Annamaria MIRANDA | TOPOLOGY
Annamaria MIRANDA TOPOLOGY
cod. 0512300036
TOPOLOGY
| 0512300036 | |
| DEPARTMENT OF MATHEMATICS | |
| EQF6 | |
| MATHEMATICS | |
| 2024/2025 |
| YEAR OF COURSE 3 | |
| YEAR OF DIDACTIC SYSTEM 2018 | |
| SPRING SEMESTER |
| SSD | CFU | HOURS | ACTIVITY | |
|---|---|---|---|---|
| MAT/03 | 6 | 48 | LESSONS |
| Exam | Date | Session | |
|---|---|---|---|
| TOPOLOGIA | 23/01/2025 - 11:00 | SESSIONE DI RECUPERO | |
| TOPOLOGIA | 11/02/2025 - 11:00 | SESSIONE DI RECUPERO | |
| TOPOLOGIA | 24/02/2025 - 11:00 | SESSIONE DI RECUPERO |
| Objectives | |
|---|---|
| GENERAL OBJECTIVE THE COURSE "TOPOLOGY" AIMS TO INTRODUCE STUDENTS TO ALGEBRAIC TOPOLOGY. ALGEBRAIC TOPOLOGY, WHICH STUDIES TOPOLOGICAL SPACES USING ALGEBRAIC INVARIANTS, AND IS ONE OF THE MOST IMPORTANT DISCIPLINES IN WHICH DIFFERENT FIELDS OF MATHEMATICS ARE INTERTWINED, HELPS TO REINFORCE THE BASIC TOOLS FOR THE WORK OF A GOOD MATHEMATICIAN. THE PRIMARY OBJECTIVE OF THE COURSE IS TO USE IT AS A TOOL FOR THE DEVELOPMENT OF MATHEMATICAL MATURITY. GENERAL OBJECTIVE KNOWLEDGE AND UNDERSTANDING THE MAIN AIM OF THE COURSE IS TO DETERMINE ALGEBRAIC INVARIANTS THAT ALLOW TO CLASSIFY TOPOLOGICAL SPACES UNDER HOMEOMORPHISMS. THIS CAN BE DONE BY THE FUNDAMENTAL GROUP OR BY THE HOMOLOGY AND COHOMOLOGY GROUPS. AT THE END OF THE COURSE, STUDENTS WILL HAVE TO •KNOW THE FUNDAMENTAL CONCEPTS OF ALGEBRAIC TOPOLOGY •BE ABLE TO EXPLAIN THESE CONCEPTS CLEARLY AND BE ABLE TO APPLY THEM APPROPRIATELY •TO BE ABLE TO USE ALGEBRAIC OBJECTS THAT ARE TOPOLOGICAL INVARIANTS. ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING DURING THE COURSE, PROBLEMS WILL BE PROPOSED ON WHICH TO REFLECT, AND MATHEMATICAL APPROACHES WILL BE OUTLINED IN ORDER TO IMPROVE THE SKILLS AND ABILITIES OF INVENTION AND PROOF. AT THE END OF THE COURSE, STUDENTS SHOULD HAVE ACQUIRED THE FOLLOWING SKILLS: •BE ABLE TO ANALYZE PROBLEMS, EXPLAIN CONCEPTS, AND PROVE PROPOSITIONS •BE ABLE TO DEMONSTRATE AN EFFICIENT USE OF THE PROPOSED TOPOLOGICAL TECHNIQUES, APPLYING THEM IN PROBLEM SOLVING •BE ABLE TO INDEPENDENTLY DEVELOP AND STRUCTURE A RESOURCE ON A NEW TOPIC THAT EXTENDS OR DEEPENS THE TOPICS PRESENTED DURING THE COURSE. AUTONOMY OF JUDGEMENT THE COURSE ALSO AIMS TO PROMOTE STUDENTS' AUTONOMY OF JUDGMENT. THE DEVELOPMENT OF CRITICAL THINKING, CREATIVE THINKING, AND AUTONOMOUS REASONING ARE AMONG