Giovanni Alberti (born March 21, 1965) is an Italian

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

who is active in the fields of

calculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...

and

geometric measure theory In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfac ...

Scientific activity

Alberti has studied at

Scuola Normale Superiore The Scuola Normale Superiore (commonly known in Italy as "la Normale") is a public university in Pisa and Florence, Tuscany, Italy, currently attended by about 600 undergraduate and postgraduate (PhD) students. Together with the University of Pi ...

under the guide of Giuseppe Buttazzo and

Ennio De Giorgi Ennio De Giorgi (8 February 1928 – 25 October 1996) was an Italian mathematician who worked on partial differential equations and the foundations of mathematics. Mathematical work De Giorgi's first work was in geometric measure theory, on th ...

; he is professor of mathematics at the

University of Pisa The University of Pisa (, UniPi) is a public university, public research university in Pisa, Italy. Founded in 1343, it is one of the oldest universities in Europe. Together with Scuola Normale Superiore di Pisa and Sant'Anna School of Advanced S ...

. Alberti is mostly known for two remarkable theorems he proved at the beginning of his career, that eventually found applications in various branches of modern

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

. The first is a very general Lusin type theorem for gradients asserting that every Borel vector field can be realized as the gradient of a continuously differentiable function outside a

closed subset In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...

of a priori prescribed (small) measure. The second asserts the rank-one property of the distributional derivatives of functions with bounded variation, thereby verifying a conjecture of De Giorgi. This theorem has found several applications, as for instance in the Ambrosio's proof of an open problem posed by Di Perna and

Lions The lion (''Panthera leo'') is a large cat of the genus ''Panthera'', native to Sub-Saharan Africa and India. It has a muscular, broad-chested body; a short, rounded head; round ears; and a dark, hairy tuft at the tip of its tail. It is se ...

concerning the well-posedness of the

continuity equation A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity ...

involving BV vector fields. This result is nowadays commonly known as Alberti's rank-one theorem and its proof rests of a very delicate use of sophisticated tools from

; in particular, it makes use of the concept of tangent measure to another measure. Subsequently, Alberti has given contributions to the study of various aspects of Ginzburg-Landau vortices and of the continuity equation.

Recognitions

Alberti has been awarded the

Caccioppoli prize The Caccioppoli Prize is awarded by the Italian Mathematical Union to an Italian mathematician not exceeding the age of 38 who established a wide international reputation. The prize is entitled to the memory of the Italian mathematician Renato Cacc ...

in 2002 and has been an invited speaker at the fourth

European Congress of Mathematics The European Congress of Mathematics (ECM) is the second largest international conference of the mathematics community, after the International Congresses of Mathematicians (ICM). The ECM are held every four years and are timed precisely betwee ...

. He has also been awarded the Ordine del Cherubino by the

during the 2024 award ceremony.

References

External links

Website at the University of Pisa

Fourth European Congress of Mathematics
{{DEFAULTSORT:Alberti, Giobanni 1965 births Living people 20th-century Italian mathematicians 21st-century Italian mathematicians Academic staff of the University of Pisa Scuola Normale Superiore di Pisa alumni