Educational objectives LEARNING RESULT - ACQUIRED KNOWLEDGE:
-know the basic concepts of the differential geometry of the differentiable manifolds of R ^ n and of the differential calculus on them.
-know the basic results on curves and surfaces.
-know the notion of abstract Riemannian surfaces and manifolds, and the two fundamental theorems of Gauss and Gauss-Bonnet.
-know the basic elements of spherical and hyperbolic geometry.
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Educational objectives General objectives: to develop the knowledge of Group Theory and Field Theory and acquire the knowledge of Galois Theory, with the natural applications to the solution of algebraic equations.
Specific objectives: at the end of the course the student will have acquired basic knowledge and results concerning Galois Theory.
Apply knowledge and understanding: the student will be able to solve problems of algebraic-combinational type requiring the use of techniques related to the presented topics.
Critical and judgment skills: at the end of the course the student will have the basis for conducting him towards more complex questions in the framework of modern Algebra.
Communication skills: the student will have the ability to communicate rigorously ideas and contents shown in the course.
Learning skills: the acquired knowledge will allow to carry on an autonomous study in a possible interdisciplinary context.
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Educational objectives Knowledge and understanding: Concerning equations and systems of ordinary differential equations (ODE) students will learn under which hypotheses solutions exist and when they can be given explicitly, how to investigate the properties of solutions and their maximal existence interval when they are not explicit.
The knowledge of proofs gives students a deep understanding of the subject and of the role of hypotheses.
Skills and attributes: Techniques for explicit determination of the solutions of the most common ODEs and of linear systems of differential equations with constant coefficients. Qualitative analysis of some systems of ODEs.
Applying knowledge and understanding: Students will be able to study the behavior of solutions of some simple ODEs in different application fields.
Communication skills: In view of the oral exam students will be able to present a rigorous mathematical proof.
Learning skills: Having a complete understanding of the basic mathematical tools used in ODEs,
will help students to understand their generalizations for solving more advanced problems. The proof of the existence theorem will help in studying numerical methods for solving ODEs.
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Educational objectives General objectives: acquire knowledge in commutative algebra and algebraic number theory
Specific objectives
Knowledge and understanding
At the end of the course students will have acquired the main basic notions of Commutative Algebra,
concerning commutative rings with or without zero dividers, whole ring extensions, tensor products e
flatness, rings and Artinian and Noetherian modules, including the dimension theory for k-algebra
finitely generated, primary decomposition, Dedekind domains, ramifications, class number
Ability to apply knowledge and understanding
At the end of the course students will be able to apply the knowledge acquired competently
and thoughtful and solve simple problems that require the use of techniques related to Commutative Algebra and algebraic theory
of numbers.
Judgment autonomy
The student will have the basis for analyzing the analogies and relationships between the topics covered and
topics of Geometry or Algebraic Number Theory and will have an idea of how important these are
branches of Mathematics are also historically deeply linked.
Communication skills
• The student will be able to present the main theorems with their proofs in the context of
oral test and will be able to communicate the key ideas of Algebra to non-specialists
Commutative and algebraic number theory.
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Learning ability
The student will be able to put to use the topics of Commutative Algebra and algebraic theory of numbers learned in the numerous
mathematical contexts in which they are used, both in the context of the Master's degree courses, and in
a future research activity
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Educational objectives General objectives: to acquire basic knowledge in the theory of a complex variable.
Specific objectives:
Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to holomorphic and meromorphic functions, the problem of the calculation of residues and numerous applications, to the analysis of infinite products and some fundamental functions.
Apply knowledge and understanding: at the end of the course the student will be able to solve simple problems that require the use of techniques related to the complex variable; will be able to calculate improper integrals and to study some special examples of holomorphic and meromorphic functions.
Critical and judgmental skills: the student will have the bases to analyze the similarities and relationships between the topics covered and topics of analysis (acquired in the course of Analysis II) or geometry (which can be acquired in the course of Geometry II); will also acquire the tools that have historically led to the solution of classical problems.
Communication skills: ability to expose the contents in the oral part of the assessment and in any theoretical questions present in the written test.
Learning skills: the acquired knowledge will allow a study, individual or given in an LM course, related to more specialized aspects of the complex variable and the analytic number theory.
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Educational objectives Knowledge and understanding
Aim of the course is to provide basic notions and skills of Convex Analysis in finite dimensional spaces. Particular attention will be devoted to the analytical aspects of convexity and their use in geometric and optimization problems.
Application skills
To be able to deal with problems involving convexity, in particular: characterizations of convexity, convex inequalities, regularity and monotony properties of convex functions, subdifferentials, separation of convex sets, convex optimization.
Autonomy of judgment
To have essential tools for successive approach to functional analysis, partial differential equations, control theory and mathematical programming.
To be able to autonomously solve new problems, applying mathematical tools to phenomena or
processes to be encountered in University studies or subsequent working activities.
Communication skills
To know how to communicate using properly mathematical language.
Ability to learn
To be able to deepen autonomously some arguments introduced during the course.
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