THE TRANSVERSAL PROCESSES UNDERLYING THE PLANNED ACTIVITIES. STUDENTS, IN ADDITION TO BEING STIMULATED TO APPROACH LESSONS WITH A CRITICAL AND INTERACTIVE SENSE, ARE INVOLVED IN THE DESIGN OF RESOURCES FOR THEMSELVES AND FOR OTHER STUDENTS AND IN PEER REVIEW, AND THEREFORE STIMULATED TO MAKE DECISIONS, SELECT, EVALUATE, AND DISCERN IN LINE WITH THE OBJECTIVES OF THE PRODUCT THEY ARE CREATING. ALL THIS CONTRIBUTES TO MAKING THEM MORE RESPONSIBLE AND TO PROMOTING A GREAT BOOST TO THEIR AUTONOMY OF JUDGMENT. AUTONOMY OF JUDGEMENT THE PRODUCTION OF INDIVIDUAL OR GROUP WORKS ON TOPICS INTIMATELY RELATED TO THE COURSE CONTENTS AIMS TO SHOW THE ADVANTAGES OF USING THE TOPOLOGY AND TOOLS OF ABSTRACT ALGEBRA IN THE STUDY OF TOPOLOGICAL SPACES. A CONTINUOUS COMPARISON OF SOLUTION PATHS FINDS ITS MAXIMUM EXPRESSION IN GROUP DISCUSSIONS. WORKING IN GROUPS IN FRONT OF NON-PERMANENT VERTICAL WHITEBOARDS HELPS TO HIGHLIGHT THE ARGUMENTATIVE AND COMMUNICATIVE SKILLS OF EACH STUDENT. IN ADDITION, THE CREATION OF RESOURCES, DIGITAL OR NOT, FOR THE CLASS (INTERNAL) OR FOR OTHER CLASSES (EXTERNAL), HELPS TO ENHANCE BOTH COMMUNICATION SKILLS AND THE AUTONOMY OF JUDGMENT. 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, COOPERATION. THE LECTURES ARE SUPPORTED BY TWO TYPES OF ACTIVITIES, BOTH ESSENTIAL IN ORDER TO FACILITATE THE ACHIEVEMENT OF BOTH 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 THROUGH CONJECTURE. HOWORKING AND COOPERATIVE WORKING ACTIVITIES STIMULATE THE ACTIVATION OF PROCESSES THAT HELP THE STUDENT TO ACQUIRE BOTH 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 TOWARDS A PROBLEM OF ANY KIND - ORGANIZE YOUR STUDY AND KNOW HOW TO FRAME A TOPIC - DOING MATHEMATICS BY BUILDING NEW KNOWLEDGE ON ACQUIRED KNOWLEDGE -DISCOVER OR INVENT - APPROACH PROBLEMS CRITICALLY - UNDERSTAND MORE ADVANCED TEXTS - COMMUNICATE YOUR IDEAS - FEEL MORE AUTONOMOUS - SHOW YOUR CREATIVITY -ARGUE - REVIEW AND SELECT |
| Prerequisites | |
|---|---|
| PREVIOUS COURSES CONTAINING BASIC CONCEPTS ALGEBRA ARE PRESUPPOSED. MOREOVER FIRST BASIC NOTIONS IN GENERAL TOPOLOGY. ARE REQUIRED. |
| Contents | |
|---|---|
| 1. LOCAL COMPACTNESS AND COMPACTIFICATIONS. 2. COVERING MAPS 3. THE SEIFERT VAN-KAMPEN THEOREM 4. CLASSIFICATION OF CONNECTED AND COMPACT SURFACES. 5. CW-COMPLEXES 6. HOMOLOGY EACH TOPIC WILL BE DEVELOPED IN 8 HOURS, OF WHICH 6 ARE THEORY AND 2 ARE WORK ON PROBLEMS AND EXERCISES. |
