Gabriele Bianchi

Biography

Sono nato a Prato il 15 febbraio 1960. Mi sono laureato in Matematica nel 1984 all'Universita' di Firenze, con una tesi di geometria convessa, sotto la direzione del Prof. Carlo Pucci. Dal 1986 al 1993 ho lavorato come ricercatore all' Istituto di Analisi Globale ed Applicazioni del CNR a Firenze. Nei due anni 1989-1990 ho svolto ricerche presso la School of Mathematics dell'Universita' del Minnesota a Minneapolis. Nel 1993 sono diventato Professore Associato, risultando vincitore di un concorso nazionale. Ho insegnato dal Novembre 1993 all'Ottobre 2001 presso la Facoltà di Scienze dell'Università di Ferrara, e successivamente presso la Facoltà di Agraria dell'Universita' di Firenze. Nel 2010 ho preso l'idoneita' di Professore Ordinario in un concorso svoltosi presso il Politecnico di Torino. Sono entrato in servizio in tale ruolo presso l'Universita' di Firenze alla fine del 2012. I was born in Prato, Italy, on February 15th, 1960. I graduated in mathematics in 1984, under the guidance of Prof. Carlo Pucci, discussing a thesis in convex geometry. In 1986 I become researcher in Istituto di Analisi Globale ed Applicazioni , an institute in Florence of the Italian National Research Council. I spent the two years 1989, 1990 in the Department of Mathematics of the University of Minnesota at Minneapolis, as a visiting scholar. I became associate professor in 1993. I have been teaching at the University of Ferrara from 1993 to 2001, and at the University of Florence after since 2001. I became Full Professor at the end of 2012.

Research Interest

Convex Geometry: To each regular compact set K in R^n one can associate a function gK, called covariogram, or set covariance. It can be defined as the function that to each x in R^n associates the volume of the intersection of K with its translate K+x. It coincides with the autocorrelation of the set, that is with 1K*1-K, ( 1K denotes the characteristic function of K). Does the knowledge of gK determine K (Covariogram problem, G. Matheron 1986, R. Adler and R. Pyke, 1991 )? If gK is radially symmetric, has K the same symmetry? Can one understand from gK whether K is convex? These questions are of interest also in stochastic geometry, the Covariogram problem is an example of a Phase Retrieval problem, and is relevant also in X-ray crystallography, in the determination of certain materials, the "quasi-crystals", from their diffraction image. study of properties of sequences of Steiner symmetrals of a compact set study of convex tomography Partial Differential Equations. Problems of existence, non-existence and symmetry of solutions of some elliptic semilinear PDE which arise in differential geometry. An example is the scalar curvature problem (or Kazdan Warner problem), where one tries to understand for which functions K(x) there exists a metric on Sn conformal to the standard one and whose scalar curvature is K(x). This problem reduces to the problem of the existence of a solution to an elliptic equation on Rn with a nonlinearity related to the so-called ''critical exponent''. A positive answer to a question asked by H. Brezis e E. Lieb regarding the existence of a "remainder term" in the Sobolev inequality associated to the immersion of D^{1,2}(R^n) in L^{2n/(n-2)}(R^n): the difference between the L^2 norm of the gradient of a function u and its L^{2n/(n-2)} norm can be bounded from below by a positive constant times the distance of u from the manifold of the functions for which the Sobolev inequality holds with equality. Alcune pubblicazioni scelte/ Some selected publications G. Bianchi, M. Longinetti, Reconstructing plane sets from Projections, Discrete and Computational Geometry 5 (1990), 223-242. (file .pdf) G. Bianchi, H. Egnell, A note on the Sobolev inequality, Journal of Functional Analysis 100 (1991), 18-24. (file .pdf) G. Bianchi, H. Egnell, An ODE approach to the equation \Delta u+K u(n+2)/(n-2) =0 in Rn, Mathematische Zeitschrift 210(1992),137-166. (file .pdf) G. Bianchi-H. Egnell, A Variational approach to the equation \Delta u+K u(n+2)/(n-2) =0 in Rn, Arch. Rational Mech. Anal. 122 (1993), 159-182 (file .pdf) G. Bianchi- A. Colesanti- C. Pucci, On the second differentiability of convex surfaces, Geometriae Dedicata 60(1996), 39-48 (file .pdf) G. Bianchi, Non-existence and symmetry of solutions to the scalar curvature equation, Comm. Part. Diff. Equat. , 21(1996), 229-234. (file .pdf) G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on Rn or Rn+ through the method of moving planes, Comm. Part. Diff. Equat. 22(1997), 1671-1690. (file .pdf) G. Bianchi, P. Gronchi, Steiner symmetrals and their distance from a ball, Israel J. Math. 181 (2003), 181--193. (file .pdf) G. Bianchi, Matheron's Conjecture for the Covariogram Problem, J. London Math. Soc. 71 (2005), 203 - 220. (file .pdf) G. Averkov and G. Bianchi, Confirmation of Matheron’s Conjecture on the covariogram of a planar convex body, Journal of the European Mathematical Society 11 (2009), 1187-1202; arXiv:0711.0572 [math.MG]. G. Bianchi, The covariogram determines three-dimensional convex polytopes, Advances in Mathematics 220 (2009), 1771--1808; arXiv:0805.1605v1 [math.MG]. G. Bianchi, R. J. Gardner and M. Kiderlen, Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms, Journal of the American Mathematical Society, 24 (2011), 293-343; arXiv:1003:4486[math.MG] G. Bianchi, A. Burchard, P. Gronchi, A. Volcic, Convergence in shape of Steiner symmetrizations, Indiana University Mathematical Journal 61 (2012), 1695-1710;, arXiv:1206.2041[math.MG] G. Averkov and G. Bianchi, Covariograms generated by valuations, International Mathematics Research Notices 2015 (2015), 9277-9329; doi: 10.1093/imrn/rnu219; arXiv:1307.1529 [math.MG]. G. Bianchi, The covariogram and Fourier-Laplace transform in $C^n$, Proceedings of the London Mathematical Society 113 (2016), 1-23, doi: 10.1112/plms/pdw020 ; arXiv:1312.7816 [math.MG]. G. Bianchi, M. Kelly A Fourier analytic proof of the Blaschke--Santalò inequality, Proceedings of the American Mathematical Society 143 (2015), 4901-4912,; doi: 10.1090/proc/12785 arXiv:1312.0244v3 [math.MG]. G. Bianchi, R.J. Gardner, P. Gronchi, Symmetrization in Geometry , Advances in Mathematics 36 (2017), 51-88, doi: 10.1016/j.aim.2016.10.003 ; arXiv:1603.00643 [math.MG].