Learning objectives

MATHEMATICS
Graduates in Mathematics have an excellent knowledge of the fundamentals:

• of mathematical analysis (differential and integral calculus in one and several variables, theory of ordinary differential equations);
• of algebra (linear algebra, fundamental algebraic structures);
• of Euclidean affine and projective geometry, general topology, theory of curves and surfaces;
• of Mathematical Physics.

They also know some fundamentals

• of functional analysis;
• of the theory of holomorphic functions of a complex variable;
• of differential geometry and cohomology of varieties;
• of probability and statistics.

and have an adequate knowledge of the fundamental methods of Numerical Analysis;

Furthermore,

• they know and understand the basic applications of Mathematics to Physics and Computer Science;
• they are able to read and understand even advanced texts in Mathematics, and to consult research articles in Mathematics;
• they are able to construct and develop arguments in Mathematics with a clear identification of assumptions and conclusions.

Graduates in mathematics are able to:

• produce rigorous demonstrations of mathematical results that are not identical to, but clearly related to, those already known;
• solve problems of moderate difficulty in various fields of mathematics by taking advantage of symbolic formulation;
• formalise and model problems in mathematical form, analyse them and solve them using mathematical methods;
• use computational tools both to support mathematical processes and to acquire further information.

PHYSICS AND COMPUTERS
Graduates in Mathematics:

• know the fundamentals of programming;
• know and understand the basic applications of Mathematics to Physics and Computer Science;
• have adequate computer skills, including programming languages and specific software;
• know the theoretical foundations of classical Physics.

Graduates in mathematics are able to:

• use computer and computational tools both to support mathematical processes and to acquire further information;
• understand possible connections between different areas and subjects in mathematics and between these and the fields of other disciplines;
• evaluate and understand the formulation of elementary physical models.

The fundamental tool for the development of students' knowledge and skills is classroom lectures combined with exercise and/or laboratory sessions. This appears to be the most effective way to convey the specific contents of the individual disciplines and their mutual relationships. Exercises and workshops are essential to acquire the ability to work with knowledge and not limit oneself to purely mnemonic repetition, which makes mathematics uninteresting and boring. The repeated proposal of exercises to be carried out individually or with group activities encourages the acquisition of greater autonomy in study and the ability to independently assess the correctness of one's own work. The use of computer tools in laboratory activities, both within courses in the computer science and mathematics fields, enables the student to acquire specific skills and considerably broadens the ability to experiment independently with the applications of the knowledge acquired. The final tests of courses involving a workshop tend to test not only specific knowledge but also the ability to work independently or in small groups.
The preparation for the final examination aims to stimulate the student to come into direct contact with the mathematical literature, beyond the recommended texts or handouts used in the individual courses, and to sharpen the individual's ability to find his or her way around the consultation of texts and a scientific bibliography in both Italian and English.

The verification of learning outcomes takes place by means of:

• assessment tests during the courses;
• individual or group exercises proposed by the individual lecturers;
• written and/or oral examinations at the end of the individual courses;
• the discussion of the final examination before the Degree Committee.