa(eds.),ˇ J. Wiley (1995), 133–159. Abstract. This is the last chapter of a book devoted to a very interesting and actual problem in Mathematical Analysis. Here the geometric theory of PDE’s is considered and applied to the d’Alembert equation in its connec- tion with the problem of representation of functions by (partial) separation of variables. (34) A. Prástaro& Th. M. Rassias, A geometric approach to an equation of J.D’Alembert, Proc. Amer. Math. Soc. 123(5)(1995), 1597–1606. Abstract. By using a geometric framework of PDE’s we prove that the ∗ ∂2 lg f set of solutions of the D’Alembert equation ( ) ∂x∂y = 0 is larger than the set of smooth functions of two variables f(x, y) of the form (∗∗) f(x, y) = h(x).g(y). This agrees with a previous counterexample by Th. M. Rassias given to a statement by C. M. Stéphanos. More precisely, we have the following result: The set of 2-dimensional integral manifolds of PDE (∗) properly contains the ones representable by graphs of 2-jet-derivatives of functions f(x, y) expressed in the form (∗∗). A generalization of this result to functions of more than two variables is sketched also by considering the n equation ∂ log f = 0. ∂x1...∂xn (35) A. Prástaro, Geometry of quantized super PDE’s, The Interplay Between Differential Geometry and Differential Equations, V. Lychagin (ed.), Amer. Math. Soc. Transl. 2/167(1995), 165–192. Abstract. In this paper we announce some results on the geometrization of super PDE’s, i.e., PDE’s defined in the category of supermanifolds. These results generalize previous ones for PDE’s [29]. (36) A. Prástaro, Quantum geometry of super PDE’s, Rep. Math. Phys. 37(1)(1996), 23–140. Abstract. In order to extend to super PDEs the theory of quqntization of PDEs as contained in [29] we first develop a geometric theory for super PDEs (see also [30, 35]). Superspaces and supermanifolds are introduced by using the concept of weak differentiability as usually given for locally convex spaces. This allows us to consider in algebraic way superdual spaces and superderivative spaces and to develop a formal theory for super PDEs that directly extends the previous ones for standard manifolds of finite dimension. In particular, we give a criterion of formal superintegrability for super PDEs, and show that a geometric theory of singular supersolu- tions, with singularities of Thom–Boardman type, can be formulated in the framework of super PDEs too. These results generalize the previous ones obtained for ordinary manifolds by H.Goldschmidt and by Moscow’s mathematical school. Conservation superlaws associated to super PDEs are considered and related with some spectral sequences and wholly cohomological character of these equations. Then, the quantization of super PDEs is formulated on the ground of quantum cobordism [25, 39]. This is made in order to give an intrinsic and fully covariant geometric formulation CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 35

of unified quantum field theory. In particular, a theory of quantum supergravity is developed. We explain how canonical quantization and quantum tunneling effects arise in super PDEs. Furthermore, we explicitly extend previous results of Witten and Atiyah in topological quantum field theory to our geometric framework for super PDEs. Obstructions to existence of quantum cobords in super PDEs are given by means of supercharacteristic classes. These results can be considered as a generalization of the recent results obtained by Gibbons and Hawking. (37) A. Prástaro, (Co)bordism in PDEs and quantum PDEs, Rep. Math. Phys. 38(3)(1996), 443–455. Abstract. In this paper we announce some recent results on the quantum and integral (co)bordism in PDEs and quantum PDEs. We shall essentially prove that the tecnique of (co)bordism, introduced by Pontrjagin and Thom in algebraic topology, can be generalized in the framework of partial differential equations in order to obtain sufficient criteria that allow to decide when a p-dimensional compact closed integral manifold contained in a PDE ⊂ k Ek Jn (W ), is the boundary of a (p + 1)-dimensional integral compact manifold contained also in Ek (integral bordism) or eventually in the jet- k space Jn (W ) containing Ek (quantum bordism). Furthermore, we shall prove that such results can be extended to the category of quantum PDEs. Here, by the term “quantum manifold” (and as a consequence of “quantum PDEs”) we mean a new structure that extends globally usual concepts of quantum spaces, and that is very useful for physical applications. (38) A. Prástaro, Quantum and integral (co)bordisms in partial differential equations, Acta Appl. Math. 51(3)(1998), 243–302. Abstract. Characterizations of quantum bordisms and integral bordisms in PDEs by means of subgroups of usual bordism groups are given. More precisely, it is proved that integral bordism groups can be expressed as extensions of quantum bordism groups and these last are extensions of subgroups of usual bordism groups. Furthermore, a complete cohomological characterization of integral bordism and quantum bordism is given. Appli- cations to particular important classes of PDEs are considered. Finally, we give a complete characterization of integral and quantum singular bordisms by means of some suitable characteristic numbers. Some examples of interesting PDEs which arise in Physics are also considered where existence of solutions with change of sectional topology (tunnel effect) is proved. As an 0,n−1 application, we relate integral bordism to the spectral term E1 , that represents the space of conservation laws for PDEs. This gives, also, a general method to associate in a natural way a Hopf algebra to any PDE. (39) A. Prástaro, Quantum and integral bordism groups in the Navier-Stokes equation, New Developments in Differential Geometry, Budapest 1996, J. Szenthe (ed.), Kluwer Academic Publishers, Dordrecht (1998), 343-360. Abstract. In this paper we announce some results concerning theorems of existence and classification of solutions of the Navier-Stokes equation (NS). In particular, following our general theory for bordism groups in PDEs, introduced in [37, 38, 70], the quantum and integral bordism groups of (NS) are explicitly calculated. (See also [42, 45, 46, 63, 65].) 36 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

(40) A. Prástaro& Th.M.Rassias, On the set of solutions of the generalized d’Alembert equation, C. R. Acad. Sci. Paris 328(I-5)(1999), 389–394. Abstract. By using a geometric approach we prove that the set of so- n lutions of the generalized d’Alembert equation ∂ log f/∂x1 ··· ∂xn = 0, considered in the domain of the (x1, ··· , xn)-space Rn, is larger that the set of the functions that can be represented in the form as f(x1, ··· , xn) = 2 n 1 n−1 f1(x , ··· , x ) ··· fn(x , ··· , x ). Here the recent general method introduced by A. Prástaroto calculate integral and quantum (co)bordism groups in PDE’s [37, 38, 70] is used. This method is very useful in order to prove existence of tunneling effects in PDE’s, i.e., existence of solutions that change their sectional topology. (41) A. Prástaro& Th. M. Rassias, A geometric approach of the generalized d’Alembert equation, J. Comput. Appl. Math. 113(1-2)(2000), 93–122. ′ Abstract. The following results are obtained: 1) The set Sol(d A)n of all n solutions of the equation ∂ log f = 0,(n-d’Alembert equation), (n ≥ 2), ∂x1...∂x1 considered in domains of the (x1, . . . , xn) ∈ Rn, is larger than the set of all functions f that can be represented in the form f(x1, . . . , xn) = 2 n 1 n−1 ′ f1(x , . . . , x ) . . . fn(x , . . . , x ). 2) In the set of solutions Sol(d A)n ′ n n of the n-d’Alembert equation, (d A)n ⊂ JD (R , R), there are also some manifolds that have a change of sectional topology (tunneling effect). (42) A. Prástaro, (Co)bordism groups in PDEs, Acta Appl. Math. 59(2)(1999), 111–201. Abstract. We introduce a geometric theory of PDEs, by obtaining existence theorems of smooth and singular solutions. In this framework, following our previous results on (co)bordisms in PDEs [37, 38, 70] we give characterizations of quantum and integral (co)bordism groups and relate them to the formal integrability of PDEs. An explicitly proof that the usual Thom-Pontrjagin construction in (co)bordism theory can be generalized also to the case of singular integral (co)bordism in the category of differential equations is given. In fact, we prove the existence of a spectrum that characterizes the singular integral (co)bordism groups in PDEs. More- over, a general method that associates in a natural way Hopf algebras (full ≤ ≤ − ⊂ k p-Hopf algebras, 0 p n 1), to any PDE Ek Jn (W ), just introduced in [38, 70], is further studied. Applications to particular important classes of PDEs are considered. In particular, we carefully consider the Navier- Stokes equation (N) and explicitly calculate their quantum and integral bordism groups. An existence theorem of solutions of (NS) with change of sectional topology is obtained. Relations between integral bordism groups and causal integral manifolds, causal tunnel effects, and the full p-Hopf algebras, 0 ≤ p ≤ 3, for the Navier-Stokes equation are determined. (43) A. Prástaro, (Co)bordism groups in quantum PDEs, Acta Appl. Math. 64(2)(2000), 111–217. Abstract. In this paper we formulate a theory of noncommutative manifolds (quantum manifolds) and for such manifolds we develop a geometric theory of quantum PDEs (QPDEs). In particular, a criterion of formal integrability is given that extends to QPDEs previous one given by H. Goldschmidt for PDEs [50] and by us for super PDEs [30, 35, 36]. Quan- tum manifolds are seen as locally convex manifolds where the model has CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 37

