Prizes

Renato Caccioppoli Prize

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Renato Caccioppoli su MacTutor History of Mathematics archive

The Caccioppoli Prize is awarded every four years by the UMI to an Italian mathematician of established international reputation and who has not exceeded 38 years of age in the year preceding the award. Until 1970 the award was assigned every two years. The winner receives ten thousand euros and is nominated by a committee of five mathematicians appointed by the Main Office of the UMI.

Winners



Caccioppoli Prize 1960 – Ennio De Giorgi (Scuola Normale Superiore di Pisa)

Committee’s citation

Ennio De Giorgi’s high level scientific activity led him to gain full
professorship in 1958. Amongst his contributions, we mention, in
chronological order

  • His papers on the Cauchy problem, featuring a new and ingenious
    technique which is applied to prove both an existence theorem and
    nonexistence results
  • His exceptionally interesting works on the notion of sets of finite
    perimeter in which, building on previous contributions of Caccioppoli, he
    obtained new results that in turn allowed him to prove the
    isoperimetric property of the sphere in the same class of finite perimeter
    sets
  • A very deep paper on the analytic character of the extremals of multiple
    integrals, which is dedicated to the study of a very longstanding issue,
    that has gained international recognition and that is the starting point
    for several subsequent works on elliptic equations

The results obtained by Professor De Giorgi demonstrate his great talent
and put him in the front line of young Italian analysts.

Ennio De Giorgi on MacTutor History of Mathematics archive


Caccioppoli Prize 1962 – Edoardo Vesentini (University of Pisa)


Edoardo Vesentini

photo courtesy of MFO

Committee’s citation

Edoardo Vesentini’s high level scientific activity is mainly devoted to
Topology and to Complex Analysis, and in particular to the study of complex algebraic
varieties.

After his first publications (amongst which we mention one paper dedicated
to the study of polarity of algebraic curves and another one where an
intrinsic proof of Riemann-Roch theorem is obtained), there must be
recalled his proof of a fundamental theorem connecting, via duality, the
homology classes of canonical varieties of an algebraic variety with the
Chern’s cohomology classes, and his deep papers on jacobians of several
meromorphic functions on a complex variety, that generalize the classical
results related to algebraic varieties.

In the last years Vesentini started a fruitful cooperation with other
brilliant mathematicians that eventually led to very interesting results on
non-compact complex varieties related to Kodaira’s embedding theorem.

The brilliant and deep work of Vesentini, demonstrating a masterly use of
modern topological, geometrical and analytical methods, and the importance
of the results he achieved, clearly qualify him as one of the best young
Italian mathematicians.

Edoardo Vesentini on Wikipedia (italian)



Caccioppoli Prize 1964 – Emilio Gagliardo (University of Genova)

Committee’s citation

Gagliardo’s scientific production stands at very high levels, and gained
very high international recognition.

After some early, interesting contributions, amongst which we single out
those on second order parabolic equations, he obtained, under the guidance
of Caccioppoli, a compactness criterion with respect to mean convergence.
In turn, this led him to give a characterization of boundary traces that
was the starting point for the work of several authors, especially from
Russia.

Further remarkable results of Gagliardo are those on Sobolev spaces,
including some generalizations of a few inequalities due to Sobolev himslef.
Gagliardo obtained important results in Interpolation Theory. In
particular, he was the first to build an interpolation theory for Banach
spaces. Further, more recent results of Gagliardo are concerned with
integral transformations between Lebesgue spaces.

The results of Gagliardo show that he is a very original
analyst, and obviously put him amongst the best young Italian
mathematicians.

