Annamaria MIRANDA | GEOMETRY III
Annamaria MIRANDA GEOMETRY III
cod. 0512300009
GEOMETRY III
| 0512300009 | |
| DEPARTMENT OF MATHEMATICS | |
| EQF6 | |
| MATHEMATICS | |
| 2024/2025 |
| OBBLIGATORIO | |
| YEAR OF COURSE 2 | |
| YEAR OF DIDACTIC SYSTEM 2018 | |
| SPRING SEMESTER |
| SSD | CFU | HOURS | ACTIVITY | |
|---|---|---|---|---|
| MAT/03 | 7 | 56 | LESSONS |
| Exam | Date | Session | |
|---|---|---|---|
| GEOMETRIA III | 29/04/2025 - 16:00 | SESSIONE ORDINARIA |
| Objectives | |
|---|---|
| TRAINING GOALS GENERAL OBJECTIVE THE COURSE AIMS TO EXTEND THE PANORAMA OF GEOMETRIES AND STRUCTURES THAT CAN BE DEFINED ON A SET WITH AN INTRODUCTION TO THE THEORY OF TOPOLOGICAL SPACES, AND TO EXPLOIT THEIR STRONG EDUCATIONAL POTENTIALS, WHICH MANIFEST THEMSELVES ESPECIALLY WITH THE ACTIVATION OF PROCESSES OF ABSTRACTION AND GENERALIZATION. KNOWLEDGE AND UNDERSTANDING. THE COURSE AIMS TO DEVELOP IN THE STUDENT THE ABILITY TO KNOW HOW TO SOLVE A PROBLEM, INDEPENDENTLY PRODUCE A CONJECTURE, DEMONSTRATE, PRODUCE COUNTEREXAMPLES, DEFINE A CONCEPT, LOOKING AT THE PROBLEM WITH A "TOPOLOGICAL" EYE, AND THEREFORE ALSO WITH AN EYE OTHER THAN THE "EUCLIDEAN" ONE, WHICH IS POSSIBLE WITHIN A 'REVOLUTIONARY' GEOMETRY SUCH AS TOPOLOGY. MOREOVER, STARTING FROM A VISUO-SPATIAL REASONING, THE SUBJECT HELPS TO IDENTIFY THE KEY IDEA OF A PROOF, FAVORING INTUITION AS THE FIRST ESSENTIAL STEP FOR THE CONJECTURE AND RIGOROUS FORMALIZATION OF A PROOF. THE FIRST PART OF THE COURSE AIMS TO ACQUIRE THE FUNDAMENTAL CONCEPTS OF GENERAL TOPOLOGY, WHICH ARE ESSENTIAL WORKING TOOLS IN VARIOUS AREAS OF MATHEMATICS; IT AIMS TO PROVIDE AN IN-DEPTH KNOWLEDGE OF THE TOPOLOGICAL STRUCTURE AND THE FUNCTIONS THAT PRESERVE IT. THE INTRODUCTION OF TOPOLOGICAL SPACE IS FOLLOWED BY THE CONSTRUCTION OF NEW SPACES STARTING FROM GIVEN SPACES AND THE STUDY OF COMPACTNESS AND CONNECTION PROPERTIES. THE SECOND PART OF THE COURSE DEALS WITH THE INTRODUCTORY CONCEPTS AND RESULTS OF ALGEBRAIC TOPOLOGY, SUCH AS THE FUNDAMENTAL GROUP, AN ALGEBRAIC OBJECT THAT IS A TOPOLOGICAL INVARIANT; IT AIMS TO PROVIDE AN INTRODUCTION TO THE THEORY OF HOMOTOPY, THE DEFINITION OF THE FUNDAMENTAL GROUP AND THE CALCULATION OF THE FUNDAMENTAL GROUP OF SOME SPACES. THE DIDACTIC METHODS PUT INTO PLAY AIM TO INTEGRATE KNOWLEDGE, CURIOSITY, AUTONOMY, COOPERATION, PRODUCTION. STUDENTS WILL HAVE TO KNOW HOW NOT ONLY TO REPRODUCE BUT ALSO TO PRODUCE. IN ORDER TO IMPROVE THE APPLICATION AND INVENTION SKILLS IN THE DEMONSTRATION, SEVERAL PROBLEMS ARE PROPOSED THAT LAY THE FOUNDATIONS FOR THE CONSTRUCTION OF THE MATHEMATICAL MATURITY. ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING AT THE END OF THE COURSE, STUDENTS SHOULD HAVE ACQUIRED THE FOLLOWING SKILLS: •TO BE ABLE TO CLEARLY EXPLAIN CONCEPTS AND PROVE PROPOSITIONS OF TOPOLOGY AND ALGEBRAIC TOPOLOGY, HIGHLIGHTING THE PROOF STRATEGY AND POSSIBLE ALTERNATIVES •TO BE ABLE TO APPLY THEM APPROPRIATELY TO IDENTIFY RELATIONSHIPS BETWEEN PROPERTIES, TO CONJECTURE, TO PROVE NEW OR ALREADY KNOWN STATEMENTS IN A DIFFERENT WAY AUTONOMOUSLY OR IN GROUPS •KNOWING HOW TO SOLVE PROBLEMS •KNOWING, AND FEELING FREE TO, ALSO USE TOPOLOGICAL INVENTIVENESS IN SOLVING A PROBLEM IN A CONTINUOUS COMPARISON BETWEEN THE TOPOLOGICAL WORLD AND THE EUCLIDEAN WORLD, FAVORING THE DEVELOPMENT OF THE PROCESSES OF ABSTRACTION AND GENERALIZATION •KNOWING HOW TO GET OUT OF THE BOX BY GIVING FREE REIN TO ONE'S MATHEMATICAL CREATIVITY •TO BE ABLE TO CRITICALLY DISCUSS A PAPER CONCERNING THE TOPICS STUDIED (POSSIBLY PRESENTED BY A PEER OR BY CHAT GPT) •KNOWING HOW TO ARGUE THE ANSWERS •KNOWING HOW TO POSE A PROBLEM •TO BE ABLE TO PRODUCE NEW MATHEMATICAL OBJECTS WITHIN THE THEORY, AND NOT ONLY TO REPRODUCE THE CONTENTS OF THE COURSE. AUTONOMY OF JUDGEMENT THE COURSE AIMS TO PROMOTE STUDENTS' AUTONOMY OF JUDGMENT. AUTONOMY OF JUDGEMENT IS MANIFESTED BOTH IN THE PRODUCTION OF INDIVIDUAL WRITTEN PRODUCTIONS (HW) AND IN THOSE OF COLLECTIVE RESOLUTIONS (CW). IN THE FIRST CASE, THEY EXPRESS THEMSELVES BY ARGUING ABOUT THEIR OWN STRATEGIC CHOICES, IN THE SECOND CASE BY COMPARING THEIR OWN CHOICES WITH THOSE OF THEIR OWN GROUP AND THOSE OF THEIR OWN GROUP WITH THOSE OF THE OTHER GROUPS, IN CONSTRUCTIVE PUBLIC DISCUSSIONS, WHICH ARE PUT IN PLACE IN ORDER TO SELECT THE MOST SIGNIFICANT SOLUTIONS IN RELATION TO THE OBJECTIVES. COMMUNICATION SKILLS EACH SPECIFIC THEME OF THE DISCIPLINE IS DEALT WITH THROUGH LECTURES, GROUP ACTIVITIES (CW), INDIVIDUAL ACTIVITIES (HW), THROUGH A CONSTRUCTION CYCLE TO WHICH THE SPECIFIC CONTENT IS SUBJECT. AT THE END OF EACH ACTIVITY (CONSTRUCTION OF EXAMPLES, COUNTEREXAMPLES, PROOFS) OF HW OR CW, STUDENTS ARE INVITED TO ARGUE THEIR OWN ANSWERS, TO ANALYZE THE PRODUCTIONS AND ARGUMENTS OF THEIR CLASSMATES, TO COMPARE THEM WITH THEIR OWN IN ORDER TO CHOOSE THE MOST SIGNIFICANT ONES, FOR EXAMPLE THEY ARE INVOLVED IN THE SELECTION OF EXAMPLES THAT HIGHLIGHT THE ESSENTIAL CHARACTERISTICS OF A PROPERTY OR MORE ELEGANT DEMONSTRATION STRATEGIES, IN A