Presently the PhD programme in Geometry is run by the Mathematical Physcs sector and consists in three years of study and research. The main selection procedure consists in an entrance competition. . This is divided in two parts: in the written part applicants are asked to solve problems among a list given by the entrance committe. The oral exam consists of a discussion of the written test and in a brief exposition of the applicant's "Laurea" thesis (or any equivalent foreign title). In the final classification the applicant's scientific titles and references are also taken into account.

Foreign applicants may be admitted on the sole basis of their scientific backgrounds, publications and references, in a pre-selection procedure to be held by the end of April.

In the first year of their Ph. D. program, students attend both basic and more advanced courses. They are required to pass examinations up to obtain nine credits (a credit  is typically a cycle of 20 lectures) to be admitted to the second year. Examinations typically consist of a seminar-exam to be delivered in front of the other members of the Sector and/or some written excercises. A tutor (usually a junior member of the Faculty) is assigned to each of them, to help them to choose their curricula studiorum.

In the first months of their second year students start a research project, under the supervision of a staff member of the sector or of an external collaborator, in one of the research topics of the Sector. Second year students must submit a brief resume of the methods and perspectives of their research project and discuss it in the "Qualifying Examination". Students not passing that examination may be allowed to pursue a restricted research project, to be completed by the end of the Academic year. After submitting a thesis, they can obtain a degree called "Magister Philosophiae".

Students passing successfully the Qualifying Examination pursue their research program up to the end of their final year, their only duty being that of taking part in the scientific life of the sector, e.g.  attending the sector seminar activities.

Students obtain the Ph. D. degree in Mathematical Physics after submitting their Ph. D. thesis and defending it in front of a Ph. D. examination committee, whose members include SISSA staff members, external experts of the field and two external professors appointed by the Italian Ministry of Education and Research (MIUR). The Ph. D. degrees in Mathematical Physics awarded by  SISSA are by law equivalent to the Italian degree of "Dottore di Ricerca in Matematica".