Emilio Gagliardo on Wikipedia (italian)



Caccioppoli Prize 1966 – Enrico Bombieri (University of Pisa)

Committee’s citation

Enrico Bombieri’s scientific production is mainly concerned with Number
theory, Algebraic geometry and Analytic number theory, and includes
results of the highest level. It demonstrates Bombieri’s abilities to deal
with very modern tools of different kinds used to solve very difficult and
fundamental problems. His results have obtained clear international
recognition. Beside a few papers mainly dealing with the Geometry of
numbers, we mention: a group of papers devoted to Selberg’s formulas for
the estimation of rests in the “Prinzahlsatz” and their generalization to
a class of arithmetic functions, according to S.A Amitsur’s viewpoints;
yet, we mention a paper about Tu. V. Linnik’s “large sieve”. This last
one, in which a very ingenious Abel-type inequality on double sums is
employed, eventually leads to a theorem that in many applications can be
used to replace the extended Riemann hypothesis (see for instance a paper
on small differences of consecutive prime numbers). In the field of
Analytic functions, beside a few papers on meromorphic functions, he
mainly worked on the Bierberbach conjecture on univalent functions.

As for his work in Algebraic geometry, we mention two particularly
remarkable papers: a first one dealing with the exponential sums related
to the L-function introduced by Artin and Schreirer, and another one
(written with Swinnerton-Dyer) on the Artin’s Z-function of the cubic
hypersurface in the four dimensional projective space. For this case it is
possible to verify the validity of the Riemann-Weil’s conjecture.

Enrico Bombieri on MacTutor History of Mathematics archive

<!- ————————- 1968 —->


Caccioppoli Prize 1968 – Mario Miranda (University of Pisa)


Mario Miranda

photo courtesy of MFO

Committee’s citation

Mario Miranda’s work, building on previous DeGiorgi’s one, led to
fundamental contributions to the theory of minimal surfaces.



Caccioppoli Prize 1970 – Claudio Baiocchi (University of Pavia)

Committee’s citation

Baiocchi is a high level mathematician who obtained, in a short time,
important and original results in various fields of Analysis, and, in
particular, on abstract differential equations and their applications to
evolutionary problems and in the Interpolation Theory between Banach
spaces. He also gave contributions to the problem of determining when the
“strong” and “weak” extensions of a differential operator coincide.


Caccioppoli Prize 1974 – Alberto Tognoli (University of Pisa)


Alberto Tognoli

photo courtesy of MFO

Committee’s citation

In his wide scientific production Tognoli has systematically developed,
and has actually contributed to found, a modern theory of real analytic
spaces. This theory shows remarkable differences with that of complex
analytic spaces due to the appearance of completely new phenomena and
problems that Tognoli eventually overcame with his new methods. Amongst
his papers the one called “Su una congettura di Nash” stands out. In this
paper Tognoli shows how to give an answer to a fundamental approximation
problem by mean of cobordism theory methods.



Caccioppoli Prize 1978 – Enrico Giusti (University of Pisa)

Committee’s citation

After a few early papers on Schrödinger and wave equations, Giusti
devoted himself to the theory of nonlinear elliptic equations. His
research is mainly dealing with regularity problems for solutions to the
minimal surfaces equation, variational inequalities for minimal surfaces
with thin and discontinuous obstacles, partial regularity of solutions to
nonlinear elliptic systems. His result on the Bernstein problem, obtained
in collaboration with Bombieri and De Giorgi, is of exceptional
importance. His whole scientific production features truly remarkable
results and shows that Giusti gave important contributions to the
solution of difficult problems in nonlinear analysis.

Enrico Giusti on Wikipedia



Caccioppoli Prize 1982 – Antonio Ambrosetti (International School for Advanced Studies, Trieste)


Antonio Ambrosetti

photo courtesy of MFO

Committee’s citation

The by now very wide scientific production of Prof. Antonio Ambrosetti is
mainly concerned with the following topics:

  • Invertibility of maps between Banach spaces by mean of geometric methods
  • Critical point theory for functionals, i.e. “Calculus of Variations in
    the large”. As for this topic – which is one of those Ambrosetti mostly
    studied – it is possible to make a more detailed classification:

    1. Critical points problems on manifolds with boundary
    2. Eigenvalue problems for nonlinear operators and existence problems
      with infinitely many critical points
    3. Periodic solutions to Hamiltonian systems

    Such fields already gained contributions from several top
    researchers. After entering them Ambrosetti was able to solve difficult
    open problems (in particular, the one concerning the existence of periodic
    solutions with prescribed minimal period)

  • Nonlinear problems with discontinuities (a very promising field
    Ambrosetti recently devoted an interesting paper to)
  • Various issues motivated by applications: plasma equilibria, existence
    of vortices, plate equilibria

Professor Ambrosetti is a remarkably original researcher; when approaching
problems, rather than making use of massive technical tools he prefers
finding new and often surprising paths. At the origins of Ambrosetti’s
heuristic approaches we always find the attitude to catch the most basic
geometric and intuitive aspects which are typical of the Functional
Analysis “in the style of Caccioppoli”.

Antonio Ambrosetti on Wikipedia (italian)



Caccioppoli Prize 1986 – Corrado De Concini (La Sapienza University of Rome)

Committee’s citation

The committee proposes to award Prof. Corrado De Concini with the 1986
Caccioppoli prize for his contributions to the Theory of invariants, to
the Theory of algebraic groups, to Enumerative geometry, and, in
particular, for the solution (obtained with E. Arbarello) of the classical
Schottky’s problem.

Such works show that De Concini is a very high level mathematician, who
is able to use with great skills algebraic, topological and analytical
methods towards the solution of difficult problems.

Corrado De Concini on Wikipedia (italian)



Caccioppoli Prize 1990 – Gianni Dal Maso (International School for Advanced Studies, Trieste)

Committee’s citation

Gianni Dal Maso is a very high level mathematician and one of the most
noteworthy members of the Italian school of Calculus of Variations. His
whole scientific production, which already includes more than sixty
papers, stands at very high levels; some of its highlights include

  • The results on the asymptotic behavior of sequences of variational
    obstacle problems
  • The results on linear and quasilinear Dirichlet problems in variable
    domains with rough boundary (behavior of solutions with respect to the
    domain, determination of the asyptotic problem via the method of
    asymptotic capacities, Wiener criterion, 1985)
  • The results on functionals defined on the space of Special Functions of
    Bounded Variation (SBV) and related interesting applications to image segmentation
    problems


Caccioppoli Prize 1994 – Nicola Fusco (University of Naples)

Committee’s citation

Nicola Fusco is a refined mathematician who gave substantial contributions
to the Calculus of Variations. In his high level scientific production
some of the highlights are

  • The lower semicontinuity theorem for quasiconvex functionals with
    respect to the natural topology, that closes a research field many
    authors had previously contributed to
  • The partial regularity theorems, extending previously known results
    given by several different authors and elegantly avoiding some
    unnecessary assumptions which were usually considered in the literature
  • The recent papers on the regularity of minimizers of the Mumford-Shah
    functional, allowing to draw a complete picture of the image recognition
    problem

Nicola Fusco on Wikipedia



Caccioppoli Prize 1998 – Luigi Ambrosio (Scuola Normale Superiore di Pisa)

Committee’s citation

Luigi Ambrosio is a remarkable mathematician who gave substantial
contributions to the Calculus of Variations and to some aspects of the
Theory of partial differential equations. In his very high level
scientific production, he obtained particularly relevant results on the
following topics:

  • Existence, regularity and approximation problems in the theory of free
    discontinuity problems
  • Semicontinuity and relaxation for integral functionals
  • Singularities of convex functions
  • Curvature flow of manifolds in arbitrary codimension

Luigi Ambrosio on Wikipedia



Caccioppoli Prize 2002 – Giovanni Alberti (University of Pisa)

Committee’s citation

Giovanni Alberti is one of the most brilliant analysts of his generation.
He very soon revealed his talent with by proving a remarkable Lusin type
theorem for gradients, that eventually had important applications to
semicontinuity and relaxation problems for integral functionals. He
proved a deep structure theorem on the distributional derivative of BV
maps, which is nowadays known as Rank-one theorem, thereby solving in the
positive a conjecture of De Giorgi. He eventually studied subtle problems
in the theory of Gamma convergence and in the theory of multiple-scale
problems, and developed a general convergence theory for problems of
Ginzburg-Landau type in arbitrary dimension and codimension.