m1 × · · · × ms ≡ × · · · × the structure A1 As , with A A1 As a noncommutative algebra that satisfies some particular axioms (quantum algebras). A general theory of integral (co)bordism for QPDEs is developed, that extends our previous for PDEs [37, 38, 42, 70]. Then, non-commutative Hopf algebras, (full quantum p-Hopf algebras, 0 ≤ p ≤ m−1), are canonically associated to ˆ ⊂ ˆk any QPDE Ek Jm(W ), whose elements represent all the possible invari- ants that can be recognized for such a structure. Many examples of QPDEs are considered where we apply our theory. In particular, we carefully study QPDEs for supergravity. We show that the corresponding regular solutions, observed by means of quantum relativistic frames, give curvature, torsion, gravitino and electromagnetic fields as A-valued distributions on space-time, where A is a quantum algebra. For such equations canonical quantizations are obtained and the quantum and integral bordism groups and the full quantum p-Hopf algebras, 0 ≤ p ≤ 3, are explicitly calculated. Then, the existence of (quantum) tunnel effects for quantum superstrings in supergravity is proved. (44) A. Prástaro, Theorems of existence of local and global solutions of PDEs in the category of noncommutative quaternionic manifolds, Quaternionic Structures in Mathematics and Physics, Rome 1999, S. Marchiafava, P. Piccinni & M. Pontecorvo (eds.), World Scientific Publishing, Singapore (2001), 329–337. Abstract. In this paper we apply our recent geometric theory of non- commuttive (quantum) manifolds and noncommutative (quantum) PDEs [37, 38, 42, 70, 43] to the category of quantum quaternionic manifolds. These are manifolds modelled on spaces built starting from quaternionic algebras. For PDEs considered in such category we determine theorems of existence of local and global quaternionic solutions. We show also that such a category of quantum quaternionic manifolds properly contains that of manifolds with (almost) quaternionic structure. So our theorems of existence of quantum quaternionic manifolds for PDEs produce a cascade of new solutions with nontrivial topology. (45) A. Prástaro, Local and global solutions of the Navier-Stokes equation, Steps in Differential Geometry, Proceedings of the Colloquium on Differential Geometry, 25–30 July, 2000, Debrecen, Hungary, L. Kozma, P. T. Nagy & L. Tomassy (eds.), Univ. Debrecen (2001), 263–271. Abstract. A brief report is given on our recent results [42, 46] proving existence of (smooth) global solutions of the 3D nonisothermal Navier- Stokes equation (NS), and (non) uniqueness of such solutions for (smooth) boundary value problems. (46) A. Prástaro, Navier-Stokes equation. Global existence and uniqueness, (A geometric way to solve the “(NS)-problem”.), published as: Addendum I: Bordism Groups and the (NS)-Problem, in Quantized. Partial Differential Equations, World Scientific Publishing, Singapore, (2004), 377–434. Abstract. Here we report on our recent results on the integral bordism groups of the 3D nonisothermal Navier-Stokes equation (NS) [39, 42, 45] that proved the existence of global (smooth) solutions. In particular, we go in some further results emphasizing surgery techniques that allow us to better understand this geometric proof of existence of (smooth) global 38 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

solutions for any (smooth) boundary condition. Furthermore, a theorem of nonuniqueness of such solutions for general boundary conditions is given on the ground of the symmetry properties of (NS) and just by using our results on the integral bordism groups of (NS). A comparison with the isothermal case, in the zero viscosity limit condition, (Euler equation), is considered. (See also [63].) (47) A. Prástaro& Th. M. Rassias, A geometric approach to a noncommutative generalized d’Alembert equation, C. R. Acad. Sc. Paris 330(I-7)(2000), 545–550. Abstract. In this paper the authors provide an account of some of their results concerning the J. D’Alembert equation especially in a suitable category of noncommutative manifolds, proving that the geometric theory of PDE’s introduced by A. Prástarois an handable framework where problems in the theory of partial differential equations find their natural solutions. In fact, the J. d’Alembert equation is one such applications. (48) A. Prástaro& Th. M. Rassias, Results on the J.d’Alembert equation, Ann. Acad. Paed. Cracoviensis. Studia. Math. 1(2001), 117–128. Abstract. A new m-d’Alembert equation, m ≥ 2, is introduced in the category of quantum manifolds [37, 43, 70], that extends the commutative generalized d’Alembert equation just considered in [47]. For such a new equation we give theorems of existence of local and global solutions. (49) A. Prástaro, Quantum manifolds and integral (co)bordism groups in quantum partial differential equations, Nonlinear Anal. Theory Methods Appl. 47/4(2001), 2609–2620. Abstract. In this paper it is given a short account of some recent theorems by A. Prástaroon the existence of local and global solutions of QPDEs, i.e. partial differential equations built in the category of quantum manifolds [43]. This theory is then applied to the quantum Navier-Stokes equation, obtaining a generalization of the previous results by Prástaroon the Navier- Stokes equation [37, 42, 45, 46]. (50) A. Prástaro, Dirac quantization, Encyclopaedia Math. Suppl.III., M. Hazwinkel (ed.), Kluwer Academic Publishers, Dordrecht (2002), 127–129. Abstract. A panorama on the modern developments of quantizations of PDEs is given. In particular it is emphasized that on the framework of a geometric theory of PDEs, A. Prástarohas given a formulation of canonical quantization of partial differential equations, without assuming that these should be of variational type and/or linear. Furthermore, the generalization of this geometric approach to PDEs in the category of quantum manifolds, given more recently by A. Prástaro,has been considered. Relations with other recent works in noncommutative geometry are given. (51) A. Prástaro, Integral bordisms and Green kernels in PDEs, Cubo Matem- atica Educacional 4(2)(2002), 316–370. Abstract. Integral bordisms of (nonlinear) PDEs are characterized by means of geometric Green kernels and prove that these are invariant for the classic limit of statistical sets of formally integrable PDEs. Such geometric characterization of Green kernels is related to the geometric approach of canonical quantization of (nonlinear) PDEs, previously introduced by us [29]. Some applications are given where particle fields on curved space-times CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 39

