ALFONSO GIUSEPPE TORTORELLA | ADVANCED GEOMETRY

cod. 0522200040

ADVANCED GEOMETRY

	0522200040
	DEPARTMENT OF MATHEMATICS
	EQF7
	MATHEMATICS
	2023/2024

CURRICULUM GENERAL MATHEMATICS AND DIDACTICS OF MATHEMATICS

	YEAR OF COURSE 2
	YEAR OF DIDACTIC SYSTEM 2018
	SPRING SEMESTER

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/03	6	48	LESSONS	SUPPLEMENTARY COMPULSORY SUBJECTS

PROFESSORS

	ALFONSO GIUSEPPE TORTORELLA T Curriculum
	CHIARA ESPOSITO Curriculum

	Objectives
	THE COURSE ON HIGHER GEOMETRY AIMS AT INTRODUCING THE STUDENT TO LIE THEORY. IN PARTICULAR, BASIC DEFINITIONS OF LIE GROUPS AND LIE ALGEBRAS WILL BE DISCUSSED AND EXAMPLES WILL BE GIVEN. THE COURSE LIES AT AN INTERSECTION OF DIFFERENTIAL GEOMETRY AND ALGEBRA. THE MAIN OBJECTIVES OF THE COURSE "HIGHER GEOMETRY" ARE DESCRIBED AS FOLLOWS. - KNOWLEDGE AND UNDERSTANDING SKILLS: THE GOAL OF THIS COURSE WILL BE FOR STUDENTS TO ACHIEVE MASTERY IN THE USE OF TECHNIQUES RELATED TO LIE THEORY AND RELATED TOPICS. TO THIS END, THE COURSE WILL BE STRUCTURED BY FACE-TO-FACE LECTURES IN WHICH THE THEORETICAL FOUNDATIONS WILL BE DISCUSSED AND EXERCISE SESSIONS IN WHICH STUDENTS WILL DISCUSS THE EXERCISES. - APPLYING KNOWLEDGE AND UNDERSTANDING: THE STUDENT IS EXPECTED TO ACHIEVE A THOROUGH KNOWLEDGE OF THE CONCEPTS DISCUSSED AND AN ABILITY TO SOLVE USEFUL PROBLEMS IN MATHEMATICS AND PHYSICS. - AUTONOMY OF JUDGMENT: AMONG THE OBJECTIVES OF THE COURSE IS TO MAKE THE STUDENT AS AUTONOMOUS AS POSSIBLE, TO HELP HIM DEVELOP HIS CRITICAL SENSE AND THE ABILITY TO IDENTIFY FOR HIMSELF WHAT ARE IMPORTANT IDEAS AND TOPICS THAT WOULD MERIT FURTHER STUDY IN LATER STUDIES. - COMMUNICATION SKILLS: THE STUDENT WILL BE ABLE TO EXPOUND AND ARGUE THE SOLUTION OF EXERCISES; HE WILL ALSO BE ABLE TO DISCUSS AND CORRECTLY DEMONSTRATE THE MOST RELEVANT RESULTS RELATED TO LIE THEORY, THROUGH RIGOROUS LOGICAL-MATHEMATICAL LANGUAGE. - LEARNING SKILLS: THE COURSE AIMS TO SHARPEN THE STUDENT'S ANALYTICAL FACULTIES, HIS MENTAL FLEXIBILITY IN DEALING WITH NEW AND CHALLENGING CONCEPTS, PLACING THEM IN CONNECTION WITH THE ALGEBRAIC STRUCTURES LEARNED FROM THE COURSES OF THE BACHELOR'S DEGREE.

THE COURSE ON HIGHER GEOMETRY AIMS AT INTRODUCING THE STUDENT TO LIE THEORY.
IN PARTICULAR, BASIC DEFINITIONS OF LIE GROUPS AND LIE ALGEBRAS WILL BE DISCUSSED AND EXAMPLES WILL BE GIVEN.
THE COURSE LIES AT AN INTERSECTION OF DIFFERENTIAL GEOMETRY AND ALGEBRA.

THE MAIN OBJECTIVES OF THE COURSE "HIGHER GEOMETRY" ARE DESCRIBED AS FOLLOWS.

- KNOWLEDGE AND UNDERSTANDING SKILLS: THE GOAL OF THIS COURSE WILL BE FOR STUDENTS TO ACHIEVE MASTERY IN THE USE OF TECHNIQUES RELATED TO LIE THEORY AND RELATED TOPICS.
TO THIS END, THE COURSE WILL BE STRUCTURED BY FACE-TO-FACE LECTURES IN WHICH THE THEORETICAL FOUNDATIONS WILL BE DISCUSSED AND EXERCISE SESSIONS IN WHICH STUDENTS WILL DISCUSS THE EXERCISES.

- APPLYING KNOWLEDGE AND UNDERSTANDING: THE STUDENT IS EXPECTED TO ACHIEVE A THOROUGH KNOWLEDGE OF THE CONCEPTS DISCUSSED AND AN ABILITY TO SOLVE USEFUL PROBLEMS IN MATHEMATICS AND PHYSICS.

- AUTONOMY OF JUDGMENT: AMONG THE OBJECTIVES OF THE COURSE IS TO MAKE THE STUDENT AS AUTONOMOUS AS POSSIBLE, TO HELP HIM DEVELOP HIS CRITICAL SENSE AND THE ABILITY TO IDENTIFY FOR HIMSELF WHAT ARE IMPORTANT IDEAS AND TOPICS THAT WOULD MERIT FURTHER STUDY IN LATER STUDIES.

