Luca VITAGLIANO | ADVANCED GEOMETRY

cod. 0522200040

ADVANCED GEOMETRY

	0522200040
	DEPARTMENT OF MATHEMATICS
	EQF7
	MATHEMATICS
	2024/2025

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
	LUCA VITAGLIANO Curriculum

EXAMS

Exam	Date	Session
GEOMETRIA SUPERIORE	07/01/2025 - 10:00	SESSIONE DI RECUPERO
GEOMETRIA SUPERIORE	27/01/2025 - 10:00	SESSIONE DI RECUPERO
GEOMETRIA SUPERIORE	18/02/2025 - 10:00	SESSIONE DI RECUPERO

	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 (12 HOURS) 2. LIE ALGEBRAS WITH EXAMPLES (8 HOURS) 3. RELATION BETWEEN LIE GROUPS AND LIE ALGEBRAS (12 HOURS) 4. ELEMENTS OF SYMPLECTIC GEOMETRY (16 HOURS)

	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 FINAL TESTING HAS THE PURPOSE OF ASSERTING THE LEARNING OF THE THEORY ILLUSTRATED DURING THE TEACHING, THE UNDERSTANDING OF ITS ROLE IN THE LANDSCAPE OF CONTEMPORARY MATHEMATICS, AS WELL AS THE STUDENT'S ABILITY TO APPLY IT TO THE SOLUTION OF SIMPLE EXERCISES, EVEN IN ANALYTICAL AND PHYSICAL-MATHEMATICAL FIELD. THE EXAM WILL CONSIST OF TWO TESTS: 1. AN ORAL DISCUSSION OF THE PROPOSED "HOMEWORK", 2. AN ORAL EXAM. THE ORAL EXAM CAN BE CARRIED OUT IN THE FORM OF AN INTERVIEW ON THE COURSE TOPICS OR IN THE FORM OF A SEMINAR ON OPTIONAL ADVANCED TOPICS IF THE PROPOSED "HOMEWORKS" HAVE BEEN SUCCESSFULLY SOLVED AND DISCUSSED. THE TWO TESTS WILL TAKE PLACE IN TWO SEPARATE SESSIONS. IN THE FINAL ASSESSMENT, EXPRESSED IN THIRTIETHS, THE EVALUATION OF TEST 1 WILL WEIGH FOR 20% WHILE THE ORAL INTERVIEW FOR THE REMAINING 80%. THE MINIMUM GRADE FOR PASSING (18/30) IS GUARANTEED TO THE STUDENT WHO DEMONSTRATES KNOWLEDGE OF THE FUNDAMENTAL CONCEPTS (DEFINITIONS) AND HAS UNDERSTOOD THEM THROUGH THE CONSTRUCTION OF EXAMPLES. PRAISE MAY BE AWARDED TO STUDENTS WHO DEMONSTRATE THEY ARE ABLE TO INDEPENDENTLY APPLY KNOWLEDGE AND SKILLS ACQUIRED EVEN IN CONTEXTS DIFFERENT THAN THOSE PROPOSED IN LESSONS.

Verification of learning

THE FINAL TESTING HAS THE PURPOSE OF ASSERTING THE LEARNING OF THE THEORY ILLUSTRATED DURING THE TEACHING, THE UNDERSTANDING OF ITS ROLE IN THE LANDSCAPE OF CONTEMPORARY MATHEMATICS, AS WELL AS THE STUDENT'S ABILITY TO APPLY IT TO THE SOLUTION OF SIMPLE EXERCISES, EVEN IN ANALYTICAL AND PHYSICAL-MATHEMATICAL FIELD. THE EXAM WILL CONSIST OF TWO TESTS:
1. AN ORAL DISCUSSION OF THE PROPOSED "HOMEWORK",
2. AN ORAL EXAM.
THE ORAL EXAM CAN BE CARRIED OUT IN THE FORM OF AN INTERVIEW ON THE COURSE TOPICS OR IN THE FORM OF A SEMINAR ON OPTIONAL ADVANCED TOPICS IF THE PROPOSED "HOMEWORKS" HAVE BEEN SUCCESSFULLY SOLVED AND DISCUSSED. THE TWO TESTS WILL TAKE PLACE IN TWO SEPARATE SESSIONS. IN THE FINAL ASSESSMENT, EXPRESSED IN THIRTIETHS, THE EVALUATION OF TEST 1 WILL WEIGH FOR 20% WHILE THE ORAL INTERVIEW FOR THE REMAINING 80%.
THE MINIMUM GRADE FOR PASSING (18/30) IS GUARANTEED TO THE STUDENT WHO DEMONSTRATES KNOWLEDGE OF THE FUNDAMENTAL CONCEPTS (DEFINITIONS) AND HAS UNDERSTOOD THEM THROUGH THE CONSTRUCTION OF EXAMPLES.
PRAISE MAY BE AWARDED TO STUDENTS WHO DEMONSTRATE THEY ARE ABLE TO INDEPENDENTLY APPLY KNOWLEDGE AND SKILLS ACQUIRED EVEN IN CONTEXTS DIFFERENT THAN THOSE PROPOSED IN LESSONS.

	Texts
	LECTURE NOTES PROVIDED BY THE INSTRUCTOR. AS FURTHER READINGS ONE CAN CONSULT: - KIRILLOV, AN INTRODUCTION TO LIE GROUPS AND LIE ALGEBRAS, CAMBRIDGE UNIVERSITY PRESS, - BERNDT, AN INTRODUCTION TO SYMPLECTIC GEOMETRY, AMS

	More Information
	CLASSES ARE GIVEN IN ITALIAN. ADDITIONAL INFORMATION ABOUT THIS COURSE CAN BE FOUND AT THE ADDRESSES: - HTTPS://DOCENTI.UNISA.IT/039256/RISORSE - HTTPS://SITES.GOOGLE.COM/VIEW/AGTORTORELLA/TEACHING/GEOMETRIA-SUPERIORE

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Luca VITAGLIANO | ADVANCED GEOMETRY