CRITICAL DEBATE THAT STIMULATES AND BRINGS OUT THE FULL COMMUNICATION SKILLS. DISCUSSIONS ARE ALSO A STRONG STIMULUS TO DO AND ASK QUESTIONS AND TO EXPLORE NEW TOPICS INDEPENDENTLY. LEARNING ABILITY THE COURSE ALSO AIMS TO DEVELOP SKILLS THAT CAN BE TRANSFERRED TO OTHER CONTEXTS, SUCH AS METACOGNITION, ABSTRACTION, CRITICAL THINKING, CREATIVE THINKING, COMMUNICATION SKILLS, COLLABORATION, AND COOPERATION. THE LECTURES ARE SUPPORTED BY TWO TYPES OF ACTIVITIES, BOTH OF WHICH ARE ESSENTIAL IN ORDER TO FACILITATE THE ACHIEVEMENT OF DISCIPLINARY AND CROSS-CURRICULAR LEARNING OBJECTIVES, ONE INDIVIDUAL, THE OTHER COLLECTIVE. PERIODICALLY, ALTERNATELY, STUDENTS ARE INVOLVED IN ACTIVITIES THAT INVOLVE THE INDIVIDUAL PERFORMANCE AND DELIVERY OF A HOMEWORK AND IN COOPERATIVE WORKING ACTIVITIES ON PROBLEM SOLVING RANGING FROM THE CONSTRUCTION OF EXAMPLES TO THE DEMONSTRATION OF STATEMENTS. HOWORKING AND COOPERATIVE WORKING ACTIVITIES ACTIVELY INVOLVE THE STUDENT IN THE DEVELOPMENT OF PROCESSES THAT STIMULATE BOTH THE PRODUCTION OF SPECIFIC SKILLS AND TRANSFERABLE SKILLS, SUCH AS REFLECTION, CRITICAL SENSE, AND METACOGNITION, NECESSARY CONDITIONS FOR THE DEVELOPMENT OF MEANINGFUL LEARNING. AT THE END OF THE COURSE THE STUDENT WILL BE ABLE TO: - HAVE A SERENE AND PROACTIVE ATTITUDE TOWARD A PROBLEM OF ANY KIND - ORGANIZE YOUR STUDY AND KNOW HOW TO FRAME A TOPIC - DOING MATHEMATICS BY BUILDING NEW KNOWLEDGE ON ACQUIRED KNOWLEDGE - LOOK AT PROBLEMS CRITICALLY -ARGUE - UNDERSTAND MORE ADVANCED TEXTS - COMMUNICATE YOUR IDEAS - FEEL MORE AUTONOMOUS - SHOW YOUR CREATIVITY |
| Prerequisites | |
|---|---|
| BASIC CONCEPTS ACQUIRED IN THE PREVIOUS COURSES IN ANALYSIS, GEOMETRY AND ALGEBRA. PARTICULARLY USEFUL ARE THE FOLLOWING TOPICS: CONVERGENCE, CONTINUITY, ALGEBRAIC STRUCTURES. VECTOR SPACES. LINEAR EQUATIONS. PLANE AND SPACE AFFINITIES AND ISOMETRIES. |
| Contents | |
|---|---|
| I-GENERAL TOPOLOGY (35 HOURS): 1.TOPOLOGICAL SPACES. 2.CONTINUITY AND HOMEOMORPHISMS, TOPOLOGICAL INVARIANCE. 3.CONSTRUCTION OF NEW TOPOLOGICAL SPACES FROM OLD (SUBSPACE TOPOLOGY, PRODUCT TOPOLOGY, QUOTIENT TOPOLOGY). 4.SEPARATION PROPERTIES, COUNTABILITY PROPERTIES, COMPACTNESS, CONNECTEDNESS. II- INTRODUCTION TO ALGEBRAIC TOPOLOGY (21 HOURS): 6. HOMOTOPY 7. FUNDAMENTAL GROUP 8. CALCULATION OF THE FUNDAMENTAL GROUP OF THE CIRCLE AND SOME APPLICATION. EACH TOPIC LISTED WILL BE DEVELOPED IN 7 HOURS, 5 OF WHICH WILL BE THEORY AND 2 OF WORK ON PROBLEMS AND EXERCISES. |
| Teaching Methods | |
|---|---|