having physical or unphysical masses, (i.e., bradions, luxons and massive neutrinos), are canonically quantized respecting microscopic causality. (52) A. Prástaro& Th. M. Rassias, On the Ulam stability in geometry of PDE’s, Functional Equations Inequality and Applications, Th. M. Rassias (ed.), Kluwer Academic Publishers, Dordrecht (2003), 139–147. Abstract. The article is concerned with the problem of unstability of flows corresponding to solutions of the Navier-Stokes equation in relation with the stability of a new functional equation that is stable as well as superstable in an extended Ulam sense. In such a framework a natural characterization of global stable laminar flow is given also. (53) A. Prástaro& Th. M. Rassias, Ulam stability in geometry of PDE’s, Non- linear Funct. Anal. Appl. 8(2)(2003), 259–278. Abstract. The unstability of characteristic flows of solutions of PDE’s is related to the Ulam stability of functional equations. In particular, we consider, as master equation, the Navier-Stokes equation. The integral (co)bordism groups, that have recently been introduced by A. Prástaroto solve the problem of existence of global solutions of the Navier-Stokes equation [37, 38, 39, 42, 70], lead to a new application of the Ulam stability for functional equations. This allowed us here to prove that the characteristic flows associated to perturbed solutions of global laminar solutions of the Navier-Stokes equation, can be characterized by means of a stable (as well superstable) functional equation (functional Navier-stokes equation). In such a framework a natural criterion to recognize stable laminar solutions is given also. (54) A. Prástaro, Quantum super Yang-Mills equations: Global existence and mass-gap, Dynamic Syst. Appl. 4(2004), 227–232. (Eds. G. S. Ladde, N. G. Madhin and M. Sambandham), Dynamic Publishers, Inc., Atlanta, USA. ISBN:1-890888-00-1. Abstract. Quantum super Yang-Mills equations are considered in the framework of some noncommutative manifolds (quantum supermanifolds) and for such equations existence theorems of local and global solutions are obtained by using some geometric methods recently introduced by A.Prástaro[37, 43, 44, 70]. In particular, global properties of solutions are characterized by means of integral bordism groups. A criterion to recognize global solutions with mass gap is given. (See also the book quoted in [74].) (55) A. Prástaro, Geometry of PDE’s.I: Integral Bordism Groups in PDE’s, J. Math. Anal. Appl. 319(2)(2006), 547–566. Abstract. We improve some our previous theorems on the calculation of integral bordism groups of formally integrable and completely integrable PDE’s, emphasizing the role played by singular solutions and weak solutions. Some applications to interesting PDE’s, defined on finite dimensional manifolds, are also considered. (56) A. Prástaro, Geometry of PDE’s.II: Variational PDE’s and integral bordism groups, J. Math. Anal. Appl. 321(2)(2006), 930–948. Abstract. In the framework of the geometry of PDE’s, we classify variational equations of any order with respect to their formal properties. Fol- lowing our previous results [55], we relate constrained variational PDEs to 40 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

their integral bordism groups. In this way we are able to characterize global solutions of constrained variational PDEs and to relate them to the structure of global solutions for the corresponding constraint equations. Some applications are also considered. (57) A. Prástaro, Conservation laws in quantum super PDE’s, Proceedings of the Conference on Differential & Difference Equations and Applications, (eds. R. P. Agarwal & K. Perera), Hindawi Publishing Corporation, New York (2006), 943–952. Abstract. Conservation laws are considered for PDE’s built in the category QS of quantum supermanifolds. These are functions defined on the integral bordism groups of such equations and belonging to suitable Hopf algebras (full quantum Hopf algebras). In particular, we specialize our calculations on the quantum super Yang-Mills equations and quantum black holes. (58) A. Prástaro, (Co)bordism groups in quantum super PDE’s.I: Quantum supermanifolds, Nonlinear Anal. Real World Appl. 8(2)(2007), 505–538. Abstract. Following our previous works on noncommutative manifolds and noncommutative PDE’s [37, 43, 44, 54, 57, 70, 74, 76], we consider in these series of three papers, some further results on quantum supermanifolds and quantum super PDE’s. In particular, in this first part we focus our attention on quantum supermanifolds. These structures globalize the notion of quantum superalgebras, obtaining noncommutative manifolds that are useful to give a fully covariant description of noncommutative geometric structures, hence of quantum physics. We study the geometry of quantum supermanifolds and characterize their (co)homological and (co)bordism properties. Covariant quantizations of super PDE’s are given, obtaining examples of quantum supermanifolds justifying their definitions. (59) A. Prástaro, (Co)bordism groups in quantum super PDE’s.II: Quantum super PDE’s, Nonlinear Anal. Real World Appl. 8(2)(2007), 480–504. Abstract. Following our previous works on the integral (co)bordism groups of quantum PDE’s [37, 43, 44, 59, 70, 74], we specialize, now, on quantum super partial differential equations, i.e., partial differential equations built in the category of quantum supermanifolds. These are manifolds modeled on locally convex topological vector spaces built starting from quantum algebras endowed also with a Z2-gradiation, and a Z2-graded Lie algebra structure, (quantum superalgebra). Then, we extend to these manifolds, with such richer structure, our previous results, and build a geometric theory of quantum super PDEs, that allows us to obtain theorems of existence of (smooth) local and global solutions in the category of quantum supermanifolds. Some quantum (super) PDE’s, arising from the Dirac quantization of some classical (super) PDE’s, are considered in some details. (60) A. Prástaro, (Co)bordism groups in quantum super PDE’s.III: Quantum super Yang-Mills equations, Nonlinear Anal. Real World Appl. 8(2)(2007), 447–479. Abstract. In this third part of a series of three papers devoted to the study of geometry of quantum super PDE’s [58, 59], we apply our theory, developed in the first two parts, to quantum super Yang-Mills equations and quantum supergravity equations. For such equations we determine their CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 41

integral bordism groups, and by using some surgery techniques, we obtain theorems of existence of global solutions, also with nontrivial topology, for Cauchy problems and boundary value problems. Quantum tunnelling effects are described in this context. Furthermore, for quantum supergravity equations we prove existence of solutions of the type quantum black holes evaporation processes just by using an extension to quantum super PDEs of our theory of integral (co)bordism groups. Our proof is constructive, i.e., we give geometric methods to build such solutions. In particular a criterion to recognize quantum global (smooth) solutions with mass-gap, for the quantum super Yang-Mills equation, is given. Finally it is proved that quantum super PDE’s contain also solutions that come from Dirac quantization of their superclassical counterparts. This proves that quantum super PDE’s are (nonlinear) generalizations of Dirac quantized superclassical PDE’s. Applications of this result to free quantum super Yang-Mills equations are given. (61) R. Agarwal & A. Prástaro, Geometry of PDE’s.III(I): Webs on PDE’s and integral bordism groups. The general theory, Adv. Math. Sci. Appl. 17(1)(2007), 239–266. Abstract. Web structures are recognized on any partial differential equation (PDE) that bring new insights in the geometric theory of PDE’s. Re- lations with integrability properties of PDE’s, and their integral bordism groups, are obtained also, emphasizing the role played by singular and weak solutions. Applications to some important PDE’s of the Mathemati- cal Physics are given too. (62) R. Agarwal & A. Prástaro, Geometry of PDE’s.III(II): Webs on PDE’s and integral bordism groups. Applications to Riemannian geometry PDE’s, Adv. Math. Sci. Appl. 17(1)(2007), 267–285. Abstract. By using previous results by A.Prástaroon integral bordism groups of PDE’s, and some issues of the companion paper in [61], we characterize in a geometric way local and global solutions of (generalized) Yam- abe equations and Ricci-flow equations. We prove that such results help to find natural linear and parallel webs on a large category of PDE’s, that are important in order to find regular and singular solutions on such PDE’s. In particular, by applying algebraic topologic methods on the Ricci-flow equation we definitively prove that the Poincaréconjecture on the 3-dimensional manifolds is true. (63) A. Prástaro, Geometry of PDE’s.IV: Navier-Stokes equation and integral bordism groups, J. Math. Anal. Appl. 338(2)(2008), 1140–1151. Abstract. Following our previous results on this subject [39, 42, 45, 46], integral bordism groups of the Navier-Stokes equation are calculated for smooth, singular and weak solutions respectively. Then a characterization of global solutions is made on this ground. Enough conditions to assure existence of global smooth solutions are given and related to nullity of integral charecteristic numbers of the boundaries. Stability of global solutions are related to some characteristic numbers of the space-like Cauchy data. Global solutions of variational problems constrained by (NS) are classified by means of suitable integral bordism groups too. 42 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