| Teaching Methods | |
|---|---|
| TEACHING METHODS ARE BASED ON LESSONS, DISCUSSIONS AND INDIVIDUAL OR COLLECTIVE ACTIVITIES ORGANIZED TO SHOW THE ADVANTAGES OF USING TOPOLOGY AND TOOLS FROM ABSTRACT ALGEBRA TO STUDY TOPOLOGICAL SPACES. THE BASIC GOAL IS TO FIND ALGEBRAIC INVARIANTS THAT CLASSIFY TOPOLOGICAL SPACES UP TO HOMEOMORPHISM, THOUGH USUALLY MOST CLASSIFY UP TO HOMOTOPY EQUIVALENCE. STUDENTS APPRECIATE HOW FUNDAMENTALS RESULTS, SUCH AS THE FUNDAMENTAL ALGEBRA THEOREM, THE BROWER FIXED POINT THEOREM, ARE EASILY OBTAINED BY USING ALGEBRAIC CONCEPTS THAT ARE TOPOLOGICAL INVARIANTS. THE TEACHING METHODS ARE TARGETED AT INTEGRATING EXPERIENCE, KNOWLEDGE, LEARNING, CURIOSITY AND SKILLS. THE TEACHING METHODS AIM TO INTEGRATE KNOWLEDGE, CURIOSITY, AUTONOMY, COOPERATION, PRODUCTION. MANY PROBLEMS ARE PROPOSED TO IMPROVE APPLYING ABILITIES AND INVENTION SKILLS IN THE PROOF. |
| 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 AN ORAL EXAMINATION AIMING TO VALUE NOT ONLY THE ACQUIRED KNOWLEDGES BUT ALSO THE UNDERSTANDING LEVEL AND THE COMMUNICATIONS SKILLS AND, EVENTUALLY OF A SEMINAR AND/OR SOME HOMEWORKS ABOUT SIGNIFICANT PROBLEMS. THE PROFESSOR WILL VERIFY ALL THE GOALS REACHED BY THE STUDENT AND HE WILL EXPRESS THE STUDENT'S LEVEL BY AN OPPORTUNE GRADE. PASSING THE EXAM WITH A SCORE OF AT LEAST 18/30 MAY BE ACHIEVED BY STUDENTS WHO DEMONSTRATE THAT THEY HAVE ACQUIRED THE MINIMUM KNOWLEDGE AND SKILLS IN TERMS OF THEORETICAL AND PRACTICAL KNOWLEDGE, WITH PARTICULAR REFERENCE TO THE PRIORITY CONTENTS TO ENCOURAGE SIGNIFICANT LEARNING DURING THE CONTINUATION OF THE UNIVERSITY COURSE. 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 1989 [3] A. HATCHER, "ALGEBRAIC TOPOLOGY", CAMBRIDGE UNIVERSITY PRESS [4]C. KOSNIOWSKI -"INTRODUZIONE ALLA TOPOLOGIA ALGEBRICA" - ZANICHELLI. [5]J. M. LEE- "INTRODUCTION TO TOPOLOGICAL MANIFOLDS", SECOND EDITION, SPRINGER [6] W.S.MASSEY -" ALGEBRAIC TOPOLOGY: AN INTRODUCTION"- SPRINGER-VERLAG 1991. [7] .MUNKRES -" TOPOLOGY: "-SECOND EDITION PEARSON 2000. [8]S. WILLARD -"GENERAL TOPOLOGY"- ADDISON -WESLEY PUBLISHING COMPANY 1970. |
| More Information | |
|---|---|
| •E-MAIL ADDRESS OF THE PROFESSOR: AMIRANDA@UNISA.IT •WEBSITE ADDRESS OF THE PROFESSOR: HTTP://WWW.UNISA.IT/DOCENTI/ANNAMARIAMIRANDA/INDEX |
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