Learning objectives

The syllabus of the second-cycle course requires students to acquire in-depth knowledge and methodologies relating to one or more specific areas of mathematics and to demonstrate their independent study skills through extensive preparation of the final paper, which constitutes almost a quarter of the overall workload.
The second-cycle degree course in Mathematics allows you to delve into the theoretical aspects of mathematics, to work in the field of mathematics applications in an application-modelling context and finally to acquire the fundamental knowledge for launching a career in teaching. Consequently, students must be able to: initiate research in a field of specialization; analyse and solve complex problems, including in applied contexts; understand problems and extract their substantive elements. In addition, they must be able to: present arguments and their conclusions in mathematical terms, clearly and accurately and in a manner appropriate to the intended audience, both orally and in writing; be familiar with the teaching and learning processes of mathematics.
The structure of the degree course provides for a large number of ECTS credits allocated to both characterising and related-integrative activities and free-choice courses. This makes it possible to diversify the course catalogue by offering different curricula, within which the student can deepen his or her knowledge and specialise in particular areas of mathematics.
The degree course includes some curricula of a more theoretical nature and some of a more applied nature. The proposed curricula, as well as any individual programmes that the Course Council may approve, have the characterising activities in common and differ in the choice of related-integrative activities and free-choice courses. The characterising activities include an appropriate number of ECTS credits for advanced theoretical training and an appropriate number of ECTS credits for modelling-application training. This choice makes it possible for students on the second-cycle degree in Mathematics to acquire a sound knowledge of the mathematical disciplines.
Within the applied curricula there are also course units provided by other degree course units that enable students to place the specific characterising skills of the class in a general scientific-technological, cultural, social and economic context.

Second-cycle degree graduates in Mathematics have

• a thorough knowledge of and are able to use advanced mathematical tools with a high degree of autonomy;
• they have acquired a level of understanding of the language, techniques and content of a broad spectrum of modern mathematical theories, such that they are able to elaborate original ideas and pursue personal research paths in specific contexts;
• they have adequate computer and IT skills, including knowledge of programming languages and specific software;
• they are able to read and understand advanced texts and research articles in Mathematics.

Graduates who have opted for the Teaching Programme

• have developed in-depth knowledge of the topics covered in their future teaching, both through conceptual reworking and through the historical framing of the contents.
• They have skills in the fields of Numerical Analysis and Mathematical Physics that are certainly able to teach the basic principles of these disciplines in secondary schools;
• they also have knowledge in the fields of Mathematical Analysis, Algebra and Geometry, from a higher point of view, and Statistics, having delved in particular into some important topics for a future secondary school Mathematics teacher;
• on the basis of the choices made for the study plan, they can acquire knowledge in the pedagogical field.

Graduates who have opted for the General Plan have excellent specialist skills in the fields

• of Algebra (fundamentals of group theory, rings, algebra and their representation theory, finite and infinite Galois theory);
• of Mathematical Analysis (theory of partial derivative equations of the elliptic type in Sobolev spaces);
• of Numerical Analysis (advanced knowledge of both the theoretical and algorithmic aspects of the subject);
• of Mathematical Physics (knowledge of the main equations of Mathematical Physics with assigned initial and/or edge data);
• of Geometry (differential geometry, complex geometry, complex analysis) and of Probability (stochastic processes at discrete times, martingale, limit theorems);
• depending on the choices made within the course catalogue, they may also acquire knowledge of mathematical-financial, physical and engineering subjects.

Second-cycle degree graduates in Mathematics are able

• to produce rigorous demonstrations of mathematical results, including original ones;
• to tackle and solve new and unfamiliar problems in various applied contexts of mathematics, understanding their nature and formulating solution proposals, also with the aid of advanced computer and computational tools;
• to propose and analyse mathematical models, even very elaborate ones, associated with concrete situations deriving from other disciplines (economics, physics, computer science, engineering), and to use these models to facilitate the study of the original situation.

Graduates who have opted for the Teaching Programme are also able to design flexible curricula adapted to the school context in which they will be working and know how to foster critical and conscious learning of mathematics, in order to facilitate the acquisition of mathematical skills by students.

The fundamental teaching tool used to attain knowledge and stimulate comprehension skills is frontal classes combined with tutorial sessions. Additional teaching tools used to achieve specific objectives include the computer labs. Seminar activities as well as computer-laboratory activities are also planned, aimed in particular at developing the ability to understand, tackle and solve problems.

The results are verified through:

• written and/or oral examinations at the end of the courses;
• development of projects;
• discussion of the final examination before the Graduation Committee.