Matematica | GEOMETRY III

Matematica GEOMETRY III

 0512300009 DIPARTIMENTO DI MATEMATICA EQF6 MATHEMATICS 2021/2022

 OBBLIGATORIO YEAR OF COURSE 2 YEAR OF DIDACTIC SYSTEM 2018 SPRING SEMESTER
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/03 7 56 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 ANNAMARIA MIRANDA T
Objectives

TRAINING PURPOSES

THE COURSE "GEOMETRY III" AIMS TO INTRODUCE STUDENTS TO THE FUNDAMENTAL CONCEPTS OF GENERAL TOPOLOGY AND REAL PROJECTIVE PLANE GEOMETRY. ITS TRAINING PURPOSE, WORKING EFFICIENTLY TO DROPPING EUCLIDEAN PREJUDICES, IS TO INTERPRETE THE SAME GEOMETRIC REALITY BY USING THE "TOPOLOGICAL EYE" OR THE “ PROJECTIVE EYE”, BOTH FAR THE “EUCLIDEAN EYE” . THE COURSE AIMS TO EXTEND THE PANORAMA OF THE POSSIBLE GEOMETRIES THAT CAN BE DEFINED ON A SET, AND TO DEVELOP THE ABILITY TO KNOW HOW TO SOLVE A PROBLEM WITHIN A GEOMETRY, HOW TO PRODUCE A CONJECTURE, HOW TO PROVE, ALSO WITH AN EYE DIFFERENT FROM THE "EUCLIDEAN" ONE.THE TEACHING METHOD AIM 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.

-KNOWLEDGE AND UNDERSTANDING
THE FIRST PART IS CONCERNED WITH THE STUDY OF TOPOLOGICAL SPACES AND THEIR STRUCTURE-PRESERVING FUNCTIONS. THE STUDENTS WILL BE INTRODUCED TO THE GENERAL STRUCTURE OF A TOPOLOGICAL SPACE, THE CONSTRUCTION OF NEW TOPOLOGICAL SPACES FROM OLD, THE TOPOLOGICAL PROPERTIES INVARIANTS UNDER CONTINUOUS MAPPINGS, SUCH AS COMPACTNESS AND CONNECTION.
THE SECON PART OF THE COURSE DEALS WITH THE CONSTRUCTION OF MODELS OF PLANE GEOMETRIES OF ELLIPTIC TYPE SUCH AS THE REAL PROJECTIVE PLANE. TO THESE TOPICS THE STUDY OF CONICS WITH RELATIVE PROJECTIVE, AFFINE AND METRIC CLASSIFICATION IS ADDED.

ON SATISFYING THE REQUIREMENTS OF THIS COURSE, THE STUDENT WILL HAVE THE FOLLOWING:

- KNOWLEDGE AND SKILLS.

• TO UNDERSTAND THE FUNDAMENTAL IDEAS IN GENERAL TOPOLOGY.
• TO UNDERSTAND THE FUNDAMENTAL IDEAS IN PROJECTIVE PLANE GEOMETRY.
• TO BE ABLE TO EXPLAIN CLEARLY THE AQUIRED RESULTS AND CONCEPTS AND TO APPLY THEM TO THE PROPOSED PROBLEMS.
•TO BE ABLE TO USE ” EYES” DIFFERENT FROM THE EUCLDEAN ONE .

-APPLICATION SKILLS

THE STUDENT WILL BE ABLE:
•TO EXPLAIN CLEARLY CONCEPTS AND PROVING PROPOSITIONS FROM TOPOLOGY AND PROJECTIVE GEOMETRTY, HIGHLIGHTING THE DEMONSTRATIVE STRATEGY AND POSSIBLE ALTERNATIVES.
•TO SHOW AN EFFICIENT USE OF TOPOLOGY TECHNIQUES, BY APPLYING THEM TO PROBLEM-SOLVING, IN HOMEWORKS OR IN WORKING GROUP ACTIVITIES.
•TO DEMONSTRATE CAPACITY FOR MATHEMATICAL REASONING THROUGH ANALYZING AND ARGUMENTING PROBLEMS
• TO GIVE A PROBLEM AND TO CONSTRUCT A RESULT
Prerequisites
BASIC CONCEPTS ACQUIRED IN THE PREVIOUS COURSES IN ANALYSIS, GEOMETRY AND ALGEBRA.
PARTICULARLY USEFUL ARE THE FOLLOWING TOPICS: CONVERGENCE, CONTINUITY, ALGEBRAIC STRUCTURES, FINITE-DIMENSIONAL VECTOR SPACES. LINEAR EQUATIONS. LINEAR MAPS AND MATRICES. BILINEAR AND QUADRATIC FORMS.
DIAGONALIZATION.
PLANE AND SPACE AFFINITIES AND ISOMETRIES.
Contents
I-GENERAL TOPOLOGY:
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-REAL PLANE PROJECTIVE GEOMETRY:
5.MODELS OF THE REAL PROJECTIVE PLANE.
6.REAL PROJECTIVE CONICS (POLARITY)
7. DEGENERATION OF A CONIC
8.CONICS AND THEIR PROJECTIVE, AFFINE, METRIC CLASSIFICATION.
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 OBJECTIVES, ONE INDIVIDUAL, THE OTHER 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 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. 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] E. SERNESI -GEOMETRIA 1 BOLLATI-BORINGHIERI 2000.
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[4] S. WILLARD, GENERAL TOPOLOGY, ADDISON-WESLEY