• About SNS

    The Scuola Normale Superiore is a public institute for higher education that in its two centuries of life has earned itself a special place, both in Italy and abroad, a place characterised by merit, talent and scientific rigour.  Two types of course are available: the undergraduate course and the PhD course.The teaching activity is distributed among three academic structures: the Faculty of Humanities, the Faculty of Mathematical and Natural Sciences, placed in Pisa, and the Institute of Humanities and Social Sciences, located in Palazzo Strozzi in Florence. 

  • Admission

    The evaluation for entrance to the first year of the undergraduate course does not include the high school leaving certificate, and the bachelor's degree is not taken into consideration in the entrance examination for the fourth year course. For each PhD course, candidates’ level of competence, talent, motivations and aptitudes to scientific research will be assessed on the basis of their qualifications and research project and an interview.

  • Academics

    The Scuola Normale Superiore offers two types of course: the undergraduate course, leading to first and second level university degrees, and the PhD course, the international equivalent of the Italian Dottorato di ricerca.The teaching and research activity is distributed among three academic structures: the Faculty of Humanities, the Faculty of Mathematical and Natural Sciences, and the Institute of Humanities and Social Sciences.The first two academic structures, housed at the Pisa site, organize courses for both the  undergraduate course and the PhD course. The Institute of Humanities and Social Sciences, situated in Palazzo Strozzi in Florence, deals only with the PhD course.

  • Research

    A highly qualifying feature of the Normale way is the strong link between teaching and research that is a characteristic of both the undergraduate and the graduate programmes of the Scuola. The research structures of the two Faculties welcome students with a relevant study interest, enabling them to collaborate in a mature way with the activities of the researchers.

  • International

    The Scuola Normale is an institute of a decidedly international nature. Examinations for admission to the undergraduate degree course and for the PhD course are open to all citizens worldwide. A certain number of places on the PhD course are reserved for students from other countries. During the pre laurea and  post lauream teaching courses, study and research programmes are made available at overseas universities and research centres with which the Scuola forms an intense network of collaboration.  The doctorate course in particular is taught in a veritable graduate school in line with the highest international standards. 

Research Group in Calculus of Variations and Geometric Measure Theory

Research group in Calculus of Variations and Geometric Measure Theory

Techniques from measure theory, and particularly from geometric measure theory, have been applied to different problems in partial differential equations since (at least) the first regularity results for minimal hypersurfaces in dimension larger than two.

In more recent years, the development of Gamma-convergence, and in a different direction, the progress in the analysis of nonlinear geometric problems were based on a consistent and systematic use of geometric measure theory, epitomized by the theory of Cartesian currents, and by the theory of Young measures and H-measures (or semiclassical measures).

The research of this group will be focussed in the following directions: the first and most established one concerns the applications of geometric measure theory to the asymptotics of certain classes of geometric variational problems along the line of the Modica-Mortola result for semilinear equations of reaction-diffusion type (Allen-Cahn equation), and to similar approaches for complex-valued Ginzburg-Landau system.

On a somewhat different line, measure theoretic tools have been applied quite successfully in the last four years to classes of pde's which display intrinsically "bad" solutions - in the sense of non-regular - such as those arising in mass transport problems and first order transport equations associated to the flow of non-Lipschitz vectorfield a là Di Perna-Lions.

We point out in particular that the recent proof of L. Ambrosio of the uniqueness for BV vectorfields is based on very effective combinations of measure theoretic tools which had been developed in the past years for completely different problems. In these same areas, it has also become apparent that sharp counterexamples can sometimes be provided only through deep and non-trivial constructions from classical measure theory and real analysis.

Indeed, one of the general goals of this group is to try to bridge the gap between real analysis and "hard" geometric measure theory on the one side, and the pde's and calculus of variations on the other side, in the hope that the wealth of results and geometric constructions available in the former might shed some light on the problems of the latter.
A more specific goal of future research is understanding the optimal assumptions in the Ambrosio-Di Perna-Lions type of results: so far the regularity conditions on the vectorfields have always been expressed in terms of coefficients in certain functionals spaces, i.e., in terms of linear quantities. Yet there is no specific reason for this, and indeed there are plenty of examples which suggest that a completely different and more geometric perspective could be quite successful, specifically for a system of conservation laws. Notice that despite the remarkable progress of the last four years, the picture of existence results for optimal transport problems is also far from being complete, and we plan to address some of the very basic questions which are still unanswered.

Research areas

Calculus of Variations, Geometric Measure Theory,
Optimal Transport Theory, Harmonic maps, Evolution problems,
Analysis in Metric Spaces and in Carnot groups.

Meeting and seminars

They usually took place in the Collegio Puteano on thursday in the afternoon,
Some information about past seminars can be found here: http://gemethnes.sns.it/seminar/.
Seethe following link http://cvgmt.sns.it/ for more up-to-date informations.


Palazzo della Carovana
ph. 050 509255
fax 050 563513
e-mail: l.ambrosio@sns.it
Head of the group: Luigi Ambrosio