(64) A. Prástaro, (Un)stability and bordism groups in PDE’s, Banach J. Math. Anal. 1(1)(2007), 139–147. Abstract. In this paper, by using the theory of integral bordism groups in PDE’s, previously introduced by Prástaro,we give a new interpretation of the concept of (un)stability in the framework of the geometric theory of PDE’s. A geometric criterium to identify stable PDE’s and stable solutions of PDE’s is given. (65) A. Prástaro, Extended crystal PDE’s stability.I: The general theory, Math. Comput. Modelling. (2008). Abstract. This work, divided in two parts, follows some our previous works devoted to the algebraic topological characterization of PDE’s. In this first part, the stability of PDE’s is studied in details in the framework of the geometric theory of PDE’s, and bordism groups theory of PDE’s. In particular we identify criteria to recognize PDE’s that are stable (in extended Ulam sense) and in their regular smooth solutions do not occur finite time unstabilities, (stable extended crystal PDE’s). Applications to some important PDE’s are considered in some details. (In the second part a stable extended crystal PDE encoding anisotropic incompressible magnetohydrodynamics is obtained.) (66) A. Prástaro, Extended crystal PDE’s stability.II: The extended crystal MHD- PDE’s, Math. Comput. Modelling. (2008). Abstract. This paper is the second part of a work devoted to the algebraic topological characterization of PDE’s stability and its relation with an important class of PDE’s called extended crystals PDE’s. In fact, their integral bordism groups can be considered extensions of subgroups of crystallographic groups. This allows us to identify a characteristic class that measures the obstruction to the existence of global solutions. In part I we identified criteria to recognize PDE’s that are stable (in extended Ulam sense) and in their regular smooth solutions do not occur finite time unstabilities, (stable extended crystal PDE’s). Here we study in some details a new PDE encoding anisotropic incompressible magnetohydrodynamics. A stable extended crystal MHD-PDE’s is obtained where in its smooth solutions do not occurr unstabilities in finite times. These results are considered first for systems without body energy source and after by introducing also a contribution by energy source in order to take into account of nuclear energy production. A condition in order solutions satisfy the second principle of thermodynamics is given. (67) A. Prástaro, On the extended crystal PDE’s stability.I: The n-d’Alembert extended crystal PDE’s, Appl. Math. Comput. 204(1)(2008), 63–69. Abstract. Our recent results on extended crystal PDE’s and geometric theory on PDE’s stability, are applied to the generalized n-d’Alembert ′ PDE’s, (d A)n, n ≥ 2. We prove that these are extended crystal PDE’s ′ for any n ≥ 2. For suitable n,(d A)n becomes an extended 0-crystal PDE and also a 0-crystal PDE. An equation, having all the same smooth solu- ′ tions of (d A)n, but without unstabilities at “finite time” is obtained for each n ≥ 2. (68) A. Prástaro, On the extended crystal PDE’s stability.II: Entropy-regular- solutions in MHD-PDE’s, Appl. Math. Comput. 204(1)(2008), 82–89. CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 43

Abstract. Local and global existence theorems of entropy-regular-solutions in the geometric framework of MHD-PDE’s, recently introduced by A.Prástaro, are given. Stability properties of such solutions are studied. In particular it is shown how to stabilize the smooth entropy-regular-solutions, in order to avoid finite-times unstabilities. (69) A. Prástaro, On quantum black-hole solutions of quantum super Yang-Mills equations, Proceedings Dynamic Systems Appls. 5(2008), 407–414. (Eds. G. S. Ladde, N. G. Madhin C. Peng & M. Sambandham), Dynamic Pub- lishers, Inc., Atlanta, USA. ISBN: 1-890888-01-6. Abstract. The category Q, (resp. QS), of quantum manifolds, (resp. quantum supermanifolds), introduced by A.Prástaro,gives a natural framework where implement a geometric theory of quantum (super) partial differential equations (PDE’s). The interest for quantum supermanifolds is motivated by the fact that these structures allow us to describe the uni- fication of all the four fundamental forces (gravity, electromagnetic, weak nuclear, strong nuclear), at the quantum level. In this note we present some recent developments in this direction. In particular we will consider quantum black holes as solutions of quantum super Yang-Mills equations. These interpret very high energy level production of particles, where the effects of strong-quantum-gravity become dominant. (70) A. Prástaro, Surgery and bordism groups in quantum partial differential equations.I: The quantum Poincaréconjecture, Nonlinear Anal. Theory Methods Appl. 71(12)(2009), 502–525. Abstract. In this work, in two parts, we continue to develop the geometric theory of quantum PDE’s, introduced by us starting from 1996. (The second part is quoted in ref.[71].) This theory has the purpose to build a rigorous mathematical theory of PDE’s in the category DS of noncommutative manifolds (quantum (super)manifolds), necessary to encode physical phenomena at microscopic level (i.e., quantum level). Aim of the present paper is to report on some new issues in this direction, emphasizing an interplaying between surgery, integral bordism groups and conservations laws. In particular, a proof of the Poincaréconjecture, generalized to the category DS, is given by using our geometric theory of PDE’s just in such a category. (71) A. Prástaro, Surgery and bordism groups in quantum partial differential equations.II: Variational quantum PDE’s, Nonlinear Anal. Theory Meth- ods Appl. 71(12)(2009), 526–549. Abstract. This is the second part of a work devoted to the interplay between surgery, integral bordism groups and conservation laws, in order to characterize the geometry of PDE’s in the category QS of quantum (super)manifolds. (Part I is quoted in ref.[70].) In this paper we will consider variational problems, in the category QS, constrained by partial differential equations. We get theorems of existence for local and global solutions. The characterization of global solutions is made by means of integral bordism groups. Applications to some important examples of the Mathematical Physics, as quantum super-black-hole solutions of quantum super Yang- Mills equations, are discussed in some details. Quantum supermanifolds 44 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

allow us to unify, at the quantum level, the four fundamental forces, (gravitational, electromagnetic, weak-nuclear, strong-nuclear), in an unique geometric structure. The geometric theory of PDE’s, built in the category QS of quantum supermanifolds, gives us the right mathematic tool to describe quantum phenomena also at very high energy levels, where quantum-gravity becomes dominant. (72) R. P. Agarwal & A. Prástaro, Singular PDE’s geometry and boundary value problems, J. Nonlinear Conv. Anal. 9(3)(2008), 417–460. Abstract. Local and global existence theorems for boundary value problems in singular PDE’s are considered. In particular, surgery techniques and integral bordism groups are utilized, following previous works by A.Prástaro on PDE’s, in order to build global solutions crossing also singular points and to study their stability properties. (73) R. P. Agarwal & A. Prástaro, On singular PDE’s geometry and boundary value problems, Appl. Anal. 88(8)(2009), 1115-1131. Abstract. A geometric formulation of singular PDE’s is considered. Surgery techniques and integral bordism groups are utilized, following previous works by A.Prástaroon PDE’s, in order to build global solutions crossing also singular points and to study their stability properties. A detailed proof on the integral characterization of singular ODE’s is given. (74) A. Prástaro, Extended crystal PDE’s. Mathematics Without Boundaries: Surveys in Pure Mathematics. P. M. Pardalos and Th. M. Rassias (Eds.) Springer-Heidelberg New York Dordrecht London, (to appear). arXiv:0811.3693[math.AT]. Abstract. In this paper we show that between PDE’s and crystallographic groups there is an unforeseen relation. In fact we prove that integral bordism groups of PDE’s can be considered extensions of crystallographic subgroups. In this respect we can consider PDE’s as extended crystals. Then an algebraic-topological obstruction (crystal obstruction), characterizing existence of global smooth solutions for smooth boundary value problems, is obtained. Applications of this new theory to the Ricci-flow equation and Navier-Stokes equation are given that solve some well-known fundamental problems. These results, are also extended to singular PDE’s, introducing (extended crystal singular PDE’s). An application to singular MHD-PDE’s, is given following some our previous results on such equations, and showing existence of (finite times stable smooth) global solutions crossing critical nuclear energy production zone. (75) A. Prástaro, Quantum extended crystal PDE’s, Nonlinear Studies 18(3)(2011), 447–485. arXiv:1105.0166[math.AT]. Abstract. Our recent results on extended crystal PDE’s are generalized to PDE’s in the category QS of quantum supermanifolds. Then obstructions to the existence of global quantum smooth solutions for such equations are obtained, by using algebraic topologic techniques. Applications are considered in details to the quantum super Yang-Mills equations. Furthermore, our geometric theory of stability of PDE’s and their solutions, is also generalized to quantum extended crystal PDE’s. In this way we are able to identify quantum equations where their global solutions are stable at finite CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 45

