## Presentation

The PhD program in Mathematical Sciences comprises seven Curricula:

- Algebra, Number Theory, Logic and History of Mathematics
- Geometry (algebraic, complex, differential) and Topology
- Analysis (functional, harmonic, complex) and PDE
- Mathematical Physics, Dynamical Systems and Biomathematics
- Probability, Statistics, Mathematical Finance and Operational Research
- Numerical Analysis, Scientific Computing and Mathematical Modelling
- Mathematical Aspects of Computer Science and Technology

Furthermore, in the framework of the PhD program, we have a joint doctorate, such as ALGANT DOC with a strong emphasis on Algebra, Geometry and Number Theory, that received the Erasmus Mundus label (2010).

The goal of the PhD in Mathematical Sciences (Dottorato in Scienze Matematiche) is to provide the graduate students with research methodology of modern Mathematics, both in the fields of pure and applied mathematics, to the point of being able to improve problem solving techniques and to become independent (both scientifically and culturally) in order to produce original and significant results. Since the Doctoral Thesis is the main test of having achieved research maturity, we believe that in a three year program the main attention is to be devoted to the research work toward the completion of the Thesis.

Generalities

Once the enrolment is formally completed, each graduate student will be assigned to a tutor. The tutor will guide the student typically until he/she will choose the thesis advisor.

During the first year, the program consists of attending courses that require a final exam. The choice of the courses will be made by the student together with his/her tutor, according to the personal interests and background. During the first year, each student will be required to attend a total of 90 hours, divided in courses of 30 or 15 hours, for a total of 90 credit hours. Some of these courses could be attended at other institutions, according to a course schedule that will be defined at the beginning of each academic year, avoiding superpositions with the courses taught at the Math Department in Milano.

At the end of the first year each student is expected to have successfully completed all of his/her required courses and have chosen the thesis advisor.

In the following two years, free from the obligation of specifically attending courses, the gradutate student should concentrate the effort in developing knowledge, methodologies in the chosen area of research, possibly attending workshops, seminars, conferences even at the international level, under the guide of the thesis advisor. This time the student is considered to be just sufficient time to be able to obtain original results. The student activity will be periodically monitored by our board, through the help of the advisor and/or the tutor.

All the doctoral thesis will be evaluated by independent referees, typically from foreign well-known university, before the graduate student being admited to the final exam.

For the whole duration of the program, the graduate students will have the opportunity of widen their mathematical knowledge by attending the conferences organized by the Seminario Matematico e Fisico di Milano, and the Lezioni Leonardesche, lectures given by mathematicians of the highest level and prestige.

Organization

The dean of the PhD program is the Coordinator (Coordinatore) who typically directs the activity of the program for a period of three cycles. The Coordinator is the chief administrative officer of the PhD program. The program is organized into three year long Cycles (Cicli) in which new students enter the program and under normal progress exit the program having completed their Dissertation (PhD Thesis). The Coordinator is assisted by the Doctoral Secretary (Segreteria di Dottorato) and the board of the Graduate Faculty (Collegio Docenti) who, together with the Coordinator, are responsible for the functioning of the program and the general supervision of the doctoral students (including a regular monitoring of the progress of each student)

Courses

Doctoral Candidates (Dottorandi) of the PhD in Mathematical Sciences are required to complete an individual course of study (Piano di Studio) in the first year of the program. The required courses are to be selected from the various courses offered during the academic year and/or other seminal activities which are organized in modules of 15 or 30 hours of lectures. A complete plan of study corresponds to 90 hours of lectures and it is issued by the Committee of Studies (Commissione Piano di Studi). Same Committee is appointing a Tutor.

After the first year of study, students are free from the responsibility of attending courses but are encouraged, nonetheless, to attend additional courses and/or seminars of interest to them or as recommended by their Thesis Supervisor (Relatore) and the Graduate Faculty (Collegio Docenti). This is particularly true of the seminars given by distinguished visiting mathematicians who contribute every year to the vitality of the PhD programs.

To this aim the PhD in Mathematical Sciences is supporting a Visiting Professors program.

Rules

Successful completion of the program is represented by the production of a Dissertation (Ph.D. Thesis) judged to be original and significant. The program is designed to create and stimulate an environment suitable for the formation of future research mathematicians. When the Thesis Supervisor (Relatore) together with the Doctoral Candidate (Dottorando) deem that thesis is ready for evaluation, the Graduate Faculty (Collegio Docenti) will solicit a referee report on the thesis from an internationally recognized scholar in the subject area of the thesis. On the basis of this report, a report by the Thesis Supervisor (Relatore) and a preliminary presentation of the thesis by the Student, the Graduate Faculty will make a decision on the admission of the candidate to the final Thesis Defence.

In addition to the successful completion of the Dissertation, students are required to complete an individual course of study (Piano di Studio) in the first year of the program. A complete plan of study corresponds to 90 hours of lectures and it is issued by the Committee of Studies (Commissione Piano di Studi). Same Committee is appointing a Tutor.

Each entering student is assigned a Tutor who follows closely the progress of an individual student, typically during the first year of the program, until the choice of a Thesis Supervisor (Relatore) is made. The Thesis Supervisor (Relatore), normally selected by the end of the first year of study, has then the responsibility to guide the student towards a successful completion of the Dissertation (PhD Thesis) making reports.