Giovanni Alberti (mathematician)

Giovanni Alberti (born March 21, 1965) is an Italian mathematician who is active in the fields of calculus of variations, real analysis and geometric measure theory.

Giovanni Alberti
Born (1965-03-21) 21 March 1965
Ferrara, Italy
Nationality Italy
Alma materScuola Normale Superiore
Known forAlberti's rank-one theorem
AwardsCaccioppoli Prize (2002)
Scientific career
FieldsMathematics
InstitutionsUniversity of Pisa

Scientific activity

Alberti has studied at Scuola Normale Superiore under the guide of Giuseppe Buttazzo and Ennio De Giorgi; he is professor of mathematics at the University of Pisa. 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. 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 of a priori prescribed (small) measure.[1] The second asserts the rank-one property of the distributional derivatives of functions with bounded variation, thereby verifying a conjecture of De Giorgi.[2] This theorem has found several applications, as for instance in the Ambrosio's proof of an open problem posed by Di Perna and Lions concerning the well-posedness of the continuity equation involving BV vector fields.[3] 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 geometric measure theory; in particular, it makes use of the concept of tangent measure to another measure.[4][5] Subsequently, Alberti has given contributions to the study of various aspects of Ginzburg-Landau vortices and of the continuity equation.[6]

Recognition

Alberti has been awarded the Caccioppoli prize in 2002 and has been an invited speaker at the fourth European Congress of Mathematics.

References
  1. Alberti, Giovanni (1991). "A Lusin type theorem for gradients". Journal of Functional Analysis. 100: 110–118. doi:10.1016/0022-1236(91)90104-D.
  2. Alberti, Giovanni (1993). "Rank one property for derivatives of functions with bounded variation". Proceedings of the Royal Society of Edinburgh, Section A. 123 (2): 239–274. doi:10.1017/S030821050002566X.
  3. Ambrosio, Luigi (2004). "Transport equation and Cauchy problem for BV vector fields". Inventiones Mathematicae. 158 (2): 227–260. doi:10.1007/s00222-004-0367-2.
  4. "Alberti's rank-one Theorem". Encyclopedia of Mathematics. Retrieved June 12, 2013.
  5. De Lellis, Camillo (2008). "A Note on Alberti's Rank-One Theorem". Transport Equations and Multi-D Hyperbolic Conservation Laws. Lecture Notes of the Unione Matematica Italiana. 5. UMI Springer Lecture Notes in Mathematics. pp. 61–74. CiteSeerX 10.1.1.362.429. doi:10.1007/978-3-540-76781-7_2. ISBN 978-3-540-76780-0.
  6. "Caccioppoli prize citation". Italian Mathematical Union. Retrieved May 5, 2013.
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