- COMMUNICATION SKILLS: THE STUDENT WILL BE ABLE TO EXPOUND AND ARGUE THE SOLUTION OF EXERCISES; HE WILL ALSO BE ABLE TO DISCUSS AND CORRECTLY DEMONSTRATE THE MOST RELEVANT RESULTS RELATED TO LIE THEORY, THROUGH RIGOROUS LOGICAL-MATHEMATICAL LANGUAGE.

- LEARNING SKILLS: THE COURSE AIMS TO SHARPEN THE STUDENT'S ANALYTICAL FACULTIES, HIS MENTAL FLEXIBILITY IN DEALING WITH NEW AND CHALLENGING CONCEPTS, PLACING THEM IN CONNECTION WITH THE ALGEBRAIC STRUCTURES LEARNED FROM THE COURSES OF THE BACHELOR'S DEGREE.

	Prerequisites
	THE COURSE ASSUMES FAMILIARITY WITH: - THE CONCEPTS OF LINEAR ALGEBRA ALREADY STUDIED IN THE THREE-YEAR DEGREE, AND WITH - THE CONCEPTS OF DIFFERENTIABLE VARIETY, TANGENT VECTOR FIELD, CARTAN CALCULUS ALREADY STUDIED IN A BASIC COURSE OF DIFFERENTIAL GEOMETRY, OF WHICH THIS COURSE IS A NATURAL EXTENSION AND CONTINUATION.

	Contents
	1. LIE GROUPS WITH EXAMPLES 2. LIE ALGEBRAS WITH EXAMPLES 3. RELATION BETWEEN LIE GROUPS AND LIE ALGEBRAS 4. ELEMENTARY REPRESENTATION THEORY 5. BASICS ON STRUCTURE THEORY OF LIE ALGEBRAS

	Teaching Methods
	THE TEACHING WILL TAKE PLACE PRIMARILY THROUGH FACE-TO-FACE LECTURES. HOWEVER, EXERCISES AND PROBLEMS WILL BE PROPOSED FOR THE STUDENT TO SOLVE IN THE CLASSROOM OR AS "HOMEWORK," IN ORDER TO PROMOTE AN "ACTIVE" (AND, THEREFORE, MORE EFFECTIVE) FORM OF LEARNING, AS WELL AS AUTONOMY OF JUDGMENT ON THE TOPICS OF COURSE.

	Verification of learning
	THE EXAM IS DESIGNED TO ASSESS THE KNOWLEDGE AND ABILITY TO UNDERSTAND THE CONCEPTS PRESENTED DURING THE LECTURES, AS WELL AS THE ABILITY TO APPLY THAT KNOWLEDGE IN THE STUDY OF LIE THEORY. THE EXAM CONSISTS OF AN ORAL TEST. THE STUDENT WILL BE EXPECTED TO KNOW THE NOTIONS OF LIE GROUP, LIE ALGEBRA AND THE MOST IMPORTANT RESULTS OF LIE THEORY. ALTERNATIVELY, IF THE STUDENT HAS DONE THE HOMEWORKS ASSIGNED DURING THE COURSE, THE EXAM WILL CONSIST OF AN APPROXIMATELY 60-MINUTE SEMINAR ON A MORE ADVANCED TOPIC NOT COVERED DURING THE LECTURES. THE FINAL GRADE WILL BE GIVEN USING A 30-POINT SCALE. HONORS MAY BE AWARDED TO STUDENTS WHO DEMONSTRATE THE ABILITY TO INDEPENDENTLY APPLY ACQUIRED KNOWLEDGE AND SKILLS IN CONTEXTS OTHER THAN THOSE PROPOSED IN CLASS AND, IN THE CASE OF THE SEMINAR, DEMONSTRATE THE ABILITY TO PREPARE A SEMINAR ON AN ADVANCED TOPIC INDEPENDENTLY.

Verification of learning

THE EXAM IS DESIGNED TO ASSESS THE KNOWLEDGE AND ABILITY TO UNDERSTAND THE CONCEPTS PRESENTED DURING THE LECTURES, AS WELL AS THE ABILITY TO APPLY THAT KNOWLEDGE IN THE STUDY OF LIE THEORY.

THE EXAM CONSISTS OF AN ORAL TEST. THE STUDENT WILL BE EXPECTED TO KNOW THE NOTIONS OF LIE GROUP, LIE ALGEBRA AND THE MOST IMPORTANT RESULTS OF LIE THEORY. ALTERNATIVELY, IF THE STUDENT HAS DONE THE HOMEWORKS ASSIGNED DURING THE COURSE, THE EXAM WILL CONSIST OF AN APPROXIMATELY 60-MINUTE SEMINAR ON A MORE ADVANCED TOPIC NOT COVERED DURING THE LECTURES.
THE FINAL GRADE WILL BE GIVEN USING A 30-POINT SCALE. HONORS MAY BE AWARDED TO STUDENTS WHO DEMONSTRATE THE ABILITY TO INDEPENDENTLY APPLY ACQUIRED KNOWLEDGE AND SKILLS IN CONTEXTS OTHER THAN THOSE PROPOSED IN CLASS AND, IN THE CASE OF THE SEMINAR, DEMONSTRATE THE ABILITY TO PREPARE A SEMINAR ON AN ADVANCED TOPIC INDEPENDENTLY.

	Texts
	LECTURE NOTES PROVIDED BY THE INSTRUCTOR.

	More Information
	CLASSES ARE GIVEN IN ITALIAN. ADDITIONAL INFORMATION ABOUT THIS COURSE CAN BE FOUND AT THE ADDRESSES: - HTTPS://DOCENTI.UNISA.IT/032483/RISORSE?CATEGORIA=335&RISORSA=10015 - HTTPS://SITES.GOOGLE.COM/VIEW/AGTORTORELLA/TEACHING/GEOMETRIA-SUPERIORE

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ALFONSO GIUSEPPE TORTORELLA | ADVANCED GEOMETRY