times. These results, are also extended to quantum singular (super)PDE’s, introducing (quantum extended crystal singular (super) PDE’s). (76) A. Prástaro, Quantum extended crystal super PDE’s. Nonlinear Anal. Real World Appl. 13(6)(2012), 2491–2529. arXiv:0906.1363[math.AT]. Abstract. We generalize our geometric theory on extended crystal PDE’s and their stability, to the category QS of quantum supermanifolds. By using algebraic topologic techniques, obstructions to the existence of global quantum smooth solutions for such equations are obtained. Applications are given to encode quantum dynamics of nuclear nuclides, identified with graviton-quark-gluon plasmas, and study their stability. We prove that such quantum dynamical systems are encoded by suitable quantum extended crystal Yang-Mills super PDE’s. In this way stable nuclear-charged plasmas and nuclides are characterized as suitable stable quantum solutions of such quantum Yang-Mills super PDE’s. An existence theorem of local and global solutions with mass-gap, is given for quantum super Yang-Mills PDE’s, (\YM), by identifying a suitable constraint, (Higgs\ ) ⊂ (\YM), Higgs quantum super PDE, bounded by a quantum super partial differential relation (Goldstone\ ) ⊂ (\YM), quantum Goldstone-boundary. A global solution V ⊂ (\YM), crossing the quantum Goldstone-boundary acquires (or loses) mass. Stability properties of such solutions are characterized. (77) A. Prástaro, Exotic heat PDE’s, Commun. Math. Anal. 10(1)(2011), 64–81. arXiv:1006.4483[math.GT]. Abstract. Exotic heat equations that allow to prove the Poincaréconjec- ture, some related problems and suitable generalizations too are considered. The methodology used is the PDE’s algebraic topology, introduced by A. Prástaroin the geometry of PDE’s, in order to characterize global solutions. (78) A. Prástaro, Exotic heat PDE’s.II. Essays in Mathematics and its Appli- cations. In Honor of Stephen Smale’s 80th Birthday. P. M. Pardalos and Th. M. Rassias (Eds.) Springer-Heidelberg New York Dordrecht London (2012), 369–419. ISBN 978-3-642-28820-3 (Print) 978-3-28821-0 (Online). DOI: 10.1007/978-3-642-28821-0. arXiv: 1009.1176[math.AT]. Abstract. Exotic heat equations that allow to prove the Poincaréconjec- ture and its generalizations to any dimension are considered. The methodology used is the PDE’s algebraic topology, introduced by A. Prástaroin the geometry of PDE’s, in order to characterize global solutions. In particular it is shown that this theory allows us to identify n-dimensional exotic spheres, i.e., homotopy spheres that are homeomorphic, but not diffeomorphic to the standard Sn. (79) A. Prástaro, Exotic n-d’Alembert PDE’s and stability. Nonlinear Analysis. Stability, Approximation and Inequalities. Series: Springer Optimization and its Applications Vol 68. P. M. Pardalos, P. G. Georgiev and H. M. Srivastava (Eds.). Springer Optimization and its Applications Volume 68 (2012), 571–586. ISBN 978-1-4614-3498-6. arXiv:1011.0081[math.AT]. Abstract. In the framework of the PDE’s algebraic topology, previously introduced by A. Prástaro, exotic n-d’Alembert PDE’s are considered. These ′ ′ are n-d’Alembert PDE’s, (d A)n, admitting Cauchy manifolds N ⊂ (d A)n identifiable with exotic spheres, or such that ∂N, can be exotic spheres. For 46 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

such equations local and global existence theorems and stability theorems are obtained. (80) A. Prástaro, Exotic PDE’s. Mathematics Without Boundaries: Surveys in Interdisciplinary Research. P. M. Pardalos and Th. M. Rassias (Eds.) Springer-Heidelberg New York Dordrecht London, (to appear). arXiv:1101.0283[math.AT]. Abstract. In the framework of the PDE’s algebraic topology, previously introduced by A. Prástaro,are considered exotic differential equations, i.e., differential equations admitting Cauchy manifolds N identifiable with exotic spheres, or such that their boundaries ∂N are exotic spheres. For such equations are obtained local and global existence theorems and stability theorems. In particular the smooth (4-dimensional) Poincaréconjecture is proved. This allows to complete the previous Theorem 4.59 in [22] also for the case n = 4. (81) A. Prástaro, Quantum exotic PDE’s. Nonlinear Anal. Real World Appl. 14(2)(2013), 893–928. arXiv:1106.0862[math.AT]. Abstract. Following the previous works on the A. Prástaro’sformula- tion of algebraic topology of quantum (super) PDE’s, it is proved that a canonical Heyting algebra (integral Heyting algebra) can be associated to any quantum PDE. This is directly related to the structure of its global solutions. This allows us to recognize a new inside in the concept of quantum logic for microworlds. Furthermore, the Prástaro’sgeometric theory of quantum PDE’s is applied to the new category of quantum hypercomplex manifolds, related to the well-known Cayley-Dickson construction for algebras. Theorems of existence for local and global solutions are obtained for (singular) PDE’s in this new category of noncommutative manifolds. Finally the extension of the concept of exotic PDE’s, recently introduced by A.Prástaro,has been extended to quantum PDE’s. Then a smooth quantum version of the quantum (generalized) Poincarélemma is given too. These results extend ones for quantum (generalized) Poincarélemma, previously given by A. Prástaro. (82) A. Prástaro, Strong reactions in quantum super PDE’s. I: Quantum hypercomplex exotic super PDE’s. arXiv:1205.2984[math.AT]. (Part I and Part II are unified in arXiv.) Abstract. In order to encode strong reactions of the high energy physics, by means of quantum nonlinear propagators in the Prástaro’sgeometric theory of quantum super PDE’s, some related geometric structures are further developed and characterized. In particular super-bundles of geometric objects in the category QS of quantum supermanifolds are considered and quantum Lie derivative of sections of super bundle of geometric objects are calculated. Quantum supermanifolds with classic limit are classified with respect to the holonomy groups of these last commutative manifolds. A theorem characterizing quantum super manifolds with structured classic limit as super bundles of geometric objects is obtained. A theorem on the characterization of chi-flow on suitable quantum manifolds is proved. This solves a previous conjecture too. Quantum instantons and quantum solitons are defined are useful generalizations of the previous ones, well-known in the literature. Quantum conservation laws for quantum super PDEs are CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 47