Giovanni Alberti on Wikipedia (italian)



Caccioppoli Prize 2006 – Andrea Malchiodi (International School for Advanced Studies, Trieste)

Committee’s citation

The work of Andrea Malchiodi is devoted to important problems of
variational nature, such as

  • The Yamabe problem and the scalar curvature problem
  • Conformal geometric problems for fourth order operators
  • Concentration phenomena for singular perturbation problems
  • Solitary waves for Schrödinger nonlinear equations and related
    semiclassical states

On such topics, Malchiodi obtained several remarkable and original
results, that appeared on some of the most prestigious Mathematics
journals, and that gained international recognition. In particular, it is
worth mentioning his prestigious papers on Q-curvature and those on the
existence of solutions to singular perturbation problems concentrating on
positive dimension sets. Such results gained a wide international success,
making him one of the most brilliant figures in the nowadays Mathematical
community.



Caccioppoli Prize 2010 – Giuseppe Mingione (University of Parma)

Committee’s citation

The vast scientific production of Giuseppe Mingione, which is entirely
focused on regularity problems for vectorial integral functionals and
nonlinear systems of partial differential equations of elliptic and
parabolic type, is rich of remarkable results, that have been obtained by
mean of absolutely innovative techniques, in turn based on an original and
sometimes surprising way of using Potential theory ideas in the nonlinear
setting.

Some of the highlights are

  • The singular sets estimates for minima of vectorial integral
    functionals, which close an entire set of problems that had remained open
    since the seventies
  • The boundary regularity results for non-linear systems of p-Laplacean type
  • The Calderón-Zygmund type results on the existence of higher order
    fractional derivatives of solutions to measure data problems

Above all, there must be mentioned two recent papers featuring pointwise
gradient estimates for solutions to quasilinear equations
via Riesz and Wolff potentials of the data. Such very general estimates
allow to obtain in a unified way several results proved in the last thirty
years by very different and peculiar techniques, and to fix some
borderline cases which were still remaining an open problem.



Caccioppoli Prize 2014 – Camillo De Lellis, (Università di Zurigo)

Committee’s citation

Camillo De Lellis è un matematico di grande talento e profondità che ha dato contributi importanti al Calcolo delle Variazioni, alla Teoria Geometrica della Misura, alla Fluidodinamica e alla teoria dei sistemi di leggi di conservazione. La sua si presenta come una figura di punta nel panorama internazionale dell’Analisi Matematica.

Nella sua produzione scientifica, tutta di altissimo livello e di ampio
spettro, spiccano

— I risultati, ottenuti con L. Székelyhidi, sull’esistenza di soluzioni anomale dell’equazione di Eulero. La loro costruzione, ottenuta con un originale uso dell’integrazione convessa di Gromov, si collega in modo importante alla teoria della turbolenza e permette di fare dei passi importanti verso la soluzione della congettura di Onsager.
— Una nuova dimostrazione della regolarità della regolarità delle soluzioni delle superficie minime in codimensione maggiore di uno, ottenuta con E. Spadaro.
— I risultati sui sistemi di leggi di conservazione. In particolare, quelli con L. Ambrosio e F. Bouchut sulla buona positura per il cosiddetto sistema di Keyfitz-Kranzer, e quelli con G. Crippa, in cui si sviluppa un approccio alternativo alla teoria di DiPerna-Lions per l’unicità delle equazioni di trasporto con coefficienti molto irregolari.

Camillo De Lellis su Wikipedia



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