| THE TEACHING METHOD AIMS TO INTEGRATE KNOWLEDGE, CURIOSITY, AUTONOMY, COOPERATION, PRODUCTION. MANY PROBLEMS ARE PROPOSED TO IMPROVE APPLICATION AND INVENTION SKILLS IN THE DEMONSTRATION. STUDENTS WILL NEED TO KNOW NOT ONLY TO REPRODUCE BUT ALSO TO PRODUCE. THE COURSE IS ORGANIZED IN LECTURES AND PROBLEM SOLVING ACTIVITIES TO SUPPORT THE LEARNING OBJECTIVES AND SKILLS. LESSONS AIM TO REDUCE COMPLEXITY TO SIMPLICITY, TO MAKE ACCESSIBILE STANDARD DEMONSTRATIVE TECHNIQUES, TO APPRECIATE HOW SOME FUNDAMENTALS THEOREMS PERFORM SIMPLICITY. THE LESSONS ARE SUPPORTED BY TWO TYPES OF LEARNING ACTIVITIES, BOTH INDISPENSABLE IN ORDER TO FASTEN THE ACHIEVEMENT OF THE LEARNING GOALS, ONE INDIVIDUAL, THE OTHER ONE COLLECTIVE. PERIODICALLY, ACCORDING TO A PRECISE CALENDAR, IN AN ALTERNATING MANNER, THE STUDENTS ARE INVOLVED IN INDIVIDUAL HOMEWORKS AS WELL AS IN COOPERATIVE WORKING ACTIVITIES. |
| Verification of learning | |
|---|---|
| A FINAL EXAMINATION AIMS TO VALUE THE KNOWLEDGE OF THE ARGUMENTS TREATED IN THE COURSE, THE LEVEL OF UNDERSTANDING OF PERFORMED MATHEMATICAL APPROACHES, THE COMMUNICATION SKILLS, THE OPENING IN DISCUSSION, THE ORIGINALITY IN ARGUMENTATION AND THE INVENTION IN DEMONSTRATION. IT CONSISTS OF TWO STEPS: A SELECTIVE WRITTEN EXAMINATION LASTING TWO HOURS AND AN ORAL EXAMINATION. THE FIRST ONE CONSISTS OF SIMPLE EXERCISES AND OPEN QUESTIONS, WHILE THE SECOND ONE AIMS TO VALUE NOT ONLY THE ACQUIRED KNOWLEDGES BUT ALSO THE UNDERSTANDING LEVEL AND THE COMMUNICATIONS SKILLS. THE FINAL MARK WILL BE OBTAINED FROM AN APPROXIMATED GRADE AVERAGE. PASSING THE EXAM WITH A SCORE OF AT LEAST 18/30 IN BOTH THE WRITTEN AND ORAL TESTS WILL BE ACHIEVED BY STUDENTS WHO WILL DEMONSTRATE THAT THEY HAVE ACQUIRED THE MINIMUM THEORETICAL AND PRACTICAL KNOWLEDGE, WITH REFERENCE TO THE PRIORITY CONTENTS ENCOURAGING SIGNIFICANT LEARNING IN THE CONTINUATION OF THE ACADEMIC PATH. FULL MARKS WILL BE GIVEN TO THE STUDENTS ABLE TO APPLY WITH ORIGINALITY THE ACQUIRED KNOWLEDGES. |
| Texts | |
|---|---|
| [1] V.CHECCUCCI, A.TOGNOLI, E.VESENTINI -"LEZIONI DI TOPOLOGIA GENERALE"- FELTRINELLI [2] R.ENGELKING -"GENERAL TOPOLOGY"- HELDERMANN VERLAG [3] A. RUSSO - "TOPOLOGIA GENERALE. SPAZI TOPOLOGICI, GRUPPO FONDAMENTALE, OMOLOGIA SINGOLARE "- ARACNE [4] E. SERNESI - GEOMETRIA 1 BOLLATI-BORINGHIERI 2000. [5] S. WILLARD - GENERAL TOPOLOGY, ADDISON-WESLEY |
| More Information | |
|---|---|
| E-MAIL ADDRESS: AMIRANDA@UNISA.IT |
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