characterized. Quantum conservation laws are proved work for evaporating quantum black holes too. Characterization of observed quantum nonlinear propagators, in the observed quantum super Yang-Mills PDE, by means of conservation laws and observed energy is obtained. Some previous results by A. Prástaroabout generalized Poincaréconjecture and quantum exotic spheres, are generalized to the category Qhyper,S of hypercomplex quantum supermanifolds. (This is the first part of a work divided in two parts. For part II see [83].) (83) A. Prástaro, Strong reactions in quantum super PDE’s. II: Nonlinear quantum propagators. arXiv:1205.2984[math.AT]. (Part I and Part II are unified in arXiv.) Abstract. In this second part, of a work devoted to encode strong reactions of the high energy physics, in the algebraic topologic theory of quantum super PDE’s, (previously formulated by A. Prástaro),decomposition theorems of integral bordisms in quantum super PDEs are obtained. (For part I see [82].) In particular such theorems allow us to obtain representations of quantum nonlinear propagators in quantum super PDE’s, by means of elementary ones (quantum handle decompositions of quantum nonlinear propagators). These are useful to encode nuclear and subnuclear reactions in quantum physics. Prástaro’sgeometric theory of quantum PDE’s allows us to obtain constructive and dynamically justified answers to some important open problems in high energy physics. In fact a Regge- type relation between reduced quantum mass and quantum phenomenological spin is obtained. A dynamical quantum Gell-Mann-Nishijima formula is given. An existence theorem of observed local and global solutions with electric-charge-gap, is obtained for quantum super Yang-Mills \ \ \ PDE’s, (YM)[i], by identifying a suitable constraint, (YM)[i]w ⊂ (YM)[i], quantum electromagnetic-Higgs PDE, bounded by a quantum super partial \ \ differential relation (Goldstone)[i]w ⊂ (YM)[i], quantum electromagnetic Goldstone-boundary. An electric neutral, connected, simply connected observed quantum particle, identified with a Cauchy data of (\YM)[i], it is \ proved do not belong to (YM)[i]w. Existence of Q-exotic quantum nonlinear propagators of (\YM)[i], i.e., quantum nonlinear propagators that do not respect the quantum electric-charge conservation is obtained. By using integral bordism groups of quantum super PDE’s, a quantum crossing symmetry theorem is proved. As a by-product existence of massive photons and massive neutrinos are obtained. A dynamical proof that quarks can be broken-down is given too. A quantum time, related to the observation of any quantum nonlinear propagator, is calculated. Then an apparent quantum time estimate for any reaction is recognized. A criterion to identify solutions of the quantum super Yang-Mills PDE encoding (de)confined quantum systems is given. Supersymmetric particles and supersymmetric reactions are classified on the ground of integral bordism groups of the quantum super Yang-Mills PDE (\YM). Finally, existence of the quantum Majorana neutrino is proved. As a by-product, the existence of a new quasi-particle, that we call quantum Majorana neutralino, is recognized made by means of two 48 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

e quantum Majorana neutrinos, a couple (νee, ν¯e), supersymmetric partner of (νe, ν¯e), and two Higgsinos. (84) A. Prástaro, Strong reactions in quantum super PDE’s. III: Exotic quantum supergravity. arXiv:1206.4956[math.AT]. Abstract. of quantum super PDE’s, quantum nonlinear propagators in the observed quantum super Yang-Mills PDE, (\YM)[i], are further characterized. In particular, quantum nonlinear propagators with non-zero lost quantum electric-charge, are interpreted as exotic-quantum supergravity effects. As an application, the recently discovered bound-state called Zc(3900), is obtained as a neutral quasi-particle, generated in a Q-quantum exotic supergravity process. Quantum entanglement is justified by means of the algebraic topologic structure of quantum nonlinear propagators. Ex- istence theorem for solutions of (\YM)[i] admitting negative local tempera- tures (quantum thermodynamic-exotic solutions) is obtained too and related to quantum entanglement. (85) A. Prástaro, The Landau’s problems. I: The Goldbach’s conjecture proved. arXiv:1208.2473[math.GM]. Abstract. We give a direct proof of the Goldbach’s conjecture, (GC), in number theory, in the Euler’s form. The proof is also constructive, since it gives a criterion to find two prime numbers ≥ 1, such that their sum gives a fixed even number ≥ 2. The proof is obtained by recasting the problem in the framework of the Commutative Algebra and Algebraic Topology. Even if in this paper we consider 1 as a prime number, our proof of the GC works also for the restricted Goldbach conjecture, (RGC), i.e., by excluding 1 from the set of prime numbers. (86) A. Prástaro, The Landau’s problems. II: Landau’s problems solved. arXiv:1208.2473[math.GM]. Abstract. Three of the well known four Landau’s problems are solved in this paper. (In [85] the proof of the Goldbach’s conjecture has been already given.) (87) A. Prástaro, The Riemann hypothesis proved. arXiv:1305.6845[math.GM]. Abstract. The Riemann hypothesis is proved by extending the zeta Rie- mann function to a quantum mapping between quantum 1-spheres with quantum algebra A = C, in the sense of A. Prástaro[70, 81]. Algebraic topologic properties of quantum-complex manifolds and suitable bordism groups of morphisms in the category QC of quantum-complex manifolds are utilized. (88) A. Prástaro, Quantum Geometrodynamic Cosmology. (Submitted for publication.) Abstract. By utilizing Prástaro’squantum supergravity, it is proved that the Universe’s expansion at the Planck epoch is justified by the fact that it is encoded by a quantum nonlinear propagator having thermodynamic quantum exotic components in its boundary. This effect produces also an increasing of energy in the Universe at the Einstein epoch: Planck-epoch- legacy on the boundary of our Universe. This is the main source of the Universe’s expansion and solves the problem of the non-apparent energy- matter (dark-energy-matter) in the actual Universe. CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 49

ABSTRACTS OF MONOGRAPHS AND TEXTS (89) A. Prástaro, Geometry of PDEs and Mechanics, World Scientific Publish- ing, River Edge, NJ, 1996, 760 pp. Abstract. 1. Algebraic Geometry: Algebraic Complements, Affine spaces, Differential Manifolds, Grassmann Manifolds, Spectral Sequences. 2. Dif- ferential Equations (PDES): Geometry of Differential Equations, Ordinary Differential Equations (ODEs), Characteristics of PDEs, Affine PDEs and Green Functions, Spectral Sequences in PDEs, Tunnel Effects in PDEs, Cobordism Groups in PDEs. 3. Mechanics: Structure of Galilean Space- Time, One-Body Dynamics, Important Formulas, Fundamental Theorems of Dynamics, Lagrangian Mechanics for Perfect Holonomic Systems, Rigid- Body Dynamics. 4. Continuum Mechanics: Flow, Stress Tensor and Mo- ment of Stress Tensor, Local Dynamic Equations, Thermodynamics of Con- tinuum Media, Rheological Classification of Materials, Rheoptics, Multi- component Continuum Systems, Variational Field Theory. 5. Quantum Field Theory: Locally Convex Manifolds and Derivative Spaces, Differen- tial Geometry of Quantum Situs, Mathematical Logic and Quantization of PDEs, Formal and Dirac Quantizations of PDEs, Canonical Quantiza- tion of PDEs. 6. Geometry of Quantum PDEs: Differential Geometry of Quantum Manifolds, Cohomology of Quantum Manifolds, Formal Theory of Quantum PDEs, Cartan Spectral Sequences of Quantum PDEs, Quantum Distribution Solutions and Singular Solutions of Quantum PDEs, Tunnel Effects in Quantum PDEs, Quantum PDEs Non-holonomic Connections, Gauge Quantum PDEs, Supergravity Quantum PDEs. (90) A. Prástaro, Elementi di Meccanica Razionale, VII edizione, Aracne Ed- itrice, Roma, 2010,446 pp. Abstract. 1. This monography is addressed to Italian university students in Mathematics, Physics and Engineering. It develops with a modern geometric language the methods of classical mechanics and geometry of (partial) differential equations. The presentation, even if elementary, gives the actual mathematics situation in classical mechanics. Indice. Algebra: matrici ed applicazioni lineari; prodotto tensoriale fra spazi vettoriali; tensori; spazi vettoriali Euclidei; componenti covari- anti e controvarianti; tensori simmetrici ed antisimmetrici; orientazione di spazi vettoriali; gruppi GL(V ), O(V ), SO(V ); equazione agli autovalori; spazio affine; gruppo affine; soluzioni di equazione affine. 2. Equazioni differenziali: varietàdifferenziale; spazio tangente; campi tensoriali e loro immagine reciproca; forma volume ed orientazione di varietàdifferenziale; integrazione su varietàdifferenziale; varietàRiemanniana; gruppi di Lie e loro caratterizzazione tramite costanti di struttura; elementi di teoria geo- metrica delle equazioni differenziali. 3. Connessioni differenziali: derivata covariante; connessione di Levi-Civita; curvatura di una connessione; com- plesso di de Rham e gruppi di comologia; importanti operatori differenziali su varietàRiemanniane; componenti fisiche di oggetti geometrici. 4. Spazio-tempo Galileiano: struttura dello spazio-tempo Galileiano; moto; velocitàed accelerazione del moto; osservatore e moto osservato; gruppo 50 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

di Galileo; formule di Frenet; espressione della velocitàed accelerazione tramite prametro di linea; teorema di Coriolis; osservatori inerziali ed osservatori rigidi; moti rigidi e teorema fondamentale della cimematica dei corpi rigidi; angoli di Euler; precessioni; traiettorie polari (base e ruletta). 5. Dinamica di un elemento materiale: vincoli di ordine 0 ≤ n ≤ 2; forze ed equazione di Newton; forze conservative; equazione di Lagrange; equazione di Hamilton e campo vettoriale di Euler; struttura simplettica della meccanica; simmetrie dinamiche; teorema di Noether generalizzato; sistemi di- namici con un numero finito N di particelle; vincoli con attrito. 6. Teoremi fondamentali della dinamica: teorema dell’impulso; teorema della conservazione dell’impulso; momento totale e sue proprietà; momento assiale; coppia; lavoro; potenza; energia cinetica; stabilitàdell’equilibrio; equazioni cardinali della dinamica; teorema di conservazione; principio dei lavori vir- tuali; teorema di Köenig. 7. Meccanica Lagrangiana: Equazione di La- grange e sue forme particolari. Potenziali generalizzati e forza di Lorentz; stabilitàed equazioni di Lagrange linearizzate. Leggi di conservazione e coordinate ignorabili; equazione di Lagrange e calcolo variazionale; con- figurazioni di stato stazionario e loro stabilità. 8. Meccanica dei sistemi rigidi: baricentri di sistemi continui; tensore momento d’inerzia; equazioni cardinali per corpo rigido; moto alla Poinsot e sue leggi di conservazione. 9. Esercizi: (ventuno esercizi completamente risolti). 10. Meccanica dei continui: flusso; osservatore proprio di un sistema continuo; oggetti geometrici associati ad un flusso; tensore degli sforzi e suo momento; equazioni di Euler; equazione dinamica dei sistemi continui; termodinamica covariante dei sistemi continui; classificazione reologica dei sistemi continui; esempi di flussi e deformazioni. 11. Reottica. Bibliografia. 12. Equazioni di Maxwell e relativitàristretta. 13. Esercizi complementari (completamente svolti): Esercizi di Geometria; Esercizi di Meccanica dei sistemi rigidi; Esercizi di Meccanica dei sistemi continui. Indice analitico. Indice dei simboli. (91) A. Prástaro, Quantized Partial Differential Equations, World Scientific Pub- lishing, River Edge, NJ, 2004, 500 pp. Abstract. This book contains three chapters and two addenda. Quan- tized PDE’s.I. In this first part we consider quantum (super) manifolds as topological spaces locally identified with open sets of some locally convex topological vector spaces built starting from suitable topological algebras A, quantum (super)algebras. The noncommutative character of such quantum (super)manifolds is given by the underlying noncommutative algebras A. In fact, here A plays the role of “fundamental algebra of numbers”, like K = R, C does for usual commutative manifolds. Therefore, quantum (super)manifolds are the natural generalizations of manifolds , when one sub- stitutes commutative numbers with noncommutative ones. Commutative manifolds are contained into quantum (super)manifolds, as quantum (super)algebras A are required to contain K. This aspect is also reflected by the k fact that the class of differentiability Qw for pseudogroup structures defin- ing quantum (super)manifolds, contains the usual Ck differentiability for manifolds. In fact, the class of differentiability of such topological manifolds is defined by requiring weak differentiability and Z-linearity of the derivatives, where Z is the center of the underlying quantum (super)algebras. CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 51

We give (co)homological characterizations of quantum (super)algebras and quantum (super)manifolds, by applaying to these noncommutative topological manifolds standard methods of algebraic topology. In particular, we calculate also (co)bordism groups in quantum (super)manifolds. Quantized PDE’s.II. Here we give a geometric theory of PDE’s in the category of quantum (super)manifolds. This theory is the natural extension of the geometric theory of PDE’s in the category of commutative (super)manifolds. Empha- sis is put on some new algebraic topological techniques that allow us to calculate the integral (co)bordism groups of quantum (super)PDE’s, hence to characterize global properties of solutions of quantum (super)PDE’s. Many applicatio ns to important equations of quantum field theory are considered also. Quantized PDE’s.III. Here we consider a process that allows us to associate to a (super)PDE, defined in the category of (super)commutative manifolds, a quantum (super)PDE. This process is the covariant quantization. We describe it in some steps. In fact, we first define quantizations of PDE’s in the framework of the mathematical logic, by means of evalua- tions of the logic of a PDE Ek, that is the Boolean algebra of subsets of the classic limit Ω(Ek)c of the quantum situs Ω(Ek) of Ek, into quantum logics A ⊂ L(H), that are algebras of (self-adjoint) operators on a locally convex topological vector (Hilbert) space H, in such a way to define (pre-)spectral measures on Ω(Ek)c: Ω(Ek)c◦→L(H). We show that these quantizations can be obtained by means of a geometric process called covariant quantization, (or canonical quantization), of PDE’s, that is, roughly speaking the covariant quantization observed by a physical frame. In fact, in a purely geometric context, we prove that any physical observable deforms k the classical PDE, Ek ⊂ JD (W ), around its solutions. In this way we can associate to the Lie filtered (super)algebra of the (super)classical observ- ˆ ables, B, of Ek, a filtered quantum (super)algebra B, defined by means of distributive kernels, G˜q, propagators, canonically associated to Ek. We characterize also the propagators of PDE’s by means oftheir integral bordism groups. The final step is the relation between the formal properties E∞ · · · → Ek+1 → Ek → · · · of the classical equation Ek, with quantum ones Eˆ∞ · · · → Eˆk+1 → Eˆk → · · · . These are obtained in the category of QPDE’s, where the quantum (super)algebra Bˆ , so obtained as covariant quantization of Ek, identifies a quantum (super)PDE. Addendum I. In refs.[38, 41] are calculated, for the first time, the integral bordism groups of the 3D nonisothermal Navier-Stokes equation (NS). A direct consequence of these results is the proof of existence of global (smooth) solutions for (NS). Here we go in some further results emphasizing surgery techniques that allow us to better understand this geometric proof of existence of (smooth) global solutions for any (smooth) boundary condition. Adden- dum II. In the framework of the geometry of PDE’s, we classify variational equations of any order with respect to their formal properties. A variational sequence is introduced for constrained variational PDE’s that extends previous ones for variational calculus on fiber bundles. Such extended variational sequence allows us to locally and globally solve variational prob- ⊂ k lems, constrained by PDE’s of any order, Ek Jn (W ), by means of some 52 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

cohomological properties of Ek. Moreover, we relate constrained variational PDE’s to the integral (co)bordism groups for PDE’s. In this way we are able to characterize the structure properties of global solutions of constrained variational PDE’s and to relate them to the structure of global solutions for the corresponding constraint equations. Contents: Quantized PDE’s.I. Noncommutative Manifolds: Algebraic topology; Quantum algebras; Quantum manifolds; Quantum supermanifolds. Quantized PDE’s.II. Noncommutative PDE’s: Quantum PDE’s; The quantum Navier-Stokes equation; Quantum super PDE’s; The quantum super Yang-Mills equations. Quantized PDE’s.III. Quantizations of commutative PDE’s: Inte- gral (co)bordism groups in PDE’s; Algebraic geometry of PDE’s; Spectral measures of PDE’s; Quantizations of PDE’s; Covariant and canonical quantizations of PDE’s. Addendum I: Bordism groups and the (NS)-problem. Addendum II: Bordism groups and variational PDE’s. References. Index. CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 53

ABSTRACTS OF BOOKS (EDITOR AND COAUTHOR)

(92) A. Prástaro, Geometrodynamics Proceedings 1983, Pitagora Ed., Bologna 1984. Abstract. The Hamilton-Jacobi Equation for a Hamiltonian Action: S. Benenti. Twisteurs Sans Twisteurs: A. Crumeyrolle. General Covariance and Minimal Gravitational Coupling in Newtonian Space-Time: H. P. Kun- zle. Canonical Cartan Equations for Higher Order Variational Problems: J. M. Masqué. Bäcklund Problem and Group Theory: J. F. Pommaret. Group Structure of Non-linear Field Theories: J. F. Pommaret. Geometry and Existence Theorems for Incompressible Fluids: A. Prástaro.Relativis- tic Hydrodynamics as a Symplectic Field Theory: W. M. Tulczyjew. (93) A. Prástaro, Geometrodynamics Proceedings 1985, World Scientific Pub- lishing, Singapore 1985. Abstract. A Geometrical Interpretation of the 1-cocycles of a Lie Group: S. Benenti, W. M. Tulzyjew. Supermanifolds and Supergravity: Y. Choquet- Bruhat. Self-dual Yang-Mills Fields and the Penrose Trasnform: A. Crumey- rolle. On Smooth and Analytic Functions in Gauge Field Theory: J. Czyz. An Application of Topological Methods to the Study of Periodic Solutions of Hamiltonian Systems: G. F. Dell’Antonio. Introducing Spinors, Isospinors, etc. in Globally Nontrivial Space-times: L. Dabrowski. The Radon Trans- form on Compact Symmetric Spaces: H. Goldschmidt. Unconstrained De- grees of Freedom of Gravitational Field and the Positivity of Gravitational Energy: J. Kijowski. Polynomial Identities Satisfied by Realizations of Lie Algebras: M. Iosifescu, H. Scutaru. Free Motions in Multidimensional Universes: G. Marmo. Harmonically Immersed Lorentz Surfaces: T. Mil- nor. On the Symmetry Properties of Constrained Hamiltonian System: M. Mintchev. On a Property of Higher Order Poincaré-CartanForms in the Constructive Approach: J. M. Masqué. Covariant Canonical Formal- ism for Gravity Theories: J. E. Nelson, T. Regge. Invariant Differential Techniques: A. Nijenhuis. A Utiyama Type Theorem in the C-K-S- Gauge Approach to Gravity: A. Pérez-Rendòn.Dynamic Conservation Laws: A. Prástaro.The Doulbeault-Kostant Complex and Geometric Quantization: M. Puta. Symplectic Origin of Some Properties of Generally Covariant Field Theories: A. Smolski. Symplectic Scattering Theory: S. Sternberg. (94) A. Prástaro& Th. M. Rassias, Geometry in Partial Differential Equations, World Scientific Publishing, River Edge, NJ, 1994. Abstract. Some Applications of the Corea Formula to Partial Differential Equations: F. Bethuel, J.-M. Chidaglia. Large Solutions for the Equation of Surfaces of Prescribed Mean Curvature: F. Bethuel, O. Rey. Optical Hamiltonian Functions: M. Bialy, L. Polterovich. On the Geometry of the Hodge-de Rham Laplace Operator: M. Craioveanu, M. Puta, Th. M. Rassias. Minimal Surfaces in Economic Theory: J. Donato. The Morimoto Problem: B. Doubrov, A. Hushner. Asymptotic Expansions in Spectral Ge- ometry: P. B. Giley. Deformations and Recursion Operators for Evolution Equations: I. S. Krasil’shchik, P. H. M. Kersten. Geometric Hamiltonian 54 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

Forms for the Kadomtsev-Petviashvili and Zabolotskaya-Khokhlov Equa- tions, B. A. Kupershmidt. Classification of Mixed Type Momnge-Ampere Equations: A. Kushner. Non-Holonomic Filtration: Algebraic and Geo- metric Aspects of Non-Integrability: V. Lychagin, V. Rubtsov. Spencer Cohomologies: V. Lychagin, L. Zilbergleit. Hawking’s Relation via Fourier Integral Operators: P. E. Parker. Geometry of Super PDE’s: A. Prástaro. On a Geometric Approach to an Equation of J.d’Alembert: A. Prástaro, Th. M. Rassias. Geometric Prequantization of the Einstein’s Vacuum Field: M. Puta. On Differential Equations and Cartan’s Projective Con- nections: Y. R. Romanovsky. Smooth Marginal Analysis of Bifurcation of Extremals: Yu I. Sapronov. On the SchrödingerEquation for an N- Electron Atom: C. S. Sharma. Higher Symmetries and Conservation Laws of Euler-Darboux Equations: V. E. Shemarulin. Strings and Menbranes: K. S. Stelle. Methods for Solving Two-Dimensional Nonstationary MHD Equations at Small Alfven-Mach Numbers: V. S. Titov. CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 55

ABSTRACTS OF PATENTS (95) A. Prástaroet al., 28/5/1975 - No 23.797 A/75. Process of production of fibrous structures with high degree of birefringence. Abstract. The present invention refers to a process to produce fibrous structures having an high degreee of monoaxial orientation, by means of controlled extrusion of solutions, emulsions, suspensions of fibrogenous thermoplastic polymers. In particular, one fixes the optimum conditions of the extrusor design in order to obtain a birefringence higher than 0.1 10−4. (96) A. Prástaroet al., 11/7/1975 - No 25.334 A/75. Process of production of synthetic polymers by means of flash-spinning of polymers solutions. Abstract. The present invention refers to a process to produce fibrous structures of synthetic polymers, in the form of plexus-filament of single little fibers, by means of the technique of the “flash-spinning” of polymers solutions. In particular, one fixes the optimum thermodynamical conditions to obtain plexus-filaments that extend ranges previously found by USA-patents. This has been possible by using a new mathematical model of flash-spinning, purposely formulated, and put on informatic support. Of particular importance has been the discovery of metastable states in the extruded solution, similar to ones used in the bubble-chambers for dedect- ing sub-atomic collisions. These thermodynamical conditions, beside some suitable dynamical conditions, allow us to get optimum control conditions in such a process. 56 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

FIG. 8. Pr´astaro’smathematicians poster 2005-2006. CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 57

Fig. 9. Pr´astaro’smathematicians poster 2007-2008. 58 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

Fig. 10. Pr´astaro’smathematicians poster 2009-2012. CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014 59

5. INDEX

General Informations...... 4 Summary of Principal results...... 7 Publications...... 21 Papers...... 21 Monographs and Texts...... 25 Books (Editor and Coauthor)...... 25 Patents...... 25 Abstracts of Publications...... 27 Abstracts of Papers...... 27 Abstracts of Monographs and Texts...... 49 Abstracts of Books (Editor and Coauthor)...... 53 Abstracts of Patents...... 55 Photos and Posters. Title and A.Prástaro’spicture...... 1 University Address...... 2 Home Address...... 2 Acknowledgments (Prástaro’smathematicians poster 2009-2013)...... 3 School of Engineering, University of Roma La Sapienza...... 6 Florida Institute of Technology, Melbourne, FL - USA...... 20 ICM 2006 SC: Advances in PDE’s Geometry, Madrid - poster...... 26 Prástaro’smathematicians poster 2005-2006...... 56 Prástaro’smathematicians poster 2007-2008...... 57 Prástaro’smathematicians poster 2009-2012...... 58 Prástaro’smathematicians poster 2013-2014...... 3 Roma - Spagna Square...... 59

Fig. 11. Roma - Spagna Square. 60 CURRICULUM VITAE AGOSTINO PRASTARO´ – APRIL 2014

6. CONTENTS 1 - General Informations...... 4 2 - Summary of Principal Results...... 7 3 - Publications...... 21 4 - Abstracts of Publications...... 27 5 - Index...... 59

Curriculum Vitae Agostino Pr´astaro